Combinatorial Number Theory,

De Gruyter Proceedings in Mathematics

Combinatorial Number Theory Proceedings of the “Integers Conference 2011” Carrollton, Georgia, October 26–29, 2011 edited by Bruce Landman Melvyn B. Nathanson Jaroslav Nešetˇril Richard J. Nowakowski Carl Pomerance Aaron Robertson

De Gruyter

Mathematics Subject Classification 2010: 11-06. Editors Prof. Dr. Bruce Landman University of West Georgia Department of Mathematics 1601 Maple Street Carrollton, GA 30118, USA [email protected]

Prof. Dr. Melvyn B. Nathanson The City University of New York Lehman College (CUNY) Department of Mathematics 250 Bedford Park Boulevard West Bronx, NY 10468, USA [email protected]

Prof. Dr. Jaroslav Nešetˇril Charles University Department of Applied Mathematics Malostranské nam. 25 118 00 Prague, Czech Republic [email protected]

Prof. Dr. Richard J. Nowakowski Dalhousie University Department of Mathematics & Statistics Chase Building Halifax, NS B3H 3J5, Canada [email protected]

Prof. Dr. Carl Pomerance Dartmouth College Department of Mathematics 6188 Kemeny Hall Hanover, NH 03755-3551; USA [email protected]

Prof. Dr. Aaron Robertson Colgate University Department of Mathematics 13 Oak Drive Hamilton, 13346, USA [email protected]

ISBN 978-3-11-028048-7 e-ISBN 978-3-11-028061-6 Set-ISBN 978-3-11-028062-3 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the internet at http://dnb.dnb.de. © 2013 Walter de Gruyter GmbH, Berlin/Boston Typesetting: PTP-Berlin Protago-TEX-Production GmbH, www.ptp-berlin.eu Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen Printed on acid-free paper Printed in Germany www.degruyter.com

Preface

The Integers Conference 2011 was held October 26–29, 2011, at the University of West Georgia in Carrollton, Georgia, United States. This was the fifth Integers Conference, held bi-annually since 2003. It featured plenary lectures presented by Ken Ono, Carla Savage, Laszlo Szekely, Frank Thorne, and Julia Wolf, along with 60 other research talks. This volume consists of ten refereed articles, which are expanded and revised versions of talks presented at the conference. These ten articles will appear as a special volume of the journal Integers. They represent a broad range of topics in the areas of number theory and combinatorics including multiplicative number theory, additive number theory, game theory, Ramsey theory, enumerative combinatorics, elementary number theory, the theory of partitions, and integer sequences. The conference was made possible with the generous support of the Number Theory Foundation and the University of West Georgia. Carrollton, Georgia, USA, March 2013

The Editors

Contents

Preface 1

2

3

4

v

Rebecca Milley, Richard J. Nowakowski, and Paul Ottaway The Misère Monoid of One-Handed Alternating Games

1

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2

1.2

Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3

Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4

The Misère Monoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Neil Hindman and John H. Johnson Images of C-Sets and Related Large Sets under Nonhomogeneous Spectra

15

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2

The Various Notions of Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3

The Functions f˛ and h˛ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4

Preservation of J -Sets, C -Sets, and C  -Sets . . . . . . . . . . . . . . . . . . . 27

2.5

Preservation of Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Daniel A. Goldston and Andrew H. Ledoan On the Differences Between Consecutive Prime Numbers, I

37

3.1

Introduction and Statement of Results . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2

The Hardy–Littlewood Prime k-Tuple Conjectures . . . . . . . . . . . . . . 38

3.3

Inclusion–Exclusion for Consecutive Prime Numbers . . . . . . . . . . . . 39

3.4

Proof of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Pär Kurlberg, Jeffrey C. Lagarias, and Carl Pomerance On Sets of Integers Which Are Both Sum-Free and Product-Free

45

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2

The Upper Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

viii

5

6

7

Contents

4.3

An Upper Bound for the Density in Z=nZ . . . . . . . . . . . . . . . . . . . . . 50

4.4

Examples With Large Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Frank Thorne Four Perspectives on Secondary Terms in the Davenport–Heilbronn Theorems

55

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2

Counting Fields in General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.2.1 Counting Torsion Elements in Class Groups . . . . . . . . . . . . . 59

5.3

Davenport–Heilbronn, Delone–Faddeev, and the Main Terms . . . . . . 60 5.3.1 The Work of Belabas, Bhargava, and Pomerance . . . . . . . . . . 61

5.4

The Four Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.5

The Shintani Zeta-Function Approach . . . . . . . . . . . . . . . . . . . . . . . . 63 5.5.1 Nonequidistribution in Arithmetic Progressions . . . . . . . . . . . 66

5.6

A Refined Geometric Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.6.1 Origin of the Secondary Term . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.6.2 A Correspondence for Cubic Forms . . . . . . . . . . . . . . . . . . . . 69

5.7

Equidistribution of Heegner Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.7.1 Heegner Points and Equidistribution . . . . . . . . . . . . . . . . . . . 71

5.8

Hirzebruch Surfaces and the Maroni Invariant . . . . . . . . . . . . . . . . . . 73

5.9

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Brian Hopkins Spotted Tilings and n-Color Compositions

79

6.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.2

n-Color Composition Enumerations . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.3

Conjugable n-Color Compositions . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Aviezri S. Fraenkel and Yuval Tanny A Class of Wythoff-Like Games

91

7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

7.2

Constant Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.2.1 A Numeration System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7.2.2 Strategy Tractability and Structure of the P -Positions . . . . . . 98

7.3

Superadditive Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Contents

ix

7.4

Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.5

Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

8

9

Takao Komatsu, Florian Luca, and Yohei Tachiya On the Multiplicative Order of FnC1 =Fn Modulo Fm

109

8.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

8.2

Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

8.3

Proof of Theorem 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

8.4

Comments and Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Neil A. McKay and Richard J. Nowakowski Outcomes of Partizan Euclid

123

9.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

9.2

Game Tree Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

9.3

Reducing the Signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 9.3.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

9.4

Outcome Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

9.5

Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

10 Thomas W. Pensyl and Carla D. Savage Lecture Hall Partitions and the Wreath Products Ck o Sn

137

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 10.2 Lecture Hall Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 10.3 Statistics on Ck o Sn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 10.4 Statistics on s-Inversion Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 10.5 From Statistics on Ck o Sn to Statistics on In;k . . . . . . . . . . . . . . . . . . 141 10.6 Lecture Hall Polytopes and s-Inversion Sequences . . . . . . . . . . . . . . . 143 10.7 Lecture Hall Partitions and the Inversion Sequences In;k . . . . . . . . . . 145 10.8 A Lecture Hall Statistic on Ck o Sn . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 10.9 Inflated Eulerian Polynomials for Ck o Sn . . . . . . . . . . . . . . . . . . . . . 150 10.10 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Index

155

Combinatorial Number Theory, 1–13

© De Gruyter 2013

Chapter 1

The Misère Monoid of One-Handed Alternating Games Rebecca Milley, Richard J. Nowakowski, and Paul Ottaway Abstract. Misère games are notoriously more difficult to analyze than their normal-play counterparts. To deal with the inherent lack of structure, Plambeck (2005) [7] considered restricted universes of games and developed the concept of indistinguishability, modulo a given universe; the resulting quotient semigroup is called the misère monoid. Results for only three general universes are known. We introduce a fourth: the universe of alternating games and their sums. We find that the misère monoid of one-handed alternating games is isomorphic to .Z; C/. Keywords. Combinatorial Games, Misère Games. Mathematics Subject Classification 2010. 91A46.

1.1

Introduction

Two players Left and Right play with a stack of pennies, which are either all heads-up or all tails-up; Left can play on a tails-up stack and Right can play on a heads-up stack by removing at least one penny and then inverting any remaining coins. The player who removes the last penny loses. This is an alternating game, since neither Left nor Right can make two consecutive moves in one stack, and it is being played under misère rules. In the present chapter, we analyze individual alternating components (e.g., one stack of pennies) in order to answer the following question: under misère play, who wins a set of alternating games when they are played in a disjunctive sum? Note that a disjunctive sum of alternating components is not necessarily itself an alternating game. For a given game G, we use G L to denote a general left option and G L to denote the set of all such options. We may refer to a single left option from the position G R as G RL and to the set of all left options from the position G R as G RL . A game H is a follower of G if H can be reached from G by some sequence of (not necessarily alternating) moves. Definition 1.1. A game G is said to be alternating if G LL D ; and G RR D ; for all left and right options G L ; G R of G, and if every follower of G is also alternating.

2

Rebecca Milley, Richard J. Nowakowski, and Paul Ottaway

In PENNY NIM, each component is one-handed, i.e.„ only one player has an option from the initial position. We can specify which player, Left or Right, has the first move by the terms left-handed or right-handed, respectively. Games can also be two-handed alternating, where both players have at least one option from the start (G L 6D ;; G R 6D ;). If we placed a heads-up or tails-up stack of pennies on its side and allowed either player to move by taking some pennies and orienting the stack appropriately, then PENNY NIM would become a two-handed alternating game. This chapter considers one-handed positions; however, the majority of our one-handed results are true in the ‘universe’ of all alternating games. Since every two-handed game has exclusively one-handed followers, an understanding of one-handed games will help us to subsequently analyze all alternating games. Readers familiar with normal-play games will notice that one-handed alternating games are restricted to the normal-play values 1, 0, and 1. It is interesting to note that one-handed or ‘end’ games in general can be very problematic for misère play – Aaron Siegel calls them “significant pathologies” [9] – and yet there is simple, elegant structure in one-handed alternating games.

1.1.1 Background We begin with an overview of prerequisite material. Conventionally, the players Left and Right are female and male, respectively. We have already encountered an instance of misère-play, where the first player unable to move wins; the more standard ending condition is normal-play, where the first player unable to move loses. Games or positions are defined in terms of their options: G D ¹G L j G R º. The simplest game is the zero game, 0 D ¹ j º, where the dot indicates an empty set of options. In both play conventions, the outcome classes next (N ), previous (P ), left (L), and right (R) are partially ordered as shown in Figure 1.1, with Left preferring moves toward the top and Right preferring moves toward the bottom. In our analysis of alternating games, we often simultaneously consider the misère outcome and normal outcome of a game; to distinguish between the two, we introduce the superscripts  and C , respectively. Thus a game in N  \ RC is a next-win under misère play and a right-win under normal play. When convenient we also use the outcome functions o .G/ and oC .G/ to identify the misère or normal outcome of a game G. L 

Ÿ N 

P Ÿ R Figure 1.1. The partial order of outcome classes.

Chapter 1 The Misère Monoid of One-Handed Alternating Games

3

Many definitions from normal-play game theory1 are used without modification for misère games, including disjunctive sum, equality, and inequality. Thus, for misère games, G D H if o .G C X / D o .H C X / for all games X; G  H if o .G C X /  o .H C X / for all games X: In normal-play, the negative of a game is defined recursively as G D ¹G R jG L º, and is so called because G C .G/ D 0 for all games G. Under misère play, however, this property holds only if G is the zero game [5]. To avoid confusion and inappropriate cancellation, we write G instead of G and refer to this game as the conjugate of G. In normal-play games, there is an easy test for equality: G D 0 if and only if G 2 P C , and so G D H if and only if G  H 2 P C . In misère-play, no such test exists. Equality of misère games is difficult to prove and, moreover, is relatively rare; for example, besides ¹ j º itself, there are no games equal to the zero game under misère play [5]. Within the last ten years (see, e.g., [7–9]), a partial solution to these challenges has been presented: redefine ‘equality’ by restricting the game universe. This method has been used with much success by [2–4]. Given a set of games X, misère equivalence (modulo X) is defined by G  H (mod X) if o .G C X / D o .H C X / for all games X 2 X; while misère inequality (modulo X) is defined by G = H (mod X) if o .G C X /  o .H C X / for all games X 2 X: We use the words equivalent and indistinguishable interchangeably, and if G 6 H (mod X) then G and H are said to be distinguishable. Given a universe X, we can determine the equivalence classes under  .mod X/ and form the quotient semi-group X= . This quotient, together with the tetra-partition of elements into the sets P  , N  , R , and L , is called the misère monoid of the universe X, denoted MX . The misère monoid is by convention written multiplicatively, and its identity element, denoted e, is the equivalence class of zero. As an example, the misère monoid of the games 0, ¹0 j º, and all possible sums, is given by he; ˛ j ˛ 2 D ˛i, with N D ¹eº, R D ¹˛º, P D L D ; [2]. The closure of a set of games is the set of all disjunctive sums of those games and their followers. We denote the closure of the set of all alternating games by A and the closure of one-handed alternating games by O. Note that if X  Y then G  H (mod Y) implies G  H (mod X); in particular, equivalence in A immediately gives equivalence in O  A. Following Ottaway’s PhD thesis [6], we consider the universe A under misère play and find the misère monoid MO of one-handed alternating games. We would like to acknowledge Meghan Allen for suggesting the more difficult problem of constructing the monoid for all alternating games, of which this paper is a first step. 1A

complete overview of normal-play game theory can be found in [1].

4

Rebecca Milley, Richard J. Nowakowski, and Paul Ottaway

In Section 1.2, we determine the equivalence classes of one-handed alternating games inside the larger universe A, and in Section 1.3, we calculate outcomes and identify the equivalence classes of O alone. In Section 1.4 we present the misère monoid MO . Future goals for this research naturally include analyzing two-handed games and determining MA . Some progress has been made, but the equivalence classes of two-handed games are more complicated than those for O. It would also be interesting to classify the nonalternating games that can be written as a disjunctive sum of alternating positions; for example, the game ¹ j º can be decomposed into  C . In this way we could extend our knowledge of the alternating universe to a broader family of games. Interested readers should refer to Ottaway’s thesis [6] for an extended discussion of alternating2 games, including an analysis of subtraction games on coins (a generalization of PENNY NIM).

1.2 Equivalences In analyzing alternating games, we find it useful to classify a position by both its misère outcome and normal outcome. In general all 42 D 16 pairs of outcomes can be attained [5] (for example, there exist games which are Left-win under both normaland misère-play), but more than half of these pairs do not occur among one-handed games. Since either Left or Right has no first move in a one-handed game G, that player wins immediately under misère rules and loses immediately under normal rules, and so o .G/ 6D P  , oC .G/ 6D N C , and o .G/ 6D oC .G/. The remaining seven outcome pairs are each attained in the one-handed universe, as demonstrated in Figure 1.2 with the zero game and the games A D ¹0 j º;

B D ¹0; A j º;

C D ¹B j º;

along with their corresponding right-handed conjugates3 . R \ LC

N  \ LC

R \ P C

N \ P C

L \ RC

N  \ RC

L \ P C

A

B

C

0

A

B

C

Figure 1.2. The possible outcomes pairs for one-handed alternating games.

2 In

[6] and elsewhere, alternating games are referred to as consecutive move ban games. will find that ¹0 j º is incomparable with zero in our universe; for this and other reasons we chose not to use the normal-play name ‘1’. 3 We

Chapter 1 The Misère Monoid of One-Handed Alternating Games

5

Most of the positions in Figure 1.2 appear in PENNY NIM; if we consider stacks on which Left can play, then a one-penny stack is the game A, a two-penny stack is the game B, and all other nonzero stacks are equivalent (modulo A) to B. Throughout the chapter we make repeated use of the fact that a single one-handed game alternates between being left-handed and right-handed as players alternate turns. Furthermore, if Left has a good (misère) first move in a left-handed game, then it is to a position in L , because no one-handed game is in P  . Likewise, if Right as a good first move in an alternating game, it is to a position in R . We show in Theorem 1.4 that all one-handed games in N  \ P C are indistinguishable from the zero game, an incredibly useful result that actually holds in the larger universe A of all alternating games. We then extend this result by establishing that a single one-handed game is indistinguishable (modulo A) from any other with the same outcome pair; in particular, the seven games in Figure 1.2 are equivalent to all others in their respective classes. We first require the following two lemmas. Lemma 1.2. If X is a sum of right-handed alternating games with at least one component in L , then X 2 L . Proof. Left wins trivially playing first on X . Assume, for followers of X , Left wins playing second on a sum of right-handed games when at least one of them is in L . Let G be a component of X in L . If Right plays in G to some G R then Left either has no response and wins immediately, or responds with a good move G RL 2 L and wins by induction on the sum. If Right plays in some other component of X then Left either has no response and wins, or can play in that component to bring it back to a right-handed game, and then wins playing second on the sum, by induction, since G is still in L . Lemma 1.3. If G is a single right-handed game in P C then G = 0 (mod A), and if G is a single left-handed game in P C then G 5 0 (mod A). Proof. We prove the first statement, and the second will follow by symmetry. Let G be a right-handed game with normal outcome P C and X be a sum of alternating games. We need to show that o .G C X /  o .X /. If o .X / D R then trivially o .G C X /  o .X /. It remains to show that if Left can win X going first (second), then she can win G CX going first (second). Assume inductively that these statements hold for all followers of G C X of the form G 0 C X 0 with G 0 2 P C . Suppose now that Left can win X going first. If she has no move in X , then she has no move in G C X , and thus wins immediately. Otherwise, she has a move to some X L from which she wins moving second, and by induction she then wins G C X L moving second. Thus Left can win G C X moving first. Suppose Left wins X moving second. Note that Right always has a move at the outset. If Right moves in G, then Left has a (normal-play) winning local response to

6

Rebecca Milley, Richard J. Nowakowski, and Paul Ottaway

some G RL , which as a right-handed game must be in P C and not LC . Then Left wins G RL C X by induction. If Right moves in X to X R , then either Left wins because she has no move, or she responds with her good second move X RL and by induction wins G C X RL moving second. We are now set to prove our earlier claim that every one-handed game in N  \ P C is indistinguishable – among all alternating games – from zero. Theorem 1.4. If G is any one-handed alternating game in N  \ P C , then G  0 (mod A). Proof. Assume G is left-handed (the other case is symmetric). By Lemma 1.3 we already have G 5 0, and so it suffices to show G = 0. Let X be any sum of alternating games. Assume that for all followers X 0 of X , Left can win G CX 0 going first (second) when she can win X 0 going first (second). Suppose Left wins X playing first. If she has no move in X , then in G C X she can make a good first misère move in G 2 N  , say G L 2 L , to bring the whole position to a sum of right-handed games, one of which is in L . Lemma 1.2 then shows Left wins playing first on G C X . Otherwise, Left has a move in X to some X L , from which she wins playing second; Left then wins G C X L playing second, by induction, and so wins G C X playing first. Suppose Left wins X playing second. Then Left can win X R playing first for any Right move X R , and thus by induction wins G C X R playing first. Since Right has no choice but to play in X on the sum G C X , this shows Left wins playing second on G C X. We want to show that every one-handed alternating game not in N  \ P C is also equivalent to every other game with the same pair of outcomes. We first establish two instances of domination among left options: Left should move to a position in L \ P C over one in N  \ P C , and should choose N  \ P C over a position in N  \ RC . Naively, Left prefers one option over another if the misère or normal outcome is more favourable for Left and the other outcome is just the same. As with normal-play, if G D ¹G L j G R º, then we can remove dominated options from G L and G R to obtain a game equivalent to G under misère-play [9]. Theorem 1.5. If G 2 N  \ RC and H 2 L \ P C are one-handed alternating games, then G 5 0 5 H (mod A). Proof. Note that G and H are right-handed games. Since H 2 P C , Lemma 1.2 gives us 0 5 H . For G 5 0, let X be a sum in A. Suffices to show that if Right can win X playing first (second), then Right can win G C X playing first (second); assume that this is true for all followers X 0 of X . Now suppose Right can win X playing first. If Right has no moves in X , he can make a good first misère move in G to G R 2 R

Chapter 1 The Misère Monoid of One-Handed Alternating Games

7

and win on G R C X by Lemma 1.2. Otherwise, Right makes his good misère move in X to X R ; since Right wins X R playing second, he wins G C X R playing second by induction. This shows Right wins G C X playing first. If Right wins X playing second then Left has no good first move in X and cannot play in the right-handed game G. Left must move X to X L , which Right can win playing first, and then by induction Right can win G C X L playing first. Thus Right wins G C X playing second. The symmetric result obviously holds as well, and together these give us a ‘partial’ partial order of moves. In particular, Theorem 1.5 tells us that C 5 0 5 B and B 5 0 5 C . We will be able to establish another chain of domination (C 5 A 5 B, B 5 A 5 C ) after we prove Theorem 1.6. Theorem 1.6. If G is a one-handed alternating game, then the following statements are true modulo A: (i) (iii) (v)

If G 2 R \ LC then G  A. If G 2 N  \ LC then G  B. If G 2 R \ P C then G  C .

(ii) (iv) (vi)

If G 2 L \ RC then G  A. If G 2 N  \ RC then G  B. If G 2 L \ P C then G  C .

Proof. (i)–(ii): If G 2 R \ LC then G is left-handed, and since Left has no good first misère move, every Left option must be in N  or R . But all Left options are righthanded (or zero), and thus cannot be in R . So every Left option is in N  . Similarly, since G 2 LC , there is at least one Left option in P C or LC ; but right-handed (or zero) games cannot be in LC , so at least one Left option is in P C . Together this shows that there exists a G L  0 2 N  \ P C . By Theorem 1.5, any other options (necessarily in N  \ RC ) are dominated by 0, and so G  ¹0 j º  A. Case (ii) follows by symmetry. (iii)–(vi): We combine the remaining results and use an inductive proof. The reader can check that B is the smallest one-handed game in N  \ LC (i.e., no one-handed game with a shorter game tree is in N  \ LC ), and B, C , and C are the smallest games in R \ P C , N  \ RC , and L \ P C , respectively. We now proceed by induction on the followers. Let G and H be games in R \ P C and N  \ LC (note that G and H are left-handed; the right-handed results follow similarly). As argued in (i)–(ii), every Left option of G must be in N  because Left has no good first misère move in G 2 R . Additionally, G 2 P C forces every Left option to be in RC or N C ; but one-handed games cannot be in N C , so every option is in N  \ RC . By induction each of these options is indistinguishable from B. Thus G  ¹B j º D C . For H there is a bit more to show. Since H 2 N  , Left has a good misère move to a right-handed game in L ; since H 2 LC , Left has a good normal-play move to a right-handed game in P C . This shows that Left either has a move to L \ P C

8

Rebecca Milley, Richard J. Nowakowski, and Paul Ottaway

( C , by induction) or has moves to both L \ RC ( A by (i)) and N  \ P C ( 0). If there are any moves to N  \ RC , they are dominated by 0 or C , by Theorem 1.5. Using this and the domination of C over 0, we reduce the possibilities for H to H  ¹A; 0 j º; ¹C j º; or ¹A; C j º: In the first case we have H  B immediately. It remains to show that ¹C j º  B and ¹A; C j º  B. Claim 1. ¹C j º  B. Let J D ¹C j º. We need to show o .B C X / D o .J C X / for all sums X in A. Assume true for all followers of X . If Right wins playing first in B C X , then he must win second from some B C X R , which by induction has the same outcome as J C X R , and then Right wins first from J C X with the same move in X . If Left wins playing second in B C X , the same argument shows that Left wins playing second in J C X , since for every X R , o .B C X R / D o .J C X R /. If Right wins playing second on B C X , and Left’s first move in J C X is to some J C X L , then since Right can win playing first on B C X L he can win playing first from J C X L , by induction. Otherwise, Left’s first move in J C X is to C C X , and Right should respond with B C X , from which he wins playing second. Finally, suppose Left wins B C X playing first. If her good move is in X , then as above Left wins J C X with the same first move. If Left’s good first move is to B L C X , then Left must be able to win playing second from any position B L C X RL:::RL obtained from optimal Left play (including B L C X ). Now, in J C X , Left should move first to C C X and play her original strategy in X . When Right chooses to play in C , to B C X RL:::RL for some (not necessarily proper) follower of X , Left plays to B L C X RL:::RL and wins from there playing second. Claim 2. ¹A; C j º  B. let K D ¹A; C j º. Note that when at least one left option already exists, introducing another left option cannot create a position worse for Left; Left simply ‘ties her hand’ and ignores the extra option. In this way we see that K  J  B. It remains to show that K 5 B; i.e., that Right wins K C X whenever he wins B C X . If Right wins B CX playing first, then he wins K CX playing first, by induction, as in Claim 1. Suppose Right wins B C X playing second. If Left moves first in K C X to C C X , then Right wins by moving C to B and then playing second on B C X . If Left moves first in K C X to A C X , then Right can win first from that position, since A C X is a possible first Left move from B C X . Theorem 3.5 is a powerful result. Since the equivalences hold modulo all alternating games, and since every alternating game becomes one-handed after the first move, these equivalences are pertinent to future research in the universe A. As promised, Theorem 3.5 also allows us to prove the missing chain of domination; we can then combine the results of Theorems 1.5 and 1.7 (below) to obtain the partial orders illus-

9

Chapter 1 The Misère Monoid of One-Handed Alternating Games

L \ P C  Ÿ  C N \P L \ RC Ÿ   N \ RC

N  \ LC  Ÿ  C N \P R  \ LC Ÿ   R \ PC

C

B

 0 Ÿ

Ÿ A 

 0 Ÿ

Ÿ A 

B

C

Left options (right-handed games)

Right options (left-handed games)

Figure 1.3. The partial orders (modulo A) given by Theorems 1.5 and 1.7.

trated in Figure 1.3. As indicated by the figure, the game 0 is incomparable or ‘fuzzy’ with both A and A. To see this, note that 0 C  2 P  while A C  2 N  . Theorem 1.7. If G 2 N  \ RC ; H 2 L \ P C ; and K 2 L \ RC are one-handed alternating games, then G 5 K 5 H modulo A. Proof. By Theorem 1.6 we need only show B 5 A 5 C . As in the proof of Claim 2 above, we immediately have B 5 A, since Right can ‘tie his hand’ in B D ¹ j 0; Aº and pretend he is playing with the position A D ¹ j 0º; that is, B is at least as good as A for Right. To see that A 5 C , let X be any sum of alternating games, and suppose Left wins A C X playing first. Then we know Left must be able to win playing second from A C X LR:::L , for any X LR:::L obtained from X under optimal play. In C C X , Left plays as usual until Right moves from C CX LR:::L to B CX LR:::L ; Left then moves to ACX LR:::L and wins playing second from there. A similar argument shows Left wins C C X playing second whenever she wins A C X playing second, and so A 5 C . We have nearly finished our analysis of one-handed games in the alternating universe. The final result of this section addresses one of the main difficulties in misère game theory: the lack of inverses under disjunctive sum. As mentioned in the introduction, we do not generally have G D G in misère games as we do for normal games. Luckily for us, one-handed alternating games do possess this very convenient property. Theorem 1.8. If G is a one-handed alternating game, then G C G  0 (mod A).

10

Rebecca Milley, Richard J. Nowakowski, and Paul Ottaway

Proof. This is trivially true for games in N  \ P C . For any other (say, left-handed) game G, we have G is equivalent (modulo A) to A; B; or C . Thus, it suffices to show A C A  0, B C B  0, and C C C  0. Let X be be any sum of alternating games, and suppose without loss of generality that Left wins X . Then in A C A C X , Left plays his winning strategy in X , responding in A C A (and thereby bringing it to zero) only if Right plays there. If this happens before Left runs out of move in X , then play resumes in X and Left wins as usual. Otherwise Right does not play in A C A before Left wins X , at which point Left plays in A, bringing the game to a sum of right-handed components including A 2 L . Left then wins by Lemma 1.2. Notice this argument works whether Left wins X playing first or playing second or both. For B C B C X , the argument works as above; if Right plays in B, then Left copies her in B, bringing those components to 0 C 0 or A C A  0, and resumes play in X . If Left runs out of moves in X before this happens, she plays in B to A and wins by Lemma 1.2. Finally, in C C C C X , if Right plays in C , then Left copies in C to get B C B  0. If Right does not play in C , then after Left runs out of moves in X he plays in C to B. The game is now a sum of right-handed positions including B 2 L , so once again Left wins by Lemma 1.2. Theorem 1.8 shows that a disjunctive sum forms the set of one-handed alternating games into a group; G is in fact G. We use the notations interchangeably for the remainder of the paper and write kG to represent k copies of the game G. Together, Theorems 1.6 and 1.8 show that any sum of one-handed games in the alternating universe A can be written as aACbB CcC for integers a; b; c. In Section 1.3 we determine the outcome of such a sum, and in Section 1.4 we use this information to determine the misère monoid of O.

1.3 Outcomes If G is a sum of one handed games in A; then G  aA C bB C cC for integers a; b; c. We can therefore represent any such sum as an ordered triple .a; b; c/. Figure 1.4 illustrates the possible Left and Right options from G (with the conditions for each indicated below the horizontal line), and Theorem 1.9 establishes the misère outcome of G. Theorem 1.9. Let G D aA C bB C cC . Then 8  ˆ 0: Proof. The zero game is a next-player win under misère rules and has a C c D 0. Assume the outcomes above hold for all followers of G D .a; b; c/.

11

Chapter 1 The Misère Monoid of One-Handed Alternating Games

.a; b; c/

.a1;b;c/

.a;b1;c/;

.a;b1;c1/

.aC1;b;c/

.a1;b1;c/

.a;bC1;c/;

.a;bC1;cC1/

.aC1;bC1;c/

if a>0;

if b>0;

if c>0;

if a<0;

if b<0;

if c<0;

A7!0

B7!0;A

C 7!B

A7!0

B7!0;A

C 7!B

Figure 1.4. Left and Right options from G D aA C bB C cC .

If a C c < 0; then there exists at least one copy of A or C in the sum, so Right has a move going first to .a C 1; b; c/ or .a; b C 1; c C 1/. At best then, Right can increase the value of a C c to zero, giving Left a next-win position by induction. If Left plays first, she can play to either .a  1; b; c/ or .a; b  1; c  1/, and so can guarantee a C c remains negative and by induction leaves a Left-win position. Since Left wins playing first or second, G 2 L when a C c < 0. A symmetric argument shows G 2 R when a C c > 0. If a C c D 0; then Left going first either has no moves and wins immediately, or has a move in a copy of A; B, or C . From Figure 1.4 we see that Left can always decrease either a or c by 1, thereby moving to a C c < 0 and winning by induction. Right wins playing first by symmetry, and so G 2 N  when a C c D 0. This theorem will be very useful in analyzing the two-handed universe. For the present chapter, it shows us that the equivalence classes among one-handed games collapse even further when the universe is restricted to O. Notice that the integer b does not influence the outcome of G D aACbB CcC . Thus, for X a sum of only onehanded alternating games, we have o .X / D o .B C X /. This gives the following corollaries. Corollary 1.10. If O is the closure of one-handed alternating games, then B  0 (mod O) and C  A (mod O). Corollary 1.11. If G is a sum of one-handed games then G  aA (mod O) for some integer a. Note that we do have to restrict the universe to O to get these results. In the twohanded universe, B is distinguished from 0 with the game  D ¹0 j 0º: Left can win B C  going first, but loses playing first on . The last thing to note in our analysis of O is that the positions aA and a0 A are equivalent if and only if a D a 0 . If a 6D a0 ; then the games are distinguished by aA, since aA C .aA/ is in N  while a0 A C .aA/ is in L or R .

12

Rebecca Milley, Richard J. Nowakowski, and Paul Ottaway

1.4 The Misère Monoid The previous section gives us everything we need to describe the one-handed alternating universe O. Let ˛ represent the equivalence class of the game A; and let e represent the equivalence class of the game 0. The misère monoid of O is given by MO D he; ˛; ˛ 1 j ˛  ˛ 1 D ei; N  D ¹0º;

P  D ;;

R D ¹˛ a j a 2 Nº;

L D ¹˛ a j a 2 Nº:

As noted earlier, this monoid is actually a group; the mapping aA 7! a for a 2 Z shows MO Š .Z; C/. Acknowledgments. The first two authors would like to thank NSERC for partial support of this research.

References [1] M. H. Albert, R. J. Nowakowski, and D. Wolfe, Lessons in Play. Peters, Ltd., Nattick, MA, 2007. [2] M. R. Allen, An Investigation of Misère Partizan Games. PhD thesis, Dalhousie University, 2009. [3] M. R. Allen, Peeking at partizan misère quotients, in: Games of No Chance 4, to appear. [4] N. A. McKay, R. Milley, and R. J. Nowakowski, Hackenbush sprigs, preprint, 2012. [5] G. A. Mesdal and P. Ottaway. Simplification of partizan games in misère play, Integers 7 (2007), #G06. [6] P. Ottaway, Combinatorial Games with Restricted Options under Normal and Misère Play, PhD Thesis, Dalhousie University, 2009. [7] T. E. Plambeck, Taming the wild in impartial combinatorial games, Integers 5 (2005), #G5, Comb. Games Sect. [8] T. E. Plambeck and A. N. Siegel, Misère quotients for impartial games, Journal of Combinatorial Theory, Series A 115(4) (2008, 593 – 622. [9] A. N. Siegel. Misère canonical forms of partizan games. a.arXiv:math/0703565v1, to appear in Games of No Chance 4.

Chapter 1 The Misère Monoid of One-Handed Alternating Games

13

Author information Rebecca Milley, Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada. Email: [email protected] Richard J. Nowakowski, Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada. Email: [email protected] Paul Ottaway, Department of Mathematics, Thompson Rivers University, Kamloops, British Columbia, Canada. Email: [email protected]

Combinatorial Number Theory, 15–36

© De Gruyter 2013

Chapter 2

Images of C-Sets and Related Large Sets under Nonhomogeneous Spectra Neil Hindman and John H. Johnson Abstract. Let ˛ > 0 and 0 <  < 1. Define g˛; W N ! N by g˛; .n/ D b˛ n C  c. The set ¹g˛; .n/ W n 2 Nº is called the nonhomogeneous spectrum of ˛ and  . By extension, we refer to the maps g˛; as spectra. Bergelson, Hindman, and Kra showed that if A is an IP -set, a central set, an IP  -set, or a central -set, then g˛; ŒA is the corresponding object. We extend this result to include several other notions of largeness: C -sets, J -sets, strongly central sets, and piecewise syndetic sets. Of these, C -sets are particularly interesting, because they are the sets which satisfy the conclusion of the central sets theorem (and so have many of the strong combinatorial properties of central sets) but have a much simpler elementary description than do central sets. Keywords. Central Set, Spectrum, IP-Set. Mathematics Subject Classification 2010. 05D10.

2.1

Introduction

Given a positive real number ˛, the set ¹bn˛c W n 2 Nº is called the spectrum of ˛. (We write N for the set of positive integers and ! for the set of nonnegative integers.) Numerous results about spectra were derived by Skolem [18] and Bang [1]. (For a nice presentation of these results see [16].) In [19] Skolem introduced the more general sets ¹bn˛ C c W n 2 Nº, determining for example when two such sets can be disjoint. In terminology introduced by Graham, Lin, and Lin [9], the set ¹bn˛ C c W n 2 Nº is called a nonhomogeneous spectrum. By extension, we refer to the function g˛; W N ! N defined by g˛; .n/ D bn˛ C  c as a nonhomogeneous spectrum. In [4], V. Bergelson, B. Kra, and the first author of this paper showed that if ˛ > 0, 0 <  < 1, and a subset A of N is large in any of several senses, then so is g˛; ŒA. The main reason anyone should care about this fact is that it provides explicit nontrivial examples of sets with these largeness properties. For example, a set A is an IP  -set 1 1 if and only if whenever P hxn inD1 is a sequence in N, F S.hxn inD1 / \ A ¤ ;, where 1 F S.hxn inD1 / D ¹ n2F xn W F 2 Pf .N/º. Given any set X , Pf .X / is the set of finite nonempty subsets of X . It is trivial that for each n 2 N, nN is an IP  -set – just take n terms of any sequence which are congruent mod n and add them. It is not

16

Neil Hindman and John H. Johnson

p so trivial that ¹b 7  bn   C 1e c C 23 c W n 2 Nº is an IP  -set. This is because it is gp7;2=3 Œg;1=e ŒN. The notions of largeness which were shown in [4] to be preserved by spectra were, in addition to the IP  -sets mentioned above, IP -sets, central sets, central* sets, sets, and  -sets. The set A is an IP -set if and only if it contains F S.hxn i1 nD1 / for  -set if and only if it has nontrivial . (Thus a set is an IP some sequence hxn i1 nD1 intersection with each IP -set.) The set A is a -set if and only if it contains ¹y  x W x; y 2 B and x < yº for some infinite set B  N. And a  -set is one which has nontrivial intersection with each -set. Central sets in N were introduced by Furstenberg in [8] and were defined using notions from topological dynamics. They have a much simpler characterization in terms of the algebra of ˇN. This characterization was obtained in [2] with the assistance of B. Weiss. (It was shown to be equivalent in arbitrary semigroups by H. Shi and H. Yang [17].) In order to discuss this we give a very brief overview of that algebraic structure. The reader is referred to [13] for an elementary introduction to the subject. Detailed historical references can also be found in that book. Most of the basic facts mentioned in the next two paragraphs are not due to either of the authors of [13]. Let .S; C/ be a semigroup, not necessarily commutative, with the discrete topology. (We denote the operation by C because we are primarily interested in the semigroup ˇ .N; C/.) The Stone–Cech compactification ˇS of S can be taken to be the set of ultrafilters on S , with the points of S identified with the principal ultrafilters. The topology on ˇS has a basis consisting of ¹A W A  S º where A D ¹p 2 ˇS W A 2 pº. The operation extends to ˇS making .ˇS; C/ a right topological semigroup (so for each p 2 ˇS the function p W ˇS ! ˇS defined by p .q/ D q C p is continuous) with S contained in its topological center (so for each x 2 S , the function x W ˇS ! ˇS defined by x .q/ D x C q is continuous). Given p; q 2 ˇS and A  S , A 2 p C q if and only if ¹x 2 S W x C A 2 qº 2 p where x C A D ¹y 2 S W x C y 2 Aº. The reader should be cautioned that .ˇS; C/ is very unlikely to be commutative; the center of .ˇN; C/ is N. Any compact Hausdorff right topological semigroup T has a smallest two-sided ideal, K.T /, which is the union of all of the minimal right ideals of T and is also the union of all the minimal left ideals of T . The intersection of any minimal right ideal and any minimal left ideal is a group, and any two such groups are isomorphic. (In .ˇN; C/ there are 2c minimal left ideals and 2c minimal right ideals.) In particular there are idempotents in K.T /. An idempotent in T is in K.T / if and only if it is minimal with respect to the ordering of idempotents, wherein p  q if and only if p D p C q D q C p. Idempotents in K.T / are referred to as minimal idempotents. A subset A of S is central if and only if A is a member of a minimal idempotent of ˇS . And A is central* if and only if it has nontrivial intersection with each central set; equivalently, A is central* if and only if it is a member of every minimal idempotent in ˇS . It is a trivial consequence of the definition that central sets are partition regular. That is, if A is central and A is partitioned into finitely many cells, then one of these

Chapter 2 Images of C -Sets and Related Large Sets under Nonhomogeneous Spectra

17

cells is central. In particular, whenever S is partitioned into finitely many parts, one of these must be central. From a combinatorial point of view, what is probably the most important fact about central sets is that they satisfy the central sets theorem. We state here the commutative version thereof because it is much simpler to state and we are primarily concerned with .N; C/. We write NS for the set of sequences in S . Theorem 2.1. Let .S; C/ be a commutative semigroup, and let A be a central subset of S . There exist functions ˛ W Pf .NS / ! S and H W Pf .NS / ! Pf .N/ such that (1) if F; G 2 Pf .NS / and F ¨ G, then max H.F / < min H.G/; and (2) whenever m 2 N, G1 ; G2 ; : : : ; Gm 2 Pf .NS /, G1 ¨ G2 ¨ : : : ¨ Gm , and for   P P each i 2 ¹1; 2; : : : ; mº, fi 2 Gi , one has m t2H.Gi / fi .t / 2 A. iD1 ˛.Gi /C Proof. [7, Theorem 2.2]. The following is the original central sets theorem as proved by Furstenberg [8, Proposition 8.21]. Corollary 2.2. Let l 2 N and for each i 2 ¹1; 2; : : : ; lº, let hyi;n i1 nD1 be a sequence in Z. Let A be a central subset of N. Then there exist sequences han i1 nD1 in N and in P .N/ such that hHn i1 f nD1 (1) for all n, max Hn < min HnC1 , and (2) for all F 2 Pf .N/ and all i 2 ¹1; 2; : : : ; lº,

P n2F

.an C

P t2Hn

yi;t / 2 A.

Proof. We first note that A is central in Z because by [13, Theorem 1.65 and Exercise 4.3.5], K.ˇN/ D K.ˇZ/ \ ˇN. Pick ˛ W Pf .NZ/ ! Z and H W Pf .NZ/ ! Pf .N/ as guaranteed by Theorem 2.1. 1 1 Let G1 D ¹0; hy1;n i1 nD1 ; hy2;n inD1 ; : : : ; hyl;n inD1 º, where 0 is the function constantly equal to 0. For n 2 N, pick f 2 NN n Gn and let GnC1 D Gn [ ¹f º. For n 2 N, let Hn D H.Gn / and let an D ˛.Gn /. To see that each an 2 N, P note that 0 2 Gn so an D ˛.Gn / C t2H.Gn / 0 .t / 2 A  N. Conclusion (1) holds directly. To verify conclusion (2), let F 2 Pf .N/ and i 2 ¹1; 2; : : : ; lº. Then for  P P P 2 Gn so n2F .an C t2Hn yi;t / D ˛.Gn / C each n 2 F , hyi;n i1 n2F nD1  P y 2 A. t2H.Gn / i;t From Corollary 2.2, Furstenberg deduced several facts about central subsets of N, including the fact that any partition regular system of homogeneous linear equations with coefficients from Q has solutions in any central set; see [13, Chaps. 14–16] for many other examples of the strong properties enjoyed by central subsets of N. The algebraic description of central sets is very simple and quite easy to work with. There is an elementary description of central sets which was produced in [12] (or

18

Neil Hindman and John H. Johnson

see [13, Theorem 14.25]). However, that description is based on the notion of a collectionwise piecewise syndetic family of sets – a notion which is extremely complicated. The masochistic reader is referred to [13, Definition 14.19]. For many of the facts about central sets, what is important is that they satisfy the central sets theorem. We define a set to be a C -set if and only if it satisfies the conclusion of the central sets theorem. We have not stated the general version of the central sets theorem. In the case of commutative semigroups, the definition becomes as follows. Definition 2.3. Let .S; C/ be a commutative semigroup. A set A  S is a C -set if and only if there exist functions ˛ W Pf .NS / ! S and H W Pf .NS / ! Pf .N/ such that (1) if F; G 2 Pf .NS / and F ¨ G, then max H.F / < min H.G/; and (2) whenever m 2 N, G1 ; G2 ; : : : ; Gm 2 Pf .NS /, G1 ¨ G2 ¨ : : : ¨ Gm , and for   P P each i 2 ¹1; 2; : : : ; mº, fi 2 Gi , one has m t2H.Gi / fi .t / 2 A. iD1 ˛.Gi / C There is an elementary characterization of C -sets [14] which is very similar in form to the elementary description of central sets. The crucial distinction is that the notion of a family of sets being collectionwise piecewise syndetic is replaced by the notion of a single set being a J -set. Definition 2.4. Let .S; C/ be a commutative semigroup. A set A  S is a J -set if and only if whenever F 2 Pf .NS /, there exist a 2 S and H 2 Pf .N/ such that for P each f 2 F , a C t2H f .t / 2 A. Definition 2.5. Let S be a semigroup. Then J.S / D ¹p 2 ˇS W .8A 2 p/.A is a J -set/º. Of course one defines C  -sets and J  -sets analogously to IP  -sets and central* sets. That is, a set is a C  -set if and only if it has nonempty intersection with every C -set, and a set is a J  -set if and only if it has nonempty intersection with every J -set. Other notions of size with which we will be concerned in this chapter include strongly central sets (abbreviated by S C ), piecewise syndetic sets (abbreviated by PS ), AP sets, syndetic sets, and thick sets. A subset of N is strongly central if and only if it is a member of some idempotent in every minimal left ideal of ˇN. (Since every left ideal contains a minimal left ideal, this is equivalent to being a member of an idempotent in every left ideal of ˇN.) In .N; C/, a set A is piecewise syndetic if and only if there is a bound b such that there are arbitrarily long blocks of N in which A has no gaps longer than b. That is .9b 2 N/.8k 2 N/.9x 2 N/.¹x C 1; x C 2; : : : ; x C kº  Sb tD1 t C A/. In .N; C/, a set A is an AP -set if and only if it contains arbitrarily long arithmetic progressions, is syndetic if and only if it has bounded gaps, and is thick if and only if it contains arbitrarily long integer intervals. We say that a set is PS  if

Chapter 2 Images of C -Sets and Related Large Sets under Nonhomogeneous Spectra

19

and only if it has nonempty intersection with each piecewise syndetic set and is AP  if and only if it has nonempty intersection with each AP -set. Of the notions of size that we are considering, IP -sets, central sets, C -sets, J -sets, piecewise syndetic sets, AP -sets, and -sets are all partition regular. Notice that any J -set in N is an AP -set. In fact, it must contain arbitrarily long arithmetic progressions with increment taken from the finite sums of any prespecified sequence in N. To see this, let a sequence hxn i1 nD1 in N be given and let l 2 N. Let ¯ ® F D htxn i1 nD1 W t 2 ¹1; 2; : : : ; lº : P Pick a 2 N and H 2 Pf .N/ such that for each t 2 ¹1; 2; : : : ; lº, a C t2H txn 2 A. P If d D t2H x t , then ® ¯ a C t d W t 2 ¹1; 2; : : : ; lº  A: Definition 2.6. AP D ¹p 2 ˇN W .8A 2 p/.A is an AP -set/º. In Section 2.2 we will describe the relationship among the various notions of size that we are considering and also present elementary proofs regarding the preservation of these notions by g˛; for those notions for which we have such elementary proofs. In Section 2.3 we introduce the functions f˛ W N ! T and h˛ D g˛;1=2 and present some of the basic facts about these functions. In Section 2.4 we present some background material on C -sets and J -sets, including the fact in Theorem 2.20 that in a commutative semigroup, the translate a may be taken to be a member of the J -set itself. We then present positive results about the preservation of J -sets, C -sets, and C  -sets under spectra. We then show that if P is any property which is preserved by all spectra (including any of the properties in the left two columns of Figure 2.1), ˛ > 1, and 0 <  < 1, then a set A has property P if and only if g˛; ŒA has property P . Consequently, if A is an example of something with one of those properties but not others, so is g˛; ŒA. In Section 2.5 we show the extent to which g˛; preserves certain ideals of ˇN, where g˛; W ˇN ! ˇN is the continuous extension of g˛; .

e

2.2

e

The Various Notions of Size

We begin with a theorem providing characterizations in terms of .ˇN; C/ of each of the notions of size that we are considering. (Most of these characterizations hold for arbitrary semigroups.) We let N  D ˇN n N and given T  ˇN, E.T / D ¹p 2 T W p C p D pº. Given p 2 ˇN, p is the member of ˇZ generated by ¹A W A 2 pº.

20

Neil Hindman and John H. Johnson

Theorem 2.7. Let A  N. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q)

A is a -set if and only if there is some p 2 N  such that A 2 p C p. A is an AP -set if and only if A \ AP ¤ ;. A is an IP -set if and only if A \ E.ˇN/ ¤ ;. A is a J -set if and only if A \ J.N/  ¤ ;. A is a C -set if and only if A \ E J.N/ ¤ ;. A is a PS -set if and only if A \ K.ˇN/  ¤ ;.  A is a central set if and only if A \ E K.ˇN/ ¤ ;. A is a syndetic set if and only if for every left ideal L of ˇN, A \ L ¤ ;. A is a S C -set if and only if for every left ideal L of ˇN, A \ E.L/ ¤ ;. A is a thick set if and only if there exists a left  ideal L of ˇN such that L  A. A is a central* set if and only if E K.ˇN/  A. A is a PS  -set if and only if K.ˇN/    A. A is a C  -set if and only if E J.N/  A. A is a J  -set if and only if J.N/  A. A is an IP  set if and only if E.ˇN/  A. A is an AP  set if and only if AP  A. A is a  -set if and only if for all p 2 N  , A 2 p C p.

Proof. Statements (g) and (i) are the definitions of central and S C respectively. Statements (a) and (q) follow from [3, Lemma 1.9(a) and (j)] respectively. Statement (b) follows from [13, Theorem 3.11] and the fact that AP -sets are partition regular. Statements (c), (d), (e), and (f) follow from [13, Theorems 5.12, 14.14.7, 14.15.1, and 4.40], respectively. Statements (h) and (j) follow from [5, Theorem 2.9(d) and (c)], respectively. Statements (k), (l), (m), (n), (o), and (p) follow from statements (g), (f), (e), (d), (c), and (b), respectively. Figure 2.1 shows the relations that hold among the notions of size that we are considering. After noting that, by [13, Theorems 14.4.4 and 14.5], J.N/ and AP are ideals of .ˇN; C/ and, as we have already observed, each J -set is an AP -set so that J.N/  AP , each of the implications follows quickly from the characterizations in Theorem 2.7 except for the fact that IP )  and the corresponding fact that  ) IP  . (To see that central ) S C and thick ) central, one needs to note that by [13, Corollary 2.6], each left ideal of ˇN contains a minimal left ideal with an idempotent, which is therefore an idempotent in K.ˇN/.)  The easiest way to see that any IP -set is a -set (and thus any  -set Pn is an IP 1 set) is to note that if F S.hxn inD1 /  A and for each n 2 N, yn D tD1 x t , then ¹yn  ym W n; m 2 N and m < nº  A.

Chapter 2 Images of C -Sets and Related Large Sets under Nonhomogeneous Spectra



AP 

IP 

J

↓

21

↓

↓↓

C

PS 

↓ | central* | ↓ ↓ SC

↓

thick

syndetic central

↓↓

PS

C

J

IP

AP



↓↓ ↓

↓

Figure 2.1. Diagram of implications

That none of the missing implications is valid is established by the following table which lists for each property, a subset of N with that property that does not have any of the other properties except those that it is forced to have because of the implications in the diagram. In that table, the set D is the set produced in [11] which is a C -set but has Banach density 0. In particular D \ K.ˇN/ D ;. Since D  2N, D C 1 is not a -set. Property

Set with no extra properties

 AP IP J C PS central syndetic SC thick central* PS 

¹2n  2m W n; m 2 N and m < nº ¹22n C m2n C 1 W n; m 2 N and m < nº ¹†n2F 22n W F 2 Pf .N/º DC1 D ¹2n C 2m  1 W n; m 2 N and m < nº ¹2n C 2m W n; m 2 N and m < nº 2N C 1 p p  ¯ ® S 2N \ x 2 N W . 2x  b 2x C 1=2c/ 2 1 nD1 1=.2n C 1/; 1=.2n/ ¹2n C m W n; m 2 N and m < nº 2N n D N nD

22

Neil Hindman and John H. Johnson

Property

Set with no extra properties

C J

2N n ¹†n2F 22n W F 2 Pf .N/º N n .¹22n C m2n C 1 W n; m 2 N and m < nº [ ¹†n2F 22n W F 2 Pf .N/º/ 2N n ¹2n  2m W n; m 2 N and m < nº N n ¹†n2F 22n W F 2 Pf .N/º 2N

IP  AP  

In most cases it is at least routine, if not obvious, that the set has the specified property. We will explain now why this is true for J , central, S C , C  , and J  . The set D is a member of an idempotent p 2 J.N/, so D C 1 2 p C 1 and p C 1 2 J.N/. The set ¹2n C 2m W n; m 2 N and m < nº is the intersection of a thick a central*pset, and so is central. ® set with p ¯ Theorem 3.2] S It is a consequence of [6, that x 2 N W . 2x  b 2x C 1=2c/ 2 1 1=.2n C 1/; 1=.2n/ , and its comnD1 plement are both strongly central, and thus its intersection with 2N, which is IP  , is strongly central. (We shall be giving a short P proof of this consequence of [6, Theorem 3.2] in Theorem 2.15 below.) Since ¹ n2F 22n W F 2 Pf .N/º is not an AP -set, it is not a J -set, and so P 2N n ¹ n2F 22n W F 2 Pf .N/º is a C  -set. And, by Lemma 2.22 below, ¹22n C m2n C 1 W n; m 2 N and m < nº is P not a J -set, and so N n .¹22n C m2n C 1 W n; m 2 N and m < nº [ ¹ n2F 22n W F 2 Pf .N/º/ is a J  -set. In all cases but two it is routine to verify that the listed set does not have any properties except those it is forced to have by the implications in the diagram. The exceptions are AP and S C . One needs that the set listed for AP is not J and not . It is trivially not a -set. The fact that it is not a J -set is Lemma 2.22. One needs that the set listed for S C is not central* thick. It p is trivially not thick. ¯ ® and notp S It is not central* because the complement of x 2 N W . 2x  b 2x C 1=2c/ 2 1 nD1 1=.2n C 1/; 1=.2n/ is strongly central and thus central. It was shown in [4, Theorem 6.1] that if ˛ > 0, 0 <  < 1, P is any of , IP , central, central*, IP  , or  , and A  N is a P -set, then so is g˛; ŒA. We shall show in the remainder of this chapter that the same assertion holds if P is any of the properties in the left two columns of Figure 2.1. And we shall show that as long as ˛  1, the properties in the right column of Figure 2.1 are also preserved by g˛; , but are not if ˛ > 1. In the remainder of this section, we establish those results for which we have elementary proofs, not requiring the algebra of ˇN.

Chapter 2 Images of C -Sets and Related Large Sets under Nonhomogeneous Spectra

23

Theorem 2.8. Let A  N, let ˛ > 0, and let 0 <  < 1. (1) If A is syndetic, then so is g˛; ŒA. (2) If ˛  1 and A is thick, then so is g˛; ŒA. Proof. (1) Pick b 2 N such that for all x 2 !, ¹x C 1; x C 2; : : : ; x C bº \ A ¤ ; and pick k 2 N such that ˛ < k. We shall show that for all y 2 N, there exists m 2 g˛; ŒA such that 0  m  y  bk. So let y 2 N be given and let x be the largest element of ! such that g˛; .x/ < y. Pick t 2 ¹1; 2; : : : ; bº such that x C t 2 A and let m D g˛; .x C t /. Then m  g˛; .x C 1/  y. Also m  ˛.x C t / C  and ˛x C  < y so m  y < ˛t < bk. (2) Let b 2 N. We need to show that there is some m 2 N such that ¹m; m C 1; : : : ; m C bº  g˛; ŒA. Pick r 2 N such that ˛ > 1r , and pick n 2 N such that ¹n; nC1; : : : ; nClrº  A. Let m D g˛; .n/. Since ˛  1, we have that for all x 2 N, g˛; .x C 1/  g˛; .x/ C 1, and so it suffices to show that g˛; .n C lr/  m C l. Since m  ˛ n C  and l < ˛lr, this is true. Theorem 2.9. Let ˛ > 1 and let 0 <  < 1. Then g˛; ŒN is not thick. In particular, none of the properties in the right column of Figure 2.1 are preserved by g˛; . Proof. Pick r 2 N such that 1r < ˛  1. Given n 2 N, let k D g˛; .n/. Then g˛; .n C r/  k C r C 1 so at least one of ¹k C 1; k C 2; : : : ; k C rº is not in g˛; ŒN. Theorem 2.10. Let A be a piecewise syndetic subset of N, let ˛ > 0, and let 0 <  < 1. Then g˛; ŒA is piecewise syndetic. Proof. Pick b 2 N such that for all w 2 N there exists x 2 N such that ¹x; x C S 1; : : : ; x C wº  btD0 .t C A/. Let m D d˛b C ˛e. We claim that for all y 2 N S there exists z 2 N such that ¹z; z C 1; : : : ; z C yº  m tD0 .t C g˛; ŒA/. To this 1 end, let y 2 N and let w D d ˛ ye. Pick x 2 N such that ¹x; x C 1; : : : ; x C wº  Sm Sb tD0 .t C A/. Let z D g˛; .x/. We claim that ¹z; z C 1; : : : ; z C yº  tD0 .t C g˛; ŒA/. To this end, let k 2 ¹0; 1; : : : ; yº be given. Now y  ˛w < y C ˛ so z Cy  ˛.x Cw/C < z C1Cy C˛. Thus z Cy  g˛; .x Cw/. Pick the first l 2 ! such that z C k  g˛; .x C l/, and note that l 2 ¹0; 1; : : : ; wº. Pick t 2 ¹0; 1; : : : ; bº such that x C l C t 2 A. Let v D g˛; .x C l C t /  .z C k/. Then z C k C v 2 g˛; ŒA so it suffices to show that 0  v  m. Since g˛; .x C l C t /  g˛; .x/ D z, we have v  0. By the choice of l, z C k > g˛; .x C l  1/ so ˛x C ˛l  ˛ < z C k. Consequently ˛xC˛l C˛t C < zCkC˛t C˛  zCkC˛bC˛ so g˛; .xCl Ct / < z C k C ˛b C ˛  z C k C m as required. Theorem 2.11. Let A be an AP -set, let ˛ > 0, and let 0 <  < 1. Then g˛; ŒA is an AP -set.

24

Neil Hindman and John H. Johnson

Proof. Let r 2 N n ¹1; 2º. We need to show that g˛; ŒA contains a length r arithmetic progression. By van der Waerden’s theorem, pick m 2 N such that whenever a length m arithmetic progression is two-colored it contains a length r monochromatic arithmetic progression. For i 2 ¹0; 1º, let Bi D ¹x 2 N W g˛; .x/ C 2i  x˛ C  < g˛; .x/ C iC1 2 º. Pick a length m arithmetic progression P contained in A; and pick i 2 ¹0; 1º and a length r arithmetic progression ¹a; a Cd; : : : ; a C.r 1/d º contained in P \ Bi . We claim that ¹g˛; .a/; g˛; .a C d /; : : : ; g˛; .a C .r  1/d /º is an arithmetic progression. (It is contained in g˛; ŒP  and is therefore contained in g˛; ŒA.) To see this, let k D g˛; .a/ and let c D g˛; .a C d /  k. Now, i i C1  a˛ C  < k C and 2 2 i i C1 k C c C  .a C d /˛ C  < k C c C ; 2 2 kC

./

c

1 2

< d˛ < c C

1 2

so

:

We claim that for each t 2 ¹0; 1; : : : ; r 1º, g˛; .aCt d / D kCt c. This is trivially true for t 2 ¹0; 1º, and so assume that t 2 ¹1; 2; : : : ; r 2º and that g˛; .a Ct d / D k Ct c. Let m D g˛; .a C .t C 1/d /. Then, i i C1  .a C t d /˛ C  < k C t c C and 2 2 i i C1 m C  .a C .t C 1/d /˛ C  < m C ; so 2 2

k C tc C

./

m  k  tc 

1 2

< d˛ < m  k  t c C

1 2

:

Combining ./ and ./ we have 1 1 < m  k  tc C 2 2 1 1 m  k  tc  < c C : 2 2 c

and

Thus we have that k C .t C 1/c  1 < m < k C .t C 1/c C 1 and so k C .t C 1/c  1 D m.

e

Corollary 2.12. Let ˛ > 0 and let 0 <  < 1. Then g˛; ŒAP   AP .

e

Proof. Let p 2 AP and let A 2 g˛; .p/. Pick B 2 p such that g˛; ŒB  A. Then B is an AP -set, and by Theorem 2.11, g˛; ŒB is an AP -set, and thus A is an AP -set.

Chapter 2 Images of C -Sets and Related Large Sets under Nonhomogeneous Spectra

2.3

25

The Functions f˛ and h˛

We consider the circle group T to be R=Z and represent the points of T by points in the interval Œ 12 ; 12 /. (That is, if x 2 Œ 12 ; 12 / we write x to represent the coset x C Z.) The reason we do this is that we will be dealing extensively with intervals in R, and so it is convenient to have members of T represented by elements of R. Given ˛ > 0, we let h˛ D g˛;1=2 , so that for x 2 N, h˛ is the nearest integer to ˛x (with ties broken by rounding up). We define f˛ W N ! T by, for x 2 N, f f˛ .x/ D ˛x  h˛ .x/; and we let f ˛ W ˇN ! T be the continuous extension of f˛ . f Similarly, g˛; W ˇN ! ˇN and h ˛ W ˇN ! ˇN are the continuous extensions of g˛; and h˛ , respectively.

e

f Definition 2.13. Let ˛ > 0. Then Z˛ D ¹p 2 ˇN W f ˛ .p/ D 0º. We gather some basic facts about the objects we have just defined. f Lemma 2.14. Let ˛ > 0; and let 0 <  < 1. The function f ˛ is a homomorphism f . from ˇN into T , and consequently each idempotent is in Z˛ The restriction of h ˛ to Z˛ is an isomorphism onto Z1=˛ , whose inverse is the restriction of h1=˛ to Z1=˛ . f For all p 2 Z˛ , g˛; .p/ D h ˛ .p/.

e

e

Proof. The first conclusion follows from [13, Corollary 4.22]. The second and third conclusions are [4, Theorem 5.10] and [4, Theorem 5.8], respectively. As promised earlier, we provide a short proof of the consequence of [6, Theorem 3.2] which we used in the previous section. Theorem 2.15. If U is an open subset of .0; 12 / with 0 2 c`U and ˛ > 0 is irrational, then f˛1 ŒU  is strongly central. 1 Proof. Let L be a minimal left ideal of ˇN. Pick sequences han i1 nD1 and hbn inD1 in S 1 .0; 12 / converging to 0 with anC1 < bn < an for each n such that nD1 .bn ; an /  U . For each n 2 N pick cn and dn such that bn < dn < cn < an and pick by Kronecker’s theorem xn such that dn < f˛ .xn / < cn . Pick p 2 N  \ ¹xn W n 2 Nº; and note that p 2 Z˛ . Pick by [13, Theorem 1.61] an idempotent q 2 L \ .p C ˇN/; and pick e˛ .p/ C f e ˛ .r/; and thus e˛ .q/ D f r 2 ˇN such that q D p C r. By Lemma 2.14 f r 2 Z˛ . We claim that ¹xn W n 2 Nº  ¹y 2 N W y C f˛1 ŒU  2 rº; so that f˛1 ŒU  2 p C r as required. Let n 2 N, and let  D min¹dn  bn ; an  cn º. Then ¹z 2 N W f˛ .z/ 2 .; /º 2 r and ¹z 2 N W f˛ .z/ 2 .; /º  .xn C f˛1 ŒU .

A major contrast between the preservation of largeness notions established in [4] and the results of our next section is that each of the notions considered in [4], namely

26

Neil Hindman and John H. Johnson

IP -sets, -sets, central sets, IP  -sets, central* sets, and  -sets, is determined by members of Z˛ , so that g˛; agrees with an isomorphism on Z˛ at those points. The notion of strongly central had not yet been introduced at the time of the writing of [4]. We pause to observe that the methods of that paper are sufficient to show that it is preserved by the functions g˛; .

e

Theorem 2.16. Let A be a strongly central subset of N, let ˛ > 0, and let 0 <  < 1. Then g˛; ŒA is strongly central. Proof. Let L be a minimal left ideal of ˇN. We need to show that there is an idempotent in L \ g˛; ŒA. Now L \ Z1=˛ ¤ ; because any idempotent in L is in this intersection, and so L\Z1=˛ is a minimal left ideal of Z1=˛ by [13, Theorem 1.65(2)].

e

Therefore h1=˛ ŒL \ Z1=˛  is a minimal left ideal of Z˛ by Lemma 2.14. Again by [13,

e

Theorem 1.65(2)], we may pick a minimal left ideal M of ˇN such that h1=˛ ŒL \ f Z1=˛  D M \ Z˛ . Pick an idempotent p 2 M \ A. Then p 2 Z˛ so h ˛ .p/ 2 f f h ˛ ı h1=˛ ŒL \ Z1=˛  D L \ Z1=˛ . Since A 2 p, g˛; ŒA 2 g˛; .p/ D h˛ .p/. Thus f f h ˛ .p/ is an idempotent in L and g˛; ŒA 2 h˛ .p/.

e

e

In the rest of this chapter we need to consider points of ˇN which f˛ can take anywhere in T . The following lemma will simplify some of the computations. In this lemma we give specific values for , but we only really care that exists. 1 3 f Lemma 2.17. Let ˛ > 0, let p 2 ˇN, and let D f ˛ .p/. There exist 2 ¹ 2 ; 4 º and  > 0 such that for all ı 2 .0; , ¹x 2 N W g˛; .x/ C  ı < ˛x < g˛; .x/ C C ıº 2 p.

Proof. Assume first that ¤  12 . Let D 12 and let  D min¹ 12  ; 12 C º. Let ı 2 .0;  and let A D ¹x 2 N W f˛ .x/ 2 .  ı; C ı/º. Then by the continuity of f f ˛ , A 2 p. We note that A D ¹x 2 N W g˛; .x/ C  ı < ˛x < g˛; .x/ C C ıº. Now assume that D  12 . Let D 34 and let  D 14 . Let ı 2 .0;  and let A D ¹x 2 N W f˛ .x/ 2 Œ 12 ;  12 C ı/º [ ¹x 2 N W f˛ .x/ 2 . 12  ı; 12 /º. Then by the continuity f of f ˛ , A 2 p. We claim that A  ¹x 2 N W g˛; .x/C ı < ˛x < g˛; .x/C Cıº. (Actually, equality holds, but we do not care.) To this end, let x 2 A and let k D h˛ .x/. Assume first that f˛ .x/ 2 Œ 12 ;  12 C ı/. Then  12  ˛x  k <  12 C ı, and so k < ˛x C 34 < k C1. Thus, k D g˛; .x/; and so g˛; .x/C ı < ˛x < g˛; .x/C Cı. Now assume that f˛ .x/ 2 . 12 ı; 12 /. Then 12 ı < ˛xk < 12 . Thus, kC1 D g˛; .x/; and so g˛; .x/ C  ı < ˛x < g˛; .x/ C C ı. Lemma 2.18. Let F 2 Pf .NN/, let ˛ > 0, and let  > 0. There exists H 2 Pf .N/  P such that for all k 2 F , f˛ t2H k.t / 2 .; /.

Chapter 2 Images of C -Sets and Related Large Sets under Nonhomogeneous Spectra

27

f Proof. Let A D ¹x 2 N W f˛ .x/ 2 . 2 ; 2 /º. Since f ˛ is a homomorphism, for any f idempotent in ˇN, f ˛ .p/ D 0; and thus by Theorem 2.23, A is a C -set, and therefore a J -set. Pick by Theorem 2.20 some a 2 A and H 2 Pf .N/ such that for all k 2 F ,  P  P k.t / D f˛ a C a C t2H k.t / 2 A. Then, given k 2 F , f˛ .a/ C f˛ t2H    P P     t2H k.t / 2 . 2 ; 2 / and f˛ .a/ 2 . 2 ; 2 /, so f˛ t2H k.t / 2 .; /. When we write that hHn i1 nD1 is an increasing sequence in Pf .N/; we mean that for each n 2 N, max Hn < min HnC1 . Lemma 2.19. Let F 2 Pf .NN/, let ˛ > 0, and let  > 0. There is an increasing sequence hHn i1 in P .N/ such that, for each n 2 N and each k 2 F ,  nD1  f P k.t / 2 . f˛ t2Hn 2n ; 2n /. Proof. Pick H1 as guaranteed by Lemma 2.18 for 2 . Now let n 2 N and assume that H1 ; H2 ; : : : ; Hn have been chosen. Let l D max Hn . For k 2 F define rk 2 NN by, for t 2 N, rk .t / D k.l C t /. Pick by Lemma 2.18, L 2 Pf .N/ such that for all  P   k 2 F , f˛ t2L rk .t / 2 . 2n ; 2n /. Let HnC1 D l C L.

2.4

Preservation of J -Sets, C -Sets, and C  -Sets

We begin by showing as promised that in a commutative semigroup the translate in the definition of J -sets can be taken to be in the given J -set. Theorem 2.20. Let .S; C/ be a commutative semigroup, and let A be a J -set in S . For each F 2 Pf .NS /, there exist a 2 A and H 2 Pf .N/ such that for each f 2 F , P a C t2H f .t / 2 A. Proof. Let F 2 Pf .NS / be given and pick c 2 S . Denote by c the sequence constantly equal to c. For f 2 F define gf 2 NS by, for each n 2 N, gf .n/ D c C f .n/. Let K D ¹cº [ ¹gf W f 2 F º. Then K 2 Pf .NS / so pick b 2 S and H 2 Pf .N/ P P such that b C t2H c 2 A and for each f 2 F , b C t2H gf .t / 2 A. Let P P a D b C t2H c. Then for each f 2 F , a C t2H f .t / 2 A. Lemma 2.21. Let A D ¹22n C m2n C 1 W n; m 2 N and m < nº. If a; d 2 N and ¹a; ¯ n 2 N such that ¹a; a C d; a C 2d º  ® 2na C d; an C 2d º  A, then there is some 2 C m2 C 1 W m 2 ¹1; 2; : : : ; n  1º . Proof. Suppose first we have some k; n 2 N with k < n, some r 2 ¹1; 2; : : : ; k  1º, and some s 2 ¹1; 2; : : : ; n1º such that a D 22k Cr2k C1 and aCd D 22n Cs2n C1. Then a  22n2 C .n  2/2n1 C 1  22n2 C 2n1 2n1 D 22n1 and a C d > 22n so d > 22n1 . Thus a C 2d > 22n C 22n1 > 22n C .n  1/2n C 1. Also,

28

Neil Hindman and John H. Johnson

d < a C d  22nC1 so a C 2d < 22nC2 . Since 22n C .n  1/2n C 1 < a C 2d < 22nC2 , a C 2d … A. Thus we must have some n 2 N and r; s 2 ¹1; 2; : : : ; n  1º such that a D 22n C n d D .s  r/2n and a C 2d D 22n C .2s r2 C 1 and a C d D 22n C s2n C 1. So ® ¯  r/2n C 1 < 22nC2 . Therefore a C 2d 2 22n C m2n C 1 W m 2 ¹1; 2; : : : ; n  1º . Lemma 2.22. Let A D ¹22n C m2n C 1 W n; m 2 N and m < nº. Then A is not a J -set. Proof. Suppose that A is a J -set. Let tC1 1 2tC1 1 i tD1 ; h22t i1 i tD1 º: F D ¹h2t i1 tD1 ; h2 tD1 ; h2

Pick by Theorem 2.20 some a 2 A and H 2 Pf .N/ such that for all f 2 F , a C P / 2 A. Pick n 2 N and r 2 ¹1; 2; : : : ; n  1º such that a D 22n C r2n C 1. t2H f .tP P t 2t Let b D t2H 2 and let d D t2H 2 . Then ¹a; a C b; a C 2bº  A and ¹a; a C d; a C® 2d º  A, and so by Lemma 2.21 we ¯have ¹a; a C b; a C 2b; a C d; a C 2d º  22n C m2n C 1 W m 2 ¹1; 2; : : : ; n  1º . Pick m 2 ¹1; 2; : : : ; n  1º P such that a C b D 22n C m2n C 1. Then b D .m  r/2n so 2n divides t2H 2t , and thus min H  n. But then d  22n , and so a C d > 22n C .n  1/2n C 1, a contradiction. Theorem 2.23. Let S be a semigroup. Then J.S / is a closed two-sided ideal of ˇS , and a subset A of S is a C -set if and only if there is an idempotent p 2 J.S / \ A. Proof. [7, Theorems 3.5 and 3.8]. Thus the relationship between C -sets and J -sets is very similar to the relationship between central sets and piecewise syndetic sets: however, there are a few contrasts. Firstly, J.S / is closed, while K.ˇS / is commonly not closed. (In particular K.ˇN/ is not closed.) By [13, Corollary 4.41], ¹p 2 ˇS W .8A 2 p/.p is piecewise syndetic/º D c`K.ˇS /. Secondly, K.T / makes sense in an arbitrary compact Hausdorff right topological semigroup. We know of no reasonable meaning for J.T / in that generality. Theorem 2.24. Let S be a semigroup. If B [ C is a J -set, then either B or C is a J -set. Consequently, a set A  S is a J -set if and only if A \ J.S / ¤ ;. Proof. The first conclusion is [15, Theorem 2.14]. The second then follows from [13, Theorem 3.11]. We show now that spectra preserve J -sets. As we remarked previously, this proof f is complicated by the fact that f ˛ can take members of J.N/ anywhere in T .

Chapter 2 Images of C -Sets and Related Large Sets under Nonhomogeneous Spectra

29

Theorem 2.25. Let A be a J -set in N, let ˛ > 0, and let 0 <  < 1. Then g˛; ŒA is a J -set. Proof. Let F 2 Pf .NN/ be given. Pick by Theorem 2.24 some p 2 J.N/ such that f A 2 p. Let D f ˛ .p/. Pick and  as guaranteed by Lemma 2.17. Pick ı > 0 such that ı   and, if C  … ¹0; 1º, then . C   ı; C  C ı/ \ ¹0; 1º D ;. Pick by Lemma 2.19 an increasing sequence hHn i1 nD1 in Pf .N/ such that for each  P ı ı n 2 N and each k 2 F , f1=˛ t2Hn k.t / 2 . 3˛2n ; 3˛2n /. For k 2 F and n 2 N, P let rk .n/ D t2Hn k.t /. We claim that  P ./ for each k 2 F and each L 2 Pf .N/, we have h˛ n2L .h1=˛ ı rk /.n/ D P  P ı ı n2L rk .n/ and f˛ n2L .h1=˛ ı rk /.n/ 2 . 3 ; 3 /. To see this, let k 2 F and L 2 Pf .N/ be given. For n 2 L, let xn D .h1=˛ ı   ı 1 ı rk /.n/. Then f1=˛ rk .n/ D ˛1 rk .n/  xn so xn  3˛2 n < ˛ rk .n/ < xn C 3˛2n so P P ı ı ı ˛xn  32 n < rk .n/ < ˛xn C 32n . Therefore, ˛  n2L xn  3 < n2L rk .n/ < ˛  P P P P ı ı ı n2L xn C 3 and consequently n2L rk .n/ 3 < ˛ n2L xn < n2L rk .n/C 3 as required for (). We have that ¹x 2 N W g˛; .x/ C  ı3 < ˛x < g˛; .x/ C C ı3 º 2 p and, since J.N/  ˇN n N, ¹x 2 N W ˛x > 3º 2 p. Let C1 D ¹x 2 N W ˛x > 3 and g˛; .x/ C 

ı 3

< ˛x < g˛; .x/ C º and let

C2 D ¹x 2 N W ˛x > 3 and g˛; .x/ C  ˛x < g˛; .x/ C C 3ı º : Pick j 2 ¹1; 2º such that Cj 2 p. Now A \ Cj 2 p so A \ Cj is a J -set. Pick by Theorem 2.20 some a 2 A \ Cj and some L 2 Pf .N/ such that for each k 2 F , P a C n2L .h1=˛ ı rk /.n/ 2 A \ Cj . Let x D g˛; .a/; and for k 2 F let   P zk D g˛; a C n2L .h1=˛ ı rk /.n/ , let   P wk D g˛; a C n2L .h1=˛ ı rk /.n/ , and let  P y k D h˛ n2L .h1=˛ ı rk /.n/ : We claim that it suffices to show that there is some b 2 N such S that zk D yk C b for each k 2 F . Assume that we have done this, and let M D n2L Hn . We claim P that for each k 2 F , b C t2M k.t / D zk which will suffice since zk 2 g˛; ŒA.  P Thus, let k 2 F be given. By () zk  b D yk D h˛ .h1=˛ ı rk /.n/ D n2L P P P P n2L rk .n/ D n2L t2Hn k.t / D t2M k.t / as required. We show first that there is some i 2 ¹1; 0; 1º such that for all k 2 F , zk D wk Ci . Case 1. j D 1. Then,   P wk C  ı3 < ˛  a C n2L .h1=˛ ı rk /.n/ < wk C ; and thus   P wk C C   3ı < ˛  a C n2L .h1=˛ ı rk /.n/ C  < wk C C  :

30

Neil Hindman and John H. Johnson

If C   0, then zk D wk  1. If 0 < C   1, then 0 < C   ı3 so zk D wk . If 1 < C , then 1 < C   3ı so zk D wk C 1. Case 2. j D 2. Then   P wk C  ˛  a C n2L .h1=˛ ı rk /.n/ < wk C C ı3 ; and thus   P wk C C   ˛  a C n2L .h1=˛ ı rk /.n/ C  < wk C C  C 3ı : If C  < 0, then C  C 3ı < 0 so zk D wk  1. If 0  C  < 1, then

C  C ı3 < 1 so zk D wk . If 1  C  , then zk D wk C 1. Now to complete the proof that there is some b 2 N with zk D yk C b for each k 2 F , we show that x > 2 and wk D yk C x for each k 2 F . We know that   P wk C  ı3 < ˛  a C n2L .h1=˛ ı rk /.n/ < wk C C ı3 P  ı 2ı and by (),  ı3 < ˛  n2L .h1=˛ ı rk /.n/  yk < 3 so wk  yk C  3 < ˛a < ı ı ı 2ı wk yk C C 2ı 3 . Also xC  3 < ˛a < xC C 3 . Thus xC  3 < wk yk C C 3 ı and wk  yk C  2ı 3 < x C C 3 so x  ı < wk  yk < x C ı so wk  yk D x. Also, 3 < ˛a < x C C ı3 , so x  3. The proof of Theorem 2.25 uses the fact from Theorem 2.24 that any J -set is a member of an ultrafilter in J.N/, and therefore uses the axiom of choice. We suspect that with a bit more work, the proof can be rewritten to only depend on the fact that J -sets are partition regular. The proof of that fact in [15, Theorem 2.14] is elementary, using the Hales–Jewett theorem, but is rather complicated.

e

Corollary 2.26. Let ˛ > 0, and let 0 <  < 1. Then g˛; ŒJ.N/  J.N/.

e

Proof. Let p 2 J.N/ and let A 2 g˛; .p/. Pick B 2 p such that g˛; ŒB  A. Then B is a J -set and by Theorem 2.25, g˛; ŒB is a J -set and thus A is a J -set. Corollary 2.27. Let A be a C -set in N, let ˛ > 0, and let 0 <  < 1. Then g˛; ŒA is a C -set.

e

Proof. Pick by Theorem 2.23 an idempotent p 2 A \ J.N/. Then g˛; ŒA 2 g˛; .p/. By Corollary 2.26, g˛; .p/ 2 J.N/, while by Lemma 2.14, since p is an idempotent f f and is therefore in Z˛ , g˛; .p/ D h ˛ .p/ and h˛ .p/ is an idempotent.

e

e While we know that ge is a homomorphism on Z , it is not a homomorphism on ˛;

˛

ˇN or even on ˇN n N. The following lemma is a partial result in that direction.

Lemma 2.28. Let ˛ > 0, let 0 <  < 1, let q 2 ˇN n N, and let r 2 Z˛ . Then g˛; .q C r/ D g˛; .q/ C g˛; .r/.

e

e

e

Chapter 2 Images of C -Sets and Related Large Sets under Nonhomogeneous Spectra

31

e e

Proof. We shall show that there is some X 2 q such that for all x 2 X , g˛; .x Cr/ D g˛; .x/ C g˛; .r/. This will suffice since then the continuous functions g˛; ı r and g .r/ ı g˛; agree on a member of q and therefore agree at q. Once we have defined

e ˛;

e e

X , we will let x 2 X and produce Y 2 r such that for all y 2 Y , g˛; .x C y/ D g˛; .x/Cg˛; .y/, so that the continuous functions g˛; ıx and g˛; .x/ ı g˛; agree on a member of r. 1 3 1 f Let D f ˛ .q/. Pick by Lemma 2.17, 2 ¹ 2 ; 4 º and  2 .0; 2 / such that for all ı 2 .0; , ¹x 2 N W g˛; .x/ C  ı < ˛x < g˛; .x/ C C ıº 2 q. We consider two cases (with three subcases each). Only in cases 1b and 1c does the choice of the set Y depend on the choice of x. Case 1. C  2 ¹0; 1º. Let i D C . Case 1a. ¹x 2 N W g˛; .x/C D ˛xº 2 q. Let X D ¹x 2 N W g˛; .x/C D ˛xº. Then ˛ is rational, since otherwise jX j  1, and thus Y D ¹y 2 N W f˛ .y/ D 0º 2 r. Let x 2 X and y 2 Y and let k D g˛; .x/ and l D h˛ .y/. Then k C D ˛x and l D ˛y. Then k C i D ˛x C  , k C i C l D ˛  .x C y/ C , and l < ˛y C  < l C 1 so g˛; .x C y/ D k C i C l D g˛; .x/ C g˛; .y/. Case 1b. ¹x 2 N W g˛; .x/ C   < ˛x < g˛; .x/ C º 2 q. Let X D ¹x 2 N W g˛; .x/ C   < ˛x < g˛; .x/ C º and let x 2 X be given. Let k D g˛; .x/ and let  D min¹; 1  ; k C  ˛xº. Let Y D ¹y 2 N W f˛ .y/ 2 .; /º, let y 2 Y , and let l D h˛ .y/. Now kCi 1 < kC C  < ˛xC < kC C D kCi so g˛; .x/ D kCi 1. Also l  l   C  < ˛y C  < l C  C   l C 1 so g˛; .y/ D l. Finally, note that ˛y < l C   k C l C  ˛x so that k C l C i  1 < k C l C C      < ˛  .x C y/ C  < k C l C C  D k C l C i and thus g˛; .x C y/ D k C l C i  1. Case 1c. ¹x 2 N W g˛; .x/ C < ˛x < g˛; .x/ C C º 2 q. Let X D ¹x 2 N W g˛; .x/ C < ˛x < g˛; .x/ C C º and let x 2 X be given. Let k D g˛; .x/ and let  D min¹; 1  ; ˛x  k  º. Let Y D ¹y 2 N W f˛ .y/ 2 .; /º, let y 2 Y , and let l D h˛ .y/. Now k Ci D k C C < ˛x C < k C C C < k Ci C1 so g˛; .x/ D k Ci . Exactly as in case 1b, g˛; .y/ D l. Observe that ˛y > l    l  ˛x C k C so that k C l C i D k C l C C  < ˛  .x C y/ C  < k C l C i C  C  < k C l C i C 1 and thus g˛; .x C y/ D k C l C i . Case 2. C  … ¹0; 1º. Pick ı 2 .0; / such that ı  min¹; 1  º and . C   ı; C Cı/\¹0; 1º D ;. Let X D ¹x 2 N W g˛; .x/C  2ı < ˛x < g˛; .x/C C 2ı º and let Y D ¹y 2 N W f˛ .y/ 2 . ı2 ; ı2 /º. Let k D g˛; .x/ and let l D h˛ .y/. Then l < l  2ı C  < ˛y C  < l C ı2 C  < l C 1 so g˛; .y/ D l.

e

e

Case 2a. C  < 0. Then k  1 < k C C   ı2 < ˛x C  < k C C  C 2ı < k so g˛; .x/ D k  1. Also k C l  1 < k C l C C   ı < ˛  .x C y/ C  < k C l C C  C ı < k C l so g˛; .x C y/ D k C l  1.

32

Neil Hindman and John H. Johnson

Case 2b. 0 < C < 1. Then k < kC C  2ı < ˛xC < kC C C 2ı < kC1 so g˛; .x/ D k. Also k Cl < k Cl C C ı < ˛.x Cy/C < k Cl C C Cı < k C l C 1 so g˛; .x C y/ D k C l. Case 2c. 1 < C. Then k C1 < k C C  2ı < ˛x C < k C C C ı2 < k C2 so g˛; .x/ D k C 1. Also k C l C 1 < k C l C C   ı < ˛  .x C y/ C  < k C l C C  C ı < k C l C 2 so g˛; .x C y/ D k C l C 1.

e

Theorem 2.29. Let ˛ > 0 and let 0 <  < 1. Then g˛; ŒK.ˇN/  K.ˇN/, and consequently g˛; Œc`K.ˇN/  c`K.ˇN/.

e

Proof. Of course the second conclusion follows from the first by continuity. To see that g˛; ŒK.ˇN/  K.ˇN/, let p 2 K.ˇN/. Pick a minimal left ideal L of ˇN f such that p 2 L and pick an idempotent r 2 L. Then r 2 Z˛ (since f ˛ is a homomorphism). Now K.Z˛ / D Z˛ \ K.ˇN/ by [13, Theorem 1.65]. By Lemma 2.14, f g˛; .r/ D h ˛ .r/ is an idempotent in K.Z1=˛ / and so g˛; .r/ 2 K.ˇN/. Now p 2 L D L C r so there is some q 2 L such that p D q C r. Thus, by Lemma 2.28, g˛; .p/ D g˛; .q/ C g˛; .r/ 2 K.ˇN/.

e

e e

e

e

e

We see now that spectra preserve C  -sets. We saw in Theorem 2.9 that they need not preserve thick, PS  , J  , or AP  -sets. The reason for the distinction seems to be that C  -sets depend on idempotents, which are in particular members of Z˛ . (All of the properties shown to be preserved in [4] also depended on points which are members of Z˛ for each ˛ > 0.) Theorem 2.30. Let A be a C  -set in N, let ˛ > 0, and let 0 <  < 1. Then g˛; ŒA is a C  -set. Proof. We need to show that g˛; ŒA is a member of every idempotent in J.N/, so

e

let an idempotent p 2 J.N/ be given. By Lemma 2.14 and Corollary 2.26, h1=˛ .p/   is an idempotent in J.N/ so A 2 h1=˛ .p/. Again by Lemma 2.14, g˛; h1=˛ .p/ D   f h ˛ h1=˛ .p/ D p and so g˛; ŒA 2 p.

e

e

e e

In [13, Corollary 16.43] it was shown that if ˛  1, 0 <  < 1, P is any of the properties central, central*, IP , or IP  , and A  N, then g˛; ŒA has property P if and only if A has property P . The arguments given there depended on the specific property. We see now that in fact, all that is required is the preservation of the property by all spectra. Theorem 2.31. Let A be a set of subsets of N which is closed under passage to supersets and assume that whenever ˛ > 0, 0 <  < 1, and A 2 A, one has that g˛; ŒA 2 A. Then whenever ˛  1, 0 <  < 1, and A  N, one has g˛; ŒA 2 A if and only if A 2 A.

Chapter 2 Images of C -Sets and Related Large Sets under Nonhomogeneous Spectra

33

Proof. Let ˛  1, 0 <  < 1, and A  N. The sufficiency holds by assumption. If ˛ D 1, then g˛; is the identity function so we can assume ˛ > 1. Notice that   for all x 2 N, g1=˛;.1/=˛ g˛; .x/ D x. (If y D g˛; .x/, then ˛1 y  x C ˛ < 1 1 1 < ˛1 y C 1 ˛ y C ˛ so x ˛  x C ˛ < x C 1.) Consequently, for any A  N,   g1=˛;.1/=˛ g˛; ŒA  A. Thus if g˛; ŒA 2 A, one has A 2 A. There are a few basic constructions which have been used to produce examples of C -sets which are not central. By Theorem 2.31, if A is such an example, ˛ > 1, and 0 <  < 1, then g˛; ŒA is another such example.

2.5

Preservation of Ideals

We show in this section that if ˛  1 and 0 <   1, then the ideals K.ˇN/, c`K.ˇN/, and J.N/ are preserved by g˛; but fail dramatically to be preserved if ˛ > 1.

e

Lemma 2.32. Let p 2 ˇN, let 0 < ˛ < 1, and let 0 <  < 1. There exist l 2 Z,  2 ¹ 12 ; 34 º, and A 2 p such that for all x 2 A, g˛; l C g1=˛; .x/ D x.

e

Proof. Let D f1=˛ .p/, and pick by Lemma 2.17 2 ¹ 12 ; 34 º and  > 0 such that for all ı 2 .0; , ¹x 2 N W g1=˛; .x/ C  ı < ˛1 x < g1=˛; .x/ C C ıº 2 p. Let 1 ı D min¹ 1˛ 2˛ ; º, let A D ¹x 2 N W g1=˛; .x/ C  ı < ˛ x < g1=˛; .x/ C C ıº, and let x 2 A. Now .  ˛  ı C ˛1 /  . C ı  ˛ / D ˛1  2ı  ˛1  2  1˛ 2˛ D 1; and so pick l 2 Z   1 such that C ı  ˛  l <  ˛  ı C ˛ . Then ˛ C ˛ı    ˛l < ˛    ˛ı C 1. Let k D g1=˛; .x/. We show that g˛; .l C k/ D x. Now ˛k C˛ ˛ı < x < ˛k C˛ C˛ı so x ˛ ˛ıC˛l C < ˛ .k Cl/C < x  ˛ C ˛ı C ˛l C , 0  ˛l  ˛  ˛ı C , and ˛l  ˛ C ˛ı C   1 so x < ˛  .l C k/ C  < x C 1 as required. Theorem 2.33. Let I be a subset of ˇN which is a left ideal of .ˇZ; C/ and assume that whenever 0 < ˛ and 0 <  < 1, one has g˛; ŒI   I . Let 0 < ˛  1 and let 0 <  < 1. Then g˛; ŒI  D I . In particular, g˛; ŒK.ˇN/ D K.ˇN/, g˛; Œc`K.ˇN/ D c`K.ˇN/, g˛; ŒJ.N/ D J.N/, and g˛; ŒAP  D AP .

e e

e

e

e e

Proof. If ˛ D 1, then g˛; is the identity function, and thus we assume that ˛ < 1. We need to show that I  g˛; ŒI . Thus, let p 2 I . By Lemma 2.32 we may pick   l 2 Z, 2 ¹ 12 ; 34 º, and A 2 p such that for all x 2 A, g˛; l C g1=˛; .x/ D x. By assumption g1=˛; .p/ 2 I and since I is a left ideal of ˇZ, l C g1=˛; .p/ 2 I . Since   g˛; ı l ı g1=˛; agrees with the identity on A, we have p D g˛; l C g1=˛; .p/ and so p 2 g˛; ŒI .

e

e

A A e

A e

A

34

Neil Hindman and John H. Johnson

To verify the “in particular" conclusions, by Corollary 2.26 and Theorem 2.29, we only need verify that each of K.ˇN/, c`K.ˇN/, J.N/, and AP are left ideals of .ˇZ; C/. By [13, Lemma 1.43(c)] each minimal left ideal of ˇN is a left ideal of ˇZ and so K.ˇN/ is the union of left ideals of ˇZ and is thus a left ideal of ˇZ. Then by [13, Theorem 2.17], c`K.ˇN/ is a left ideal of ˇZ. To see that J.N/ is a left ideal of ˇZ, let p 2 J.N/. Then ˇZ C p D c`.Z C p/; and thus it suffices to show that Z C p  J.N/. Then, let m 2 Z and let A 2 m C p. We need to show that A \ N is a J -set in N, and so let F 2 Pf .NN/ be given. We have that m C A 2 p and ¹x 2 N W x > mº 2 p, and thus pick by Theorem 2.20 some P a 2 .m C A/ \ ¹x 2 N W x > mº and H 2 Pf .N/ such that for all k 2 F , a C t2H k.t / 2 .m C A/. P Then m C a 2 N, and for all k 2 F , m C a C t2H k.t / 2 A. The proof that AP is a left ideal of ˇZ is nearly identical. Given p 2 AP , m 2 Z, k 2 N, and A 2 m C p, .m C A/ \ ¹x 2 N W x > mº 2 p so it contains a length k arithmetic progression ¹a; a C d; : : : ; a C .k  1/d º. Then ¹m C a; m C a C d; : : : ; m C a C .k  1/d º is a length k arithmetic progression in A \ N. We saw in Theorem 2.8 that if ˛  1 and 0 <  < 1, then g˛; preserves thick sets. Corollary 2.34. Let 0 < ˛  1, let 0 <  < 1, and let A  N. If A is a PS  set, then g˛; ŒA is a PS  -set. If A is a J  -set, then g˛; ŒA is a J  -set. If A is an AP  -set, then g˛; ŒA is an AP  -set. Proof. The proofs are essentially the same. We will do the proof for J  -sets. Let A be a J  -set. Then J.N/  A so J.N/ D g˛; ŒJ.N/  g˛; Œ A  D g˛; ŒA.

e

e

We saw in Theorem 2.9 that if ˛ > 1, then the conclusion of Corollary 2.34 fails badly. We see now that if ˛ > 1, then the conclusion of Theorem 2.33 also fails badly. Theorem 2.35. Let ˛ > 1 and let 0 <  < 1. There exists x 2 N such that for every p 2 Z1=˛ , g˛; ŒN … x C p. Consequently, if I is a left ideal of ˇN, then I n g˛; ŒˇN ¤ ;.

e

Proof. Assume first that ˛ is rational and pick m; n 2 N such that m > n, ˛ D m n, i iC1 and m and n are relatively prime. Pick i 2 ¹0; 1; : : : ; n  1º such that n   < n . Pick t 2 N such that mt  i  1 .mod n/. (See for example [10, Theorem 56].) m iC1 Pick x 2 N such that mt D x n  i  1. Then m n  t C  < n  t C n D x. Also, m m m i m1 n  .t C 1/ C   n  t C n C n D x C n  x C 1. Therefore ./

if l 2 N and

m n

 l C   x; then

m n

 l C   x C 1:

Now, let p 2 Z1=˛ D Zn=m D mN, and suppose that g˛; ŒN 2 x C p. Then .x C g˛; ŒN/ \ mN 2 p, and so pick k 2 N such that x C mk 2 g˛; ŒN and pick

Chapter 2 Images of C -Sets and Related Large Sets under Nonhomogeneous Spectra

35

r 2 N such that x C mk D g˛; .r/. Then x C mk  m n  r C  < x C mk C 1 so m x  n  .r  nk/ C  < x C 1, contradicting (). Now assume that ˛ is irrational, so that ¹˛t W t 2 Nº is dense mod 1. (See for example [10, Theorem 36].) Pick x; t 2 N such that x    .˛  1/ < ˛t < x  . Then x  .˛  1/ < ˛t C  < x and ˛  .t C 1/ C  > x C 1. Let ı D min¹x  .˛t C  /; ˛  .t C 1/ C   .x C 1/; 12 º. We claim that for any integer l, ./

if ˛l C  > x  ı, then ˛l C   x C 1 C ı :

To see this, let l 2 Z. If l  t , then ˛l C   ˛t C   x  ı. Thus, assume that l  t C 1. Then ˛l C   ˛t C ˛ C   x C 1 C ı. Now let p 2 Z1=˛ and suppose that g˛; ŒN 2 x C p. Then .x C g˛; ŒN/ \ ¹y 2 N W f1=˛ .y/ 2 . ˛ı ; ˛ı /º 2 p, and so pick y 2 N such that f1=˛ .y/ 2 . ˛ı ; ˛ı / and x C y 2 g˛; ŒN. Pick r 2 N such that x C y D g˛; .r/. Let k D h1=˛ .y/. Then k  ˛ı < ˛1  y < k C ˛ı , and so ˛k  ı < y < ˛k C ı. Therefore x C ˛k  ı < x C y  ˛r C  < x C y C 1 < x C ˛k C ı C 1 so x  ı < ˛  .r  k/ C  < x C ı C 1, which is a contradiction to (). To verify the last conclusion, pick x as guaranteed and let I be a left ideal of ˇN. Then I contains a minimal left ideal hence contains an idempotent p which is necessarily in Z1=˛ . Then x C p 2 I . Now g˛; ŒˇN D c`g˛; ŒN, and so x C p … g˛; ŒˇN.

e

e

Acknowledgments. The first author acknowledges support received from the National Science Foundation via grant DMS-0852512. Some of the results in this paper are from the second author’s Ph.D. thesis.

References [1] T. Bang, On the sequence Œn˛; n D 1; 2; : : : ; Math. Scand. 5 (1957), 69–76. [2] V. Bergelson and N. Hindman, Nonmetrizable topological dynamics and Ramsey Theory, Trans. Amer. Math. Soc. 320 (1990), 293–320. [3] V. Bergelson and N. Hindman, Partition regular structures contained in large sets are abundant, J. Comb. Theory (Series A), 93 (2001), 18–36. [4] V. Bergelson, N. Hindman, and B. Kra, Iterated spectra of numbers – elementary, dynamical and algebraic approaches, Trans. Amer. Math. Soc. 348 (1996), 893–912. [5] V. Bergelson, N. Hindman, and R. McCutcheon, Notions of size and combinatorial properties of quotient sets in semigroups, Topology Proc. 23 (1998), 23–60. [6] V. Bergelson, N. Hindman, and D. Strauss, Strongly central sets and sets of polynomial returns mod 1, Proc. Amer. Math. Soc., to appear. [7] D. De, N. Hindman, and D. Strauss, A new and stronger Central Sets Theorem, Fundamenta Mathematicae 199 (2008), 155-175.

36

Neil Hindman and John H. Johnson

[8] H. Furstenberg, Recurrence in ergodic theory and combinatorical number theory, Princeton University Press, Princeton, 1981. [9] R. Graham, S. Lin, and C. Lin, Spectra of numbers, Math. Mag. 51 (1978), 174–176. [10] G. Hardy and E. Wright, An introduction to the theory of numbers, 5th edn., Oxford University Press, London, 1979. [11] N. Hindman, Small sets satisfying the Central Sets Theorem, in: B. Landman, M. Nathanson, J. Nešetˇril, R. Nowakowski, C. Pomerance, and A. Robertso (eds.), Combinatorial Number Theory, pp. 57–64, DeGruyter, Berlin, 2009. Also published in Integers 9 (Supplement) (2007), #A5. http://www.integers-ejcnt.org/vol9supp.html. [12] N. Hindman, A. Maleki, and D. Strauss, Central sets and their combinatorial characterization, J. Comb. Theory (Series A) 74 (1996), 188–208. ˇ [13] N. Hindman and D. Strauss, Algebra in the Stone–Cech compactification: theory and applications, 2nd edn., de Gruyter, Berlin, 2011. [14] N. Hindman and D. Strauss, A simple characterization of sets satisfying the Central Sets Theorem, New York J. Math. 15 (2009), 405–413. [15] N. Hindman and D. Strauss, Cartesian products of sets satisfying the Central Sets Theorem, Topology Proceedings 35 (2010), 203–223. [16] I. Niven, Diophantine approximations, Interscience Publishers, New York, 1963. [17] H. Shi and H. Yang, Nonmetrizable topological dynamical characterization of central sets, Fund. Math. 150 (1996), 1–9. [18] T. Skolem, On certain distributions of integers in pairs with given differences, Math. Scand. 5 (1957), 57–68. [19] T. Skolem, Über einige Eigenschaften der Zahlenmengen Œ˛ n C ˇ bei irrationalem ˛ mit einleitenden Bemerkungen über einige kombinatorische probleme, Norske Vid. Selsk. Forh. 30 (1957), 42–49.

Author information Neil Hindman, Department of Mathematics, Howard University, Washington, DC, USA. Email: [email protected] John H. Johnson, Department of Mathematics, James Madison University, Harrisonburg, Virginia, USA. Email: [email protected]

Combinatorial Number Theory, 37–44

© De Gruyter 2013

Chapter 3

On the Differences Between Consecutive Prime Numbers, I Daniel A. Goldston and Andrew H. Ledoan Abstract. We show by an inclusion-exclusion argument that the prime k-tuple conjecture of Hardy and Littlewood provides an asymptotic formula for the number of consecutive prime numbers which are a specified distance apart. This refines one aspect of a theorem of Gallagher that the prime k-tuple conjecture implies that the prime numbers are distributed in a Poisson distribution around their average spacing. Keywords. Primes, k-Tuple Conjecture. Mathematics Subject Classification 2010. Primary: 11N05; Secondary: 11P32, 11N36.

3.1

Introduction and Statement of Results

In 1976, Gallagher [5,6] showed that a uniform version of the prime k-tuple conjecture of Hardy and Littlewood implies that the prime numbers are distributed in a Poisson distribution around their average spacing. Specifically, let Pr .h; N / denote the number of positive integers n less than or equal to N such that the interval .n; n C h contains exactly r prime numbers. Gallagher then proved that an appropriate form of the prime k-tuple conjecture implies, for any positive constant  and h  log N as N ! 1, that r Pr .h; N / e  N: rŠ In particular, if r D 0, then we obtain by an argument using the prime number theorem that, as N ! 1, X pnC1 N pnC1 pn  log n

1 e 

N : log N

(3.1)

Here, pn is used to denote the n-th prime number. The purpose of the present chapter is to obtain a refinement of (3.1) which shows that the Poisson distribution of the prime numbers in short intervals extends down to the individual differences between consecutive prime numbers. To obtain this result, we employ a version of the prime

38

Daniel A. Goldston and Andrew H. Ledoan

k-tuple conjecture formulated as Conjecture H in Section 3.2, which is equivalent to the form of the conjecture used by Gallagher. Theorem 3.1. Assume Conjecture H. Let d be any positive integer, and let p be a prime number. Let, further, 8 Y p  1 ˆ ˆ ; if d is even; < 2C2 p2 pjd S.d / D p>2 ˆ ˆ : 0; if d is odd; where

Y 1 C2 D p>2

1 .p  1/2

and define N.x; d / D

 D 0:66016 : : : ;

X

1;

pnC1 x pnC1 pn Dd

where pn denotes the n-th prime number. Then for any positive constant  and d even with d  log x as x ! 1, we have N.x; d / e  S.d /

x : .log x/2

(3.2)

Here we note that S.d / is the singular series in the conjectured asymptotic formula for the number of prime pairs differing by d . Our theorem shows that for consecutive prime numbers the Poisson density is superimposed onto this formula for prime pairs. Our theorem as well as its proof are implicitly contained in the 1999 paper by Odlyzko, Rubinstein, and Wolf [11] on jumping champions.1 Without claiming any originality, we think it is worthwhile to explicitly state and prove (3.2). More precise results when d= log x ! 0 will be addressed in a second paper.

3.2 The Hardy–Littlewood Prime k-Tuple Conjectures Let H D ¹h1 ; : : : ; hk º be a set of k distinct integers. Let .xI H / denote the number of positive integers n less than or equal to x for which nCh1 ; : : : ; nChk are simultaneously prime numbers. Then the prime k-tuple conjecture of Hardy and Littlewood [8] is that, for x ! 1, (3.3) .xI H / S.H /lik .x/; 1 An integer d is called a jumping champion for a given x if d is the most frequently occurring difference between consecutive prime numbers up to x.

Chapter 3 On the Differences Between Consecutive Prime Numbers, I

where S.H / D

Y p

1

1 p

39

k   H .p/ ; 1 p

H .p/ denotes the number of distinct residue classes modulo p occupied by the elements of H , and Z x dt : (3.4) lik .x/ D k 2 .log t / Note in particular that, if H .p/ D p for some prime number p, then S.H / D 0. However, if H .p/ < p for all prime numbers p, then S.H / ¤ 0, in which case the set H is called admissible. In (3.3), H is assumed to be admissible, since otherwise .xI H / is equal to 0 or 1. The prime k-tuple conjecture has been verified only for the prime number theorem. That is to say, for the case of k D 1. It has been asserted that, in its strongest form, the conjecture holds true for any fixed integer k with an error term that is Ok .x 1=2C" / at most and uniformly for H Œ1; x. (See Montgomery and Soundararajan [9, 10].) However, we do not need such strong conjectures here. Using   x kx ; (3.5) CO lik .x/ D .log x/k .log x/kC1 obtained from integration by parts, we replace lik .x/ by its main term and make the following conjecture. Conjecture H. For each fixed integer k  2 and admissible set H , we have .xI H / D S.H /

x .1 C ok .1//; .log x/k

uniformly for H Œ1; h, where h  log x as x ! 1 and  is a positive constant.

3.3

Inclusion–Exclusion for Consecutive Prime Numbers

The prime k-tuple conjecture for the case when k D 2 provides an asymptotic formula for the number of prime numbers with a given difference d . We need to find a corresponding formula where we restrict the count to prime numbers that are consecutive, and for this one can use the prime k-tuple conjecture with k D 3; 4; : : : and inclusion–exclusion to obtain upper and lower bounds for the number of consecutive prime numbers with difference d . This method has appeared in a series of papers by Brent [1–3] and was used by Erd˝os and Strauss [4] and Odlyzko, Rubinstein, and Wolf [11] in their study of jumping champions. We consider a special type of tuple Dk for which D2 D ¹0; d º

40

Daniel A. Goldston and Andrew H. Ledoan

and, for k  3,

Dk D ¹0; d1 ; : : : ; dk2 ; d º:

Here, we require d to be even. We want to count the number of consecutive prime numbers which do not exceed x and have difference d , namely N.x; d /, and for this we do inclusion–exclusion with X 2 .x; d / D 1; px pp 0 Dd

where p 0 is also a prime number and, for k  3, k .x; d1 ; : : : ; dk2 ; d / D

X

1:

px pp 0 Dd ppj Ddj ; 1j k2

Inserting the expected main term, we obtain 2 .x; d / D S.d /li2 .x/ C R2 .x; d /

(3.6)

k .x; d1 ; : : : ; dk2 ; d / D S.Dk /lik .x/ C Rk .x; Dk /:

(3.7)

and, for k  3,

We now carry out the inclusion-exclusion. We trivially have N.x; d /  2 .x; d /: The consecutive prime numbers that differ by d are those prime numbers p and p 0 satisfying p  p 0 D d such that there is no third prime number p 00 with p 0 < p 00 < p. We can exclude these nonconsecutive prime numbers differing by d by removing all triples of this form, although this will exclude the same nonconsecutive pair of prime numbers more than once if there are quadruples of prime numbers such that p 0 < p 00 < p 000 < p. Hence, writing p  p 00 D d 0 , we obtain the lower bound X 3 .x; d 0 ; d /: N.x; d /  2 .x; d /  1d 0
We next obtain an upper bound by including the quadruples eliminated in the previous step and continue in this fashion to get, for R  1, Q2RC1 .x; d /  N.x; d /  Q2R .x; d /;

(3.8)

where, for N  2, QN .x; d / D 2 .x; d / C

N X kD3

.1/k

X

k .x; d1 ; : : : ; dk2 ; d /:

1d1 <
We use the convention here that an empty sum has the value zero.

Chapter 3 On the Differences Between Consecutive Prime Numbers, I

41

To evaluate QN .x; d /, we require a special type of singular series average considered by Odlyzko, Rubinstein, and Wolf [11]. Let, for k  3, X

Ak .d / D

S.Dk /:

(3.9)

1d1 <:::
Odlyzko, Rubinstein and Wolf [11] proved that, for k  3, Ak .d / D S.d /

d k2 C Ek .d /; .k  2/Š

(3.10)

where Ek .d / D Ok

d k2 log log d

! :

(3.11)

(See also Goldston and Ledoan [7].) Thus, on substituting (3.6), (3.7), (3.9), and (3.10), we find that, for N  2, QN .x; d / D S.d /li2 .x/ C

N X

.1/k Ak .d /lik .x/ C R2 .x; d /

kD3

C

N X

X

.1/k

kD3

Rk .x; Dk /

1d1 <
" 2  # Z xX N x N X 1 d k dt dt C .1/k Ek .d / D S.d / 2 kŠ log t .log t / .log t /k 2 2 Z

kD0

C R2 .x; d / C

N X

kD3

.1/k

kD3

X

Rk .x; Dk /;

1d1 <
where we used (3.4) in the second line. We can extract a main term independent of N out of the first term on the far righthand side above by using Taylor’s theorem. With the remainder expressed in Lagrange’s form, we have that, for M  0 and x > 0,

e x D

M X 1 e  .x/k C .x/M C1 ; kŠ .M C 1/Š

kD0

42

Daniel A. Goldston and Andrew H. Ledoan

where lies in the open interval joining 0 and x. Hence, we have   # Z x "NX 2 1 d k dt kŠ log t .log t /2 2 kD0 !     Z x Z x 1 d dt d N 1 dt D exp CO log t .log t /2 .log t /2 2 2 .N  1/Š log t !  N 1   Z x x dt d 1 3d ; exp CO p D log t .log t /2 .log x/2 N N log x 2 by the Stirling formula .M  1/Š D

p

   1 : 2M M 1=2 e M 1 C O M

Therefore, we have proved the following lemma. Lemma 3.2. For N  2, we have Z QN .x; d / D S.d /I.x; d / C

C

N X

2

.1/k Ek .d /

kD3

X

.1/k

kD3

CO

N x X

Rk .x; Dk /

1d1 <
1 p N



where

3d N log x Z

I.x; d / D

x 2

N 1



dt C R2 .x; d / .log t /k

x .log x/2

d exp log t



! ;

dt : .log t /2

3.4 Proof of the Theorem If we had imposed the additional condition that n C hj is less than or equal to x, for j 2 ¹1; : : : ; kº, in the definition of .xI H / in Section 3.2, we would have that 2 .n; d / D .xI D2 / and, for k  3, k .x; d1 ; : : : ; dk2 ; d / D .xI Dk /. However, this condition has no effect on Conjecture H, since with H Œ1; h the condition removes at most h tuples, which are absorbed into the error term. Thus, assuming that Conjecture H holds true for k  N , we have that R2 .x; d / D o.S.d /x=.log x/2 /, Rk .x; Dk / D ok .S.Dk /x=.log x/k / and, by (3.11), Ek .d / D ok .d k2 /. Then by

43

Chapter 3 On the Differences Between Consecutive Prime Numbers, I

the lemma and since S.d /  1 for d even, we have, for x ! 1 and d  log x,     x x 2 QN .x; d / D S.d /I.x; d / C oN e C o S.d / .log x/2 .log x/2   N X X x ok S.Dk / C .log x/k kD3 1d1 <
Z x dt d I.x; d /  exp 2 log x  log log x x= log x .log t /

x i h

log log x i h li2 .x/  li2  exp dN 1 C O log x log x !# " N x d log log x N  e d : 1CO .log x/2 log x 

Hence, we have I.x; d / D e

dN

" x 1CO .log x/2

dN log log x log x

!# ;

and the required estimate now follows since, if d  log x, dN  as x ! 1. Hence, the proof of the theorem is completed. Acknowledgments. During the preparation of this work, the first author received support from the National Science Foundation Grant DMS-1104434. The authors would like to express their sincere gratitude to the referee for his comments on the earlier version of this paper.

44

Daniel A. Goldston and Andrew H. Ledoan

References [1] R. P. Brent, The distribution of small gaps between successive primes, Math. Comp. 28 (1974), 315–324. [2] R. P. Brent, Irregularities in the distribution of primes and twin primes, in: Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday, Math. Comp. 29 (1975), 43–56. [3] R. P. Brent, Correction to: Irregularities in the distribution of primes and twin primes (Math. Comp. 29 (1975), 43–56), Math. Comput. 30(133) (1976), 198. [4] P. Erd˝os and E. G. Straus, Remarks on the differences between consecutive primes, Elem. Math. 35(5) (1980), 115–118. [5] P. X. Gallagher, On the distribution of primes in short intervals, Mathematika 23 (1976), 4–9. [6] P. X. Gallagher, Corrigendum: On the distribution of primes in short intervals (Mathematika 23 (1976), 4–9), Mathematika 28(1) (1981), 86. [7] D. A. Goldston and A. H. Ledoan, The jumping champion conjecture, submitted for publication (2012), available at: http ://arxiv.org/pdf/1102.4879v1. [8] G. H. Hardy and J. E. Littlewood, Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes, Acta Math. 44(1) (1922), 1–70. Reprinted as pp. 561–630 in :Collected Papers of G. H. Hardy, vol. I (including joint papers with J. E. Littlewood and others; edited by a committee appointed by the London Mathematical Society), Clarendon Press, Oxford University Press, Oxford, 1966. [9] H. L. Montgomery and K. Soundararajan, Beyond pair correlation, in: Erd˝os and his mathematics, I, pp. 507–514, Budapest, 1999; Bolyai Soc. Math. Stud. 11, János Bolyai Math. Soc., Budapest, 2002. [10] H. L. Montgomery and K. Soundararajan, Primes in short intervals, Comm. Math. Phys. 252(1–3) (2004), 589–617. [11] A. Odlyzko, M. Rubinstein, and M. Wolf, Jumping champions, Experiment. Math. 8(2) (1999), 107–118.

Author information Daniel A. Goldston, Department of Mathematics, San José State University, San José, California, USA. Email: [email protected] Andrew H. Ledoan, Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, Tennessee, USA. Email: [email protected]

Combinatorial Number Theory, 45–53

© De Gruyter 2013

Chapter 4

On Sets of Integers Which Are Both Sum-Free and Product-Free Pär Kurlberg, Jeffrey C. Lagarias, and Carl Pomerance Abstract. We consider sets of positive integers containing no sum of two elements in the set and also no product of two elements. We show that the upper density of such a set is strictly smaller than 12 and that this is best possible. Further, we also find the maximal order for the density of such sets which are also periodic modulo some positive integer. Keywords. Sum-Free Set, Product-Free Set, Upper Density. Mathematics Subject Classification 2010. 11B05, 11B75.

4.1

Introduction

If A, B are sets of integers, we let A C B denote the set of sums a C b with a 2 A, b 2 B, and we let A  B denote the set of products ab with a 2 A, b 2 B. The sumproduct problem in combinatorial number theory is to show that if A is a finite set of positive integers, then either A C A or A  A is a much larger set than A. Specifically, Erd˝os and Szemerédi [2] conjecture that if  > 0 is arbitrary and A is a set of N positive integers, then for N sufficiently large depending on the choice of , we have jA C Aj C jA  Aj  N 2 : This conjecture is motivated by the cases when either jA C Aj or jA  Aj is unusually small. For example, if A D ¹1; 2; : : : ; N º, then A C A is small, namely, jA C Aj < 2N . However, A  A is large since there is some c > 0 such that jA  Aj > N 2 =.log N /c . And if A D ¹1; 2; 4; : : : ; 2N 1 º, then jA  Aj < 2N , but jA C Aj > N 2 =2. The best that we currently know towards this conjecture is that it holds with exponent 4=3 in the place of 2, a result of Solymosi [8]. (In fact, Solymosi proves this when A is a set of positive real numbers.) In this paper we consider a somewhat different question: how dense can A be if both A C A and A  A have no elements in common with A? If A \ .A C A/ D ;, we say that A is sum-free, and if A \ .A  A/ D ;, we say that A is product-free. Before stating the main results, we give some background on sets that are either sum-free or product-free.

46

Pär Kurlberg, Jeffrey C. Lagarias, and Carl Pomerance

If a 2 A and A is sum-free, then ¹aºCA is disjoint from A, and so we immediately have that the upper asymptotic density 1 d .A/ WD lim sup jA \ Œ1; nj n!1 n is at most 12 . Density 12 can be achieved by taking A as the set of odd natural numbers. Similarly, if A is a set of residues modulo n and is sum-free, then D.A/ WD jAj=n is at most 12 , and this can be achieved when n is even and A consists of the odd residues. The maximal density for D.A/ for A a sum-free set in Z=nZ was considered in [1]. 1 if n is divisible solely by primes that In particular, the maximum for D.A/ is 13  3n 1 1 are 1 modulo 3, it is 3 C 3p if n is divisible by some prime that is 2 modulo 3 and p is the least such, and it is 13 otherwise. Consequently, we have D.A/  25 if A is a sum-free set in Z=nZ and n is odd. It is worth noting that maximal densities of subsets of arbitrary finite Abelian groups are determined in [3]. For generalizations to subsets of finite non-Abelian groups, see [4]. The problem of the maximum density of product-free sets of positive integers, or of subsets of Z=nZ, only recently received attention. For subsets of the positive integers, it was shown in [5] that the upper density of a product-free set must be strictly less than 1; in fact, it cannot exceed 1  2a1 0 , where a0 is the least member of the set. Let D.n/ denote the maximum value of D.A/ as A runs over product-free sets in Z=nZ. In [7] it was shown that D.n/ < 12 for the vast majority of integers, namely for every integer not divisible by the square of a product of six distinct primes. Moreover, the density of integers which are divisible by the square of a product of six distinct primes was shown to be smaller than 1:56 108 . Somewhat surprisingly, D.n/ can in fact be arbitrarily close to 1 (see [5]), and thus there are integers n and sets of residues modulo n consisting of 99% of all residues, with the set of pairwise products lying in the remaining 1% of the residues. However, it is not easy to find a numerical example that beats 50%. In [5], an example of a number n with about 1:61 108 decimal digits was given with D.n/ > 12 ; it is not known if there are any substantially smaller examples, say, with fewer than 108 decimal digits. In [6] the maximal order of D.n/ was essentially found: there are positive constants c; C such that for all sufficiently large n, we have D.n/  1 

c .log log n/

1 2e log 2

.log log log n/1=2

;

and there are infinitely many n with D.n/  1 

C .log log n/

1 2e log 2

.log log log n/1=2

:

In this chapter we consider two related questions. First, if A is a set of integers which is both sum-free and product-free, how large may the upper asymptotic density

47

Chapter 4 On Sets of Integers Which Are Both Sum-Free and Product-Free

d .A/ be? Here a sum-free and product-free set with natural density by taking A D ¹n W n  2; 3 .mod 5/º:

2 5

is achievable

Our first result shows that for a set A to achieve upper asymptotic density close to it must omit all small integers.

1 2

Theorem 4.1. Let A be a set of positive integers that is both product-free and sumfree, and let a0 be the smallest element of A. If d .A/ > 25 ; then necessarily   1 1 : d .A/  1 2 2a0 Second, set D  .n/ WD max¹D.A/ W A is a sum-free, product-free subset of Z=nZº: What is the maximal order of D  .n/? We prove the following complementary results, showing that density 12 can be approached, and quantifying the rate of approach. Theorem 4.2. There is a positive constant  such that for all sufficiently large numbers n, 1  : D  .n/   e 1 log 2 .log log n/ 2 2 .log log log n/1=2 Theorem 4.3. There is a positive constant   and infinitely many integers n with D  .n/ 

 1  : e 2 .log log n/1 2 log 2 .log log log n/1=2

Note that D  .5/ D 25 and, if 5jn, then D  .n/  25 . A possibly interesting computational problem is to numerically exhibit some n with D  .n/ > 25 . Theorem 4.3 assures us that such numbers exist, but the least example might be very large. One might also ask for the densest possible set A for which A, A C A, and A  A are pairwise disjoint. However, Proposition 4.9 below implies immediately that any sum-free, product-free set A Z=nZ with D.A/ > 25 also has A C A and A  A disjoint. Thus, from Theorem 4.3, we may have these three sets pairwise disjoint with D.A/ arbitrarily close to 12 .

4.2

The Upper Density

Here we prove Theorem 4.1. We begin with some notation which we use in this section. For a set A of positive integers and a positive real number x, we write A.x/ for A \ Œ1; x. Set 1 jA.x/j ; so that jA.x/j D .1  ıx /x: ıx WD 1  2 x 2

48

Pär Kurlberg, Jeffrey C. Lagarias, and Carl Pomerance

Note that ıx  0 for jA.x/j  12 x: If a is an integer, we write a C A for ¹aº C A and we write aA for ¹aº  A. Lemma 4.4. Suppose that A is a sum-free set of positive integers and that a1 ; a2 2 A. Then for all x > 0, 1 j.a1 C A.x  a1 // \ .a2 C A.x  a2 //j  .1  3ıx /  .a1 C a2 /: 2 Proof. We have the sets A.x/; a1 C A.x  a1 /; a2 C A.x  a2 / all lying in Œ1; x and the latter two sets are disjoint from the first set (since A is sum-free). Thus, j.a1 C A.x  a1 // \ .a2 C A.x  a2 //j D ja1 C A.x  a1 /j C ja2 C A.x  a2 /j  j.a1 C A.x  a1 // [ .a2 C A.x  a2 //j  ja1 C A.x  a1 /j C ja2 C A.x  a2 /j  .x  jA.x/j/  .jA.x/j  a1 / C .jA.x/j  a2 / C .jA.x/j  x/ D 3jA.x/j  x  .a1 C a2 /: But 3jA.x/j  x D 12 .1  3ıx /x, so this completes the proof. For a set A of positive integers, define the difference set A WD ¹a1  a2 W a1 ; a2 2 Aº: Further, for an integer g, let Ag WD A \ .g C A/ D ¹a 2 A W a C g 2 Aº: Corollary 4.5. If A is a sum-free set of positive integers and g 2 A then, for any x > 0, 1 jAg .x/j  .1  3ıx / x C O.1/; 2 in which the implied constant depends on both g and A. Proof. Suppose that g 2 A, so that there exist a1 ; a2 2 A such that a1  a2 D g. If a 2 A.x  a1 / and a C a1 2 a2 C A.x  a2 /, then a C g D a C a1  a2 2 A, so that a 2 Ag . That is, Ag .x  a1 / contains a1 C .a1 C A.x  a1 // \ .a2 C A.x  a2 //. Thus, by Lemma 4.4, jAg .x  a1 /j  j.a1 C A.x  a1 // \ .a2 C A.x  a2 //j  from which the corollary follows.

1 .1  3ıx /  .a1 C a2 /; 2

Chapter 4 On Sets of Integers Which Are Both Sum-Free and Product-Free

49

Proposition 4.6. If A is a sum-free set of positive integers with upper density greater than 25 , then A is the set of all even integers and A consists solely of odd numbers. Proof. We first show that A is a subgroup of Z. Since A is closed under multiplication by 1, it suffices to show that if g1 ; g2 2 A, then g1 Cg2 2 A. If g1 CAg1 contains a member a of Ag2 , then a g1 2 A and a Cg2 2 A, so that g1 Cg2 2 A. Note that g1 CAg1 and Ag2 are both subsets of A. Now, by Corollary 4.5, if g1 CAg1 and Ag2 were disjoint, we would have for each positive real number x, 1 .1  3ıx /x C O.1/  jA.x/j D .1  ıx /x; 2 so that ıx  15 CO. x1 /. Hence lim inf ıx  15 , contradicting the assumption that A has upper density greater than 25 . Thus, g1 C Ag1 and Ag2 are not disjoint, which as we have seen, implies that g1 C g2 2 A. Thus, A is a subgroup of Z, say A D gZ for some positive integer g. Since each a1  a2  0 .mod g/ for all a1 ; a2 2 A, all members of A are in one residue class modulo g. Since A has upper density greater than 25 , it follows that g D 1 or 2. But A must be disjoint from A. Indeed, if a1  a2 D a3 with a1 ; a2 ; a3 2 A, then a1 D a2 C a3 , violating the condition that A is sum-free. Thus, if g D 1, A D ;, and if g D 2, A consists solely of odd numbers. The first case violates our hypothesis, so our proof is complete. Remark 4.7. Proposition 4.6 is best possible, as can be seen by taking A as the set of positive integers that are either 2 or 3 modulo 5. We now prove the following result, which immediately implies Theorem 4.1. Proposition 4.8. Suppose that A is a sum-free set of positive integers with least member a0 . Suppose in addition that a0 A is disjoint from A. Then the upper density of A is at most max¹ 25 ; 12 .1  2a1 0 /º. Proof. If the upper density of A is at most 25 , the result holds trivially, so we may assume the upper density exceeds 25 . It follows from Proposition 4.6 that A consists solely of odd numbers. Thus, for any real number x  a0 , both a0 A.x=a0 / and A.x/ consist solely of odd numbers, they are disjoint, and they lie in Œ1; x. Thus, ˇ ˇ  ˇ  ˇ ˇ ˇ x ˇˇ 1 x ˇˇ ˇ ˇ A D jA.x/j C  x C O.1/: a jA.x/j C ˇA 0 ˇ a0 ˇ a0 ˇ 2 Further A.x/ n A.x=a0 / is contained within the odd numbers in .x=a0 ; x, so that ˇ  ˇ   ˇ x x ˇˇ 1 ˇ x C O.1/:  jA.x/j  ˇA a0 ˇ 2 a0

50

Pär Kurlberg, Jeffrey C. Lagarias, and Carl Pomerance

Adding these two inequalities and dividing by 2 gives that jA.x/j  . 12  O.1/, so that A has upper density at most

1 2



1 4a0 ,

1 4a0 /x

C

giving the result.

4.3 An Upper Bound for the Density in Z=nZ In this section we prove Theorem 4.2. We begin by noting the following simple consequence of Proposition 4.6. Proposition 4.9. Suppose that n is a positive integer and A Z=nZ is sum-free. If D.A/ > 25 , then n is even and A is a subset of the odd residues classes in Z=nZ. N the set of positive numbers in the residue classes in A. Then Proof. Replace A with A, N A has density D.A/ and is sum-free. It follows from Proposition 4.6 that all members of AN are odd. If n were odd, then AN would contain both odd and even members, so we must have n even and A a subset of the odd residue classes in Z=nZ. This completes the proof. We are now ready to prove Theorem 4.2. For those n with D  .n/  25 , the result holds for any number , so assume that D  .n/ > 25 . Let A Z=nZ be a productfree, sum-free set with D.A/ D D  .n/. By Proposition 4.9 we have that n is even and that A is a subset of the odd residues modulo n. Suppose that k is an integer with n  2k < 2n. Let N D 22k n and let B be the set of positive integers of the form 2j b where j  k and b  N=2j D 22kj n, such there is some a 2 A with b  a .mod n/. Then the members of B are in Œ1; N  and jBj D

k X j D0

2kj

2

k

jAj D 2 .2

kC1

  1 2kC1  1/jAj > 1  jAj: 2 n

(4.1)

We note that B is product-free as a set of residues modulo N . Indeed, suppose 2ji bi 2 B, for i D 1; 2; 3 and 2j1 b1 2j2 b2  2j3 b3

.mod N /:

Let ai 2 A be such that bi  ai .mod n/ for i D 1; 2; 3. We have that a1 ; a2 ; a3 are odd, and since n is even, this implies that b1 ; b2 ; b3 are odd. Using j1 C j2  2k, j3  k, and 22k jN , we have j1 C j2 D j3 . Hence a1 a2  a3 .mod n/, a violation of the assumption that A is product-free modulo n. We conclude that B is product-free modulo N . It now follows from Theorem 1.1 in [6] that for n sufficiently large,   c : jBj  N 1  e .log log N /1 2 log 2 .log log log N /1=2

Chapter 4 On Sets of Integers Which Are Both Sum-Free and Product-Free

51

Further, since N is of order of magnitude n3 , we have that log log N D log log n C O.1/, and so for any fixed choice of c0 < c we have for n sufficiently large that   c0 : jBj  N 1  e .log log n/1 2 log 2 .log log log n/1=2 Thus, from our lower bound for jBj in (4.1) we have     N 1 1 c0 jAj < 2kC1 1  1 : e n 2 .log log n/1 2 log 2 .log log log n/1=2 Since N=22kC1 D n=2, it follows that for any fixed c1 < c0 and n sufficiently large, we have   n c1 jAj < : 1 e 2 .log log n/1 2 log 2 .log log log n/1=2 We thus may choose  as any number smaller than c=2. This concludes the proof of Theorem 4.2.

4.4

Examples With Large Density

In this section we prove Theorem 4.3. We follow the argument in [5] with a supplementary estimate from [6]. Let x be a large number, let `x be the least common multiple of the integers in Œ1; x, and let nx D `2x . Then nx D e.2Co.1//x as x ! 1 so that log log nx D log x C O.1/. For a positive integer m, let .m/ denote the number of prime factors of m counted with multiplicity. Let k D k.x/ D b 4e log log nx c, let Dx0 D ¹d j`x W d odd; k < .d / < 2kº ; and let A be the set of residues a modulo nx with gcd.a; nx / 2 Dx0 . Then A is productfree (cf. Lemma 2.3 in [5]), and since nx is even and every residue in A is odd, we have that A is sum-free as well. We shall now establish a sufficiently large lower bound on D.A/ to show that D  .nx / satisfies the inequality in the theorem with n D nx . For d 2 Dx0 , the number of a .mod nx / with gcd.a; nx / D d is '.nx /=d , so that ! X 1 '.nx / X 1 '.nx / X 1 : (4.2) D  D.A/ D nx d nx d d 0 d 2Dx

d j`x d odd

d j`x d odd d 62Dx0

We have    X 1 Y Y Y  1 p p 1 .x/ 1  aC1  D   1 ; d p1 p p1 x

d j`x d odd

2
2
2
52

Pär Kurlberg, Jeffrey C. Lagarias, and Carl Pomerance

and, since '.nx /=nx D 21 

Q

pjnx ; p>2 .1

 1=p/, we find that

  1 1 .x/ 1 .x/ '.nx / X 1  1 :   nx d 2 x 2 x

(4.3)

d j`x d odd

We now use Equation (6.2) in [6], which is the assertion that X P .d /x .d /62.k;2k/

e

1 .log x/ 2 log 2 :

d .log log x/1=2

Here P .d / denotes the largest prime factor of d . Since this sum includes every odd integer d j`x with d 62 Dx0 , we have e '.nx / .log x/ 2 log 2 '.nx / X 1 1



; e 1=2 1 log nx d nx .log log x/ .log x/ 2 2 .log log x/1=2 d j` x

d odd d 62Dx0

Q where we use Mertens’ theorem in the form '.nx /=nx D px .11=p/ 1= log x for the last step. Putting this estimate and Inequality (4.3) into Equation (4.2), we get D.A/ 

c0 1 .x/   e 2 x .log x/1 2 log 2 .log log x/1=2

for some positive constant c 0 . Using .x/=x 1= log x and log x D log log nx C O.1/, we have D.A/ 

1   e 2 .log log nx /1 2 log 2 .log log log nx /1=2

for any fixed constant   > c 0 and x sufficiently large. Thus, D  .nx / satisfies the condition of Theorem 4.3 for x sufficiently large, completing the proof. Acknowledgments. We thank Albert Bush, Chris Pryby, and Joseph Vandehey for raising the question of sets which are both sum-free and product-free. We also thank Imre Ruzsa for several nice suggestions that simplified our arguments. P. K. was supported in part by grants from the Göran Gustafsson Foundation, and the Swedish Research Council. J. C. L. was supported in part by NSF grant DMS-1101373. C. P. was supported in part by NSF grant DMS-1001180.

Chapter 4 On Sets of Integers Which Are Both Sum-Free and Product-Free

53

References [1] P. H. Diananda and H. P. Yap, Maximal sum-free sets of elements of finite groups, Proc. Japan Acad. 45(1) (1969), 1–5. [2] P. Erd˝os and E. Szemerédi, On sums and products of integers: To the memory of Paul Turán, in: P. Erd˝os (ed.), Studies in pure mathematics, pp. 213–218, Birkhäuser, Basel, 1983. [3] B. Green and I. Z. Ruzsa, Sum-free sets in abelian groups, Israel J. Math. 147 (2005), 157–188. [4] K. S. Kedlaya, Product-free subsets of groups, then and now, in: R. Y. Chow and D. C. Isaksen (eds.), Communicating Mathematics, pp. 179–187, Contemp. Math. 479, Amer. Math. Soc., Providence, RI, 2009. [5] P. Kurlberg, J. C. Lagarias, and C. Pomerance, Product-free sets with high density, Acta Arith. 155 (2012), 163–173. [6] P. Kurlberg, J. C. Lagarias, and C. Pomerance, The maximal density of product-free sets in Z=nZ, IMRN 2013 (2013), 827–845. First published online February 14, 2012, doi:10.1093/imrn/ms014. [7] C. Pomerance and A. Schinzel, Multiplicative properties of sets of residues, Moscow J. Combinatorics and Number Theory 1 (2011), 52–66. [8] J. Solymosi, Bounding multiplicative energy by the sumset, Adv. Math. 222 (2009), 402– 408.

Author information Pär Kurlberg, Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden. Email: [email protected] Jeffrey C. Lagarias, Department of Mathematics, University of Michigan, Ann Arbor, Michigan, USA. Email: [email protected] Carl Pomerance, Mathematics Department, Dartmouth College, Hanover, New Hampshire, USA. Email: [email protected]

Combinatorial Number Theory, 55–78

© De Gruyter 2013

Chapter 5

Four Perspectives on Secondary Terms in the Davenport–Heilbronn Theorems Frank Thorne Abstract. This paper is an expanded version of the author’s lecture at the Integers Conference 2011. We discuss the secondary terms in the Davenport–Heilbronn theorems on cubic fields and 3-torsion in class groups of quadratic fields. Such secondary terms had been conjectured by Datskovsky–Wright and Roberts, and proofs of these or closely related secondary terms were obtained independently by Bhargava, Shankar, and Tsimerman, Hough, Zhao, and Taniguchi and the author. In this chapter we discuss the history of the problem and highlight the diverse methods used in several papers to address it. Keywords. Quadratic Field, Cubic Field, Class Group. Mathematics Subject Classification 2010. 11R16, 11R29.

5.1

Introduction

This chapter concerns the following two theorems: Theorem 5.1. Let N3˙ .X / count the number of cubic fields K with 0 < ˙Disc.K/ < X . We have 1 4.1=3/ X C K˙ X 5=6 C o.X 5=6 /; 12.3/ 5.2=3/3 .5=3/ p D 1, K  D 3, and K C D 1.

N3˙ .X / D C ˙ where C  D 3, C C

(5.1)

Theorem 5.2. For any quadratic field with discriminant D, let Cl3 .D/ denote the p 3-torsion subgroup of the ideal class group Cl.Q. D//; we have X

#Cl3 .D/ D

0<˙D<X

! 1=3 C 1 Y p 8.1=3/ 3 C C˙ X 5=6 C o.X 5=6 /; (5.2) 1 X C K˙ 2 5.2=3/3 p p.p C 1/

56

Frank Thorne

and where the sum ranges over fundamental discriminants D, the product is over all primes, and the constants are as before. The main terms are due to Davenport and Heilbronn [15], and the secondary term in Theorem 5.1 was conjectured by Roberts [32] and implicitly by Datskovsky and Wright [14]. The secondary terms were expected to be difficult to prove. However, progress has recently been made in four independent works, using a variety of methods. Both results above have been proved by Bhargava, Shankar, and Tsimerman [7], using the geometry of numbers, and by Taniguchi and the present author [37], using Shintani zeta functions. Hough [24] has proved a variation of Theorem 5.2 by studying the distribution of Heegner points in the complex upper half-plane, and Zhao [45] has obtained a variation of Theorem 5.1 for function fields, using algebraic geometry.1 The author’s lecture at the 2011 Integers Conference explained (briefly) how each of these four approaches sheds light on these secondary terms, and this chapter is an expanded version of our lecture. We begin with some background on counting fields and on counting cubic fields in particular. The reader might skip to Section 5.4 for our discussion of the secondary terms.

5.2 Counting Fields in General How does one count number fields? This question has been addressed in many excellent expository accounts, such as that by Cohen, Diaz y Diaz, and Olivier [10], or Chapter 6 of Bhargava’s ICM Proceedings article [5]. Therefore, our overview will be brief. To a large extent we concentrate on the “trivial” problem of counting quadratic fields, as some observations on this problem anticipate the methods developed to count cubic fields with good error terms. Two early important theorems in the subject were proved by Minkowski and Hermite: Theorem 5.3 (Hermite). There are only finitely many number fields of bounded discriminant. Theorem 5.4 (Minkowski). The discriminant of a number field K of degree n satisfies  n 2  n  n : (5.3) jDisc.K/j  nŠ 4 If we write n D r C 2s, where r is the number of real embeddings of K, and s is the number of pairs of complex embeddings, we may replace .=4/n with .=4/2s . 1 The

error terms in [24] and [45] are larger than the secondary terms at present, but both of these approaches naturally explain the secondary term, and both authors are currently working on refining their methods.

Chapter 5 Secondary Terms in the Davenport-Heilbronn Theorems

p  n Moreover, by Stirling’s formula nŠ 2 n ne and therefore

n  n  jDisc.K/j  e 2  C o.1/ D 5:803    C o.1/ : 4

57

(5.4)

2

The constant e 4 can be improved; see Odlyzko [29]. Conversely, this bound is sharp apart from the constant; Golod and Shafarevich [21] proved the existence of an infinite class field tower of fields K for which jDisc.K/j1=n is constant.2 To prove these theorems3 , use the r embeddings K ,! R and the s pairs of embeddings K ,! C to embed the ring of integers of K as a lattice in n-dimensional space. As proved by Minkowski, any convex body centered around the origin whose volume is greater than 2rCs jDisc.K/j1=2 must contain a lattice point other than zero. However, this point must have norm at least 1, and geometric considerations allow one to conclude Minkowski’s lower bound. Hermite’s theorem can be proved in a similar way. We again use Minkowski’s convex body theorem to find an element ˛ 2 OK , the sizes of whose complex embeddings are all small, and Qwhich is guaranteed to generate K over Q. The minimal polynomial of ˛ is equal to .X   .˛//, where  ranges over all real and complex embeddings of K, and the bounds obtained from Minkowski’s theorem yield bounds on the coefficients of the minimal polynomial of ˛. For a fixed discriminant bound, there are only finitely many such polynomials and thus only finitely many possibilities for K. Remark 5.5. Although this proof of Hermite’s theorem relies on Minkowski’s work, Hermite’s paper [23] preceded Minkowski’s by over thirty years. Hermite claimed in [23] that there are finitely many fields of bounded discriminant with fixed degree; the stronger claim of Theorem 5.3 seems to depend also on Minkowski’s work. These proofs yield results about fields of each fixed degree, and so far this has been a natural way to count them. In what follows, let Nn .X / be the number of fields K of degree n with jDisc.K/j < X . (Results are also known when the sign of the discriminant, or more generally the number of real embeddings, is specified.) We have the following results (excluding n D 3): n = 1. There is only Q. n = 2. We have the equality of Dirichlet series

 .s/ X jDisc.K/js D 1 C 1  2s C 2  4s ; .2s/

(5.5)

ŒKWQ D2

and it follows that N2 .X / D

p 6 X C O. X/: 2 

(5.6)

p 1 ; 46/, which he proved to have an infinite [27] gave the example of F D Q.11 C 11 2-class field tower, for which jDisc.K/j1=n D 92:2 : : : for each field K. We refer to Lemmermeyer [25] for a discussion of related issues, along with a very thorough bibliography. 3 See [28, Chap. III.2] for complete proofs. 2 Martinet

58

Frank Thorne

The typical proof of (5.6) estimates, for each squarefree odd q, the number of integers n divisible by q 2 satisfying the 2-adic conditions implied by Equation (5.6), and then uses inclusion–exclusion to sieve for squarefreeness. The details are commonly left as an exercise for beginning analytic number theory students. However, it was not until the recent work of Belabas, Bhargava, and Pomerance [2] that the analogous method was developed to yield power-saving error terms for N3 .X /. How does one estimate the number of integers n with 0 < n < X and q 2 jn? (For simplicity of explanation, we ignore the 2-adic conditions.) This is a lattice-point counting problem in a one-dimensional vector space, so the answer is q12 times the length of the interval .0; X / plus an error of O.1/. This is trivial, but its generalization to the cubic case, where one counts GL2 .Z/-orbits on a four-dimensional lattice, satisfying local conditions .mod q 2 / and a global condition 0 < ˙Disc.f / < X , is not. (The higher dimensional analogues are even less so!) These integers may also be counted using analytic number theory. By Perron’s formula, we have Z 2Ci1 X ds (5.7) 1D q 2s .s/X s ; s 2i1 0
and these sums may be estimated by shifting the contour and using the functional equation of the zeta function. This functional equation relies on Poisson summation, and so again relies on the question’s interpretation as a lattice point counting problem. Even the most diehard fan of contour integration would likely prefer the trivial proof, but this analytic method also generalizes usefully to the cubic case, where the Riemann zeta function is replaced with a Shintani zeta function. These two counting techniques, in combination with the inclusion–exclusion sieve, are the starting points of the geometric and analytic proofs of Theorems 5.1 and 5.2, respectively! We note one additional point related to Equation (5.5). Wright [42] observed that this Dirichlet series has the beautiful representation  X X Y 1 (5.8) jDisc.K/js D jDisc.Kv /jps ; 2 p ŒKWQ D2

ŒKv WQp 2

and proved this in a much more general framework. (He obtained similar formulas for degree n cyclic extensions of any number field.) His results follow from considering a nontrivial “twist” of the adelic zeta function of Tate’s thesis [39]. This twist makes the affine line into a prehomogeneous vector space, with the action  of GL.1/ given by .t /x D t n x. Essentially, a vector space is prehomogeneous if it has an action of an algebraic group G, which is transitive over C apart from the vanishing locus of an irreducible polynomial. This “prehomogeneous” property is essential both in Bhargava’s work and in the zeta-function approach pioneered by Sato and Shintani [34]; cubic, quartic,

Chapter 5 Secondary Terms in the Davenport-Heilbronn Theorems

59

and quintic fields are parameterized by lattice points4 up to the action of G.Z/, and these may be counted geometrically or analytically. Wright’s work on (5.8) and its generalizations mirrors his work with Datskovsky [13,14,41] on the Shintani zeta function associated to cubic fields, and this latter work is at the heart of the analytic approach to counting cubic fields. In follow-up work, Wright and Yukie [43] proposed a program to enumerate quartic and quintic fields by studying the zeta functions associated to appropriate prehomogeneous vector spaces. So far their program has not succeeded. However: n = 4, 5. Bhargava [3, 6] proved the asymptotic formulas  5 Y 1 C p 2  p 3  p 4  X; N4 .X; S4 / 24 p N5 .X /

 13 Y  1 C p 2  p 4  p 5  X: 120 p

(5.9) (5.10)

The count N4 .X; S4 / includes only quartic fields with Galois group S4 , and (5.9) implies an asymptotic for N4 .X / in combination with work of Baily [1], Cohen, Diaz y Diaz, and Olivier [9], and Wong [40]. Bhargava’s geometric approach may be considered an extension of the methods described in Sections 5.3 and 5.6. n > 5. For n > 5, Bhargava conjectured [4] that Nn .X; Sn / Cn X . The explicit constants Cn have natural interpretations as Euler products; each Euler factor represents the “probability” that a field K of degree n has a certain localization. Ellenberg and Venkatesh [19] have proved the best upper bound to date, namely Nn .X / p exp.C log n/ . X It is unclear whether prehomogeneous vector spaces may be used to obtain formulas for Nn .X / for n > 5. By a classification theorem of Sato and Kimura [33], it is known that there are no reduced, irreducible, and reductive prehomogeneous vector spaces parameterizing rings of rank > 5; it remains to be seen whether there are any prehomogeneous vector spaces not satisfying these assumptions which parameterize such rings.

5.2.1

Counting Torsion Elements in Class Groups

The problem of counting 3-torsion elements in class groups of quadratic fields is related to that of counting cubic fields. If D ¤ 1 is a fundamental discriminant, then class p field theory provides a bijection betweenp subgroups of Cl.Q. D// of index 3, and unramified cyclic cubic extensions of Q. D/. Such extensions, in turn, correspond to (non-Galois) cubic extensions of Q whose discriminant is D. Therefore, counting 3-torsion elements in quadratic fields is equivalent to counting cubic fields whose discriminant is fundamental, which are those not totally ramified at any prime. 4 Not

all of the G.Z/-orbits correspond to fields, as we will see in the cubic case.

60

Frank Thorne

Therefore, this problem may be treated as a variation of the problem of counting cubic fields. This is the approach of Sections 5.3, 5.5, and 5.6; however, there are other approaches as well, such as that of Section 5.7. A general heuristic for statistics of torsion elements in class groups was proposed by Cohen and Lenstra [11]; their heuristic has inspired a great deal of interesting follow-up work, but for the most part the Cohen–Lenstra heuristics remain wide open.

5.3 Davenport–Heilbronn, Delone–Faddeev, and the Main Terms In this section we discuss the proofs of the main terms in Theorems 5.1 and 5.2. To summarize, the idea is as follows. We count cubic fields by counting their maximal orders, and so we count all cubic orders and then sieve for maximality. We count cubic orders by counting cubic rings and then subtracting the contribution of the reducible rings, and these correspond to binary cubic forms. (The space of binary cubic forms is prehomogeneous in the sense described previously.) Thus at last we have a geometric lattice-point counting problem, which can be attacked using “brute force”. This interpretation in terms of lattice points also opens the door to the use of zeta functions. We recommend the paper of Bhargava, Shankar, and Tsimerman [7] for the simplest self-contained account of the Davenport–Heilbronn theorems, with complete proofs. We begin by defining cubic rings and cubic forms. A cubic ring is a commutative ring which is free of rank 3 as a Z-module. The discriminant of a cubic ring is defined to be the determinant of the trace form hx; yi D Tr.xy/, and the discriminant of the maximal order of a cubic field is equal to the discriminant of the field. The lattice of integral binary cubic forms is defined by ® ¯ (5.11) VZ WD au3 C bu2 v C cuv 2 C dv 3 W a; b; c; d 2 Z ; and the discriminant of such a form is given by the usual formula Disc.f / D b 2 c 2  4ac 3  4b 3 d  27a2 d 2 C 18abcd:

(5.12)

There is a natural action of GL2 .Z/ (and also of SL2 .Z/) on VZ , given by .  f /.u; v/ D

1 f ..u; v/   /: det 

(5.13)

We call a cubic form f irreducible if f .u; v/ is irreducible as a polynomial over Q, and nondegenerate if Disc.f / ¤ 0. Cubic rings are related to cubic forms by the following correspondence of Delone and Faddeev, as further extended by Gan, Gross, and Savin [20]:

Chapter 5 Secondary Terms in the Davenport-Heilbronn Theorems

61

Theorem 5.6 ([16, 20]). There is a natural, discriminant-preserving bijection between the set of GL2 .Z/-equivalence classes of integral binary cubic forms and the set of isomorphism classes of cubic rings. Furthermore, under this correspondence, irreducible cubic forms correspond to orders in cubic fields. Finally, if x 2 VZ is a cubic form corresponding to a cubic ring R, we have StabGL2 .Z/ .x/ ' Aut.R/. It is therefore necessary to exclude reducible and nonmaximal rings. The nonmaximality condition is the more difficult of the two to handle, and for this Davenport and Heilbronn established the following criterion: Proposition 5.7 ([7, 15]). Under the Delone–Faddeev correspondence, a cubic ring R is maximal if any only if its corresponding cubic form f belongs to the set Up VZ for all p, defined by the following equivalent conditions:  

the ring R is not contained in any other cubic ring with index divisible by p; the cubic form f is not a multiple of p, and there is no GL2 .Z/-transformation of f .u; v/ D au3 C bu2 v C cuv 2 C dv 3 such that a is a multiple of p 2 and b is a multiple of p.

In particular, the condition Up only depends on the coefficients of f modulo p 2 . Therefore one can count cubic fields as follows. One obtains an asymptotic formula for the number of cubic rings of bounded discriminant by counting lattice points in fundamental domains for the action of GL2 .Z/, bounded by the constraint jDisc.x/j < X . The fundamental domains may be chosen so that almost all reducible rings correspond to forms with a D 0, and so these may be excluded from the count. One then multiplies this asymptotic by the product of all the local densities of the sets Up . This gives a heuristic argument for the main term in Equation (5.1), and one incorporates a sieve to obtain a proof. The main term of Theorem 5.2 may be proved p similarly. By class field theory, there is a bijection between subgroups of Cl3 .Q. D// of index 3 and cubic fields of discriminant D, which are precisely those not totally ramified at any prime. This ramification condition may also be detected by reducing cubic forms modulo p 2 , and thus may be incorporated into Proposition 5.7. The remainder of the proof is the same.

5.3.1

The Work of Belabas, Bhargava, and Pomerance

In [2], Belabas, Bhargava, and Pomerance (BBP) introduced improvements to Davenport and Heilbronn’s method and obtained an error term of O.X 7=8C / in Equations (5.1) and (5.2). They began by observing that X

.q/N ˙ .q; X /; (5.14) N3˙ .X / D q1

62

Frank Thorne

where N ˙ .q; X / counts the number of cubic orders of discriminant 0 < ˙D < X which are nonmaximal at every prime dividing q. This simple observation has proved to be quite useful! For large q, BBP prove that N ˙ .q; X / X 3!.q/ =q 2 using reasonably elementary methods. Therefore, one has X

.q/N ˙ .q; X / C O.X=Q1 /: (5.15) N3˙ .X / D qQ

For small q, BBP estimate N ˙ .q; X / with explicit error terms using geometric methods. These error terms are good enough to allow them to take the sum in Equation (5.14) up to Q D .X log X /1=8 , which yields a final error term of O.X 7=8C /. In addition, their methods extend to counting S4 -quartic fields, where they obtain a main term of C4 X with error X 23=24C .

5.4 The Four Approaches This, then, brings us to Theorems 5.1 and 5.2. Theorem 5.1 was conjectured in print by Roberts [32], and implicitly in the foundational work of Datskovsky and Wright [13, 14,41]. Roberts gave substantial and convincing numerical evidence for his conjecture, as well as the heuristic argument described in Section 5.5. The four approaches discussed in this paper are the following: Shintani Zeta Functions (Taniguchi-T. [37, 38]). Taniguchi and the author further developed the theory of Shintani zeta functions and proved Theorems 5.1 and 5.2 with error terms of O.X 7=9C / and O.X 18=23C /, respectively. These proofs closely follow the heuristic arguments of Roberts and Datskovsky–Wright. The Shintani zeta-function approach is quite flexible, and [37] contains several generalizations of the main theorems. Perhaps the most interesting of these is a generalization to counting cubic field discriminants (or 3-torsion elements of class groups) in arithmetic progressions, where a certain “nonequidistribution” phenomenon arises. For example, when counting cubic field discriminants  a .mod 7/, the secondary term in Theorem 5.1 is different for every residue class a .mod 7/. A Refined Geometric Approach (Bhargava, Shankar, and Tsimerman [7]). Bhargava, Shankar, and Tsimerman gave the first proof of Theorem 5.1, with an error term of O.X 13=16C /. Their proof follows Davenport and Heilbronn’s original work, counting lattice points via geometric arguments, with substantial simplifications and improvements. Their proof of the secondary term is based on a “slicing” argument which we describe briefly in Section 5.6. Their treatment of the error terms introduces a certain correspondence for nonmaximal cubic rings; this correspondence has also proved useful in the context of Shintani zeta functions, and a combined approach ([8], in progress) has yielded an error term of O.X 2=3C /.

Chapter 5 Secondary Terms in the Davenport-Heilbronn Theorems

63

Equidistribution of Heegner Points (Hough [24]). Hough obtained a statement closely related to Theorem 5.2. pHis scope is limited to counting 3-torsion elements in imaginary quadratic fields Q. D/ with D  2 .mod 4/; in this context, he proves a version of Theorem 5.2, but with an error term larger than X 5=6 . In follow-up work (in preparation), he proves the secondary term for a smoothed variation of this sum. Hough derives this result as a consequence of an equidistribution theorem for the Heegner points associated to the 3-part of class groups of imaginary quadratic fields; his proof of the Davenport–Heilbronn theorem appears as a bonus, and without reference to binary cubic forms. His techniques also apply to the k-parts of these class groups, for odd k  5, where he obtains secondary terms (of order X 1=2C1=k ). His error terms are currently larger than both the secondary and the main terms for k  5, but this work sheds light on a notoriously difficult problem. Trigonal Curves and the Maroni Invariant (Zhao [45]). Zhao enumerates cubic extensions of the rational function field Fq .t / using algebraic geometry. Such extensions are in bijection with isomorphism classes of smooth trigonal curves, that is, smooth 3-fold covers of P 1 , and these may be counted by embedding them in certain surfaces Fk . He obtains a secondary term as a consequence of a bound for the integer k, called the Maroni invariant; for now his error terms are larger than this secondary term. In all of these approaches, the secondary term is easier to see than to prove. In particular, each of these approaches yield the secondary term in a natural way, but it is not a priori evident that the error terms can be made smaller than X 5=6 ! In what follows we will describe, insofar as we can in a couple of pages, why each of these approaches naturally yields a secondary term. For the Shintani zeta-function approach, we will say only a little about the error term, and for the other approaches we will say nothing. Indeed, we will use the notation O.   / whenever the details are best left to the respective papers.

5.5

The Shintani Zeta-Function Approach

Taniguchi and the author [37] proved Theorems 5.1 and 5.2 using the analytic theory of Shintani zeta functions. The Shintani zeta functions associated to the space of binary cubic forms are defined5 by the Dirichlet series X 1 jDisc.x/js

˙ .s/ WD jStab .x/j GL .Z/ 2 ˙ x2GL2 .Z/nVZ

D

X

˙Disc.R/>0

1 jDisc.R/js : jAut.R/j

(5.16)

5 The Shintani zeta functions are commonly defined in terms of SL .Z/ rather than GL .Z/, multi2 2 plying (5.16) by a factor of 2.

64

Frank Thorne

In the former sum, VZ is the lattice defined in Equation (5.11), the sum is over elements of positive or negative discriminant respectively, and Stab.x/ is the stabilizer of x in GL2 .Z/. The latter sum is over isomorphism classes of cubic rings. The equality of these two series follows from the Delone–Faddeev correspondence (Theorem 5.6). This is an example of a zeta function associated to a prehomogeneous vector space. The definition of “prehomogeneous” concerns the GL2 .C/ action on VC , but here the important (related) fact is that GL2 .R/ acts transitively on the positive- and negativediscriminant loci VR˙ . Sato and Shintani [34] developed a general theory of zeta functions associated to prehomogeneous vector spaces, and for the space of binary cubic forms Shintani proved [35] that these zeta functions enjoy analytic continuation and an explicit functional equation. These zeta functions have poles at s D 1, as is common for zeta functions, and at s D 5=6, which is much more unusual. Shintani’s work opens the door to the study of cubic fields using analytic number theory. In particular, by Perron’s formula and standard techniques we have Z 2Ci1 X Xs 1

˙ .s/ ds D (5.17) jStab.x/j s 2i1 x2GL2 .Z/nVZ ˙Disc.x/<X

6 D RessD1 ˙ .s/X C RessD5=6 ˙ .s/X 5=6 C O.X 3=5C /: 5 Although the left side is not the counting function of cubic fields, for the first time we see the X 5=6 secondary term. This, therefore, gives an explanation for the secondary terms in Theorems 5.1 and 5.2, because the Shintani zeta functions have secondary poles.6 To count cubic fields, the most important step is sieving for maximality. This is possible by work of Datskovsky and Wright [13,14,41], who gave Shintani’s work an adelic formulation along the lines of Tate’s thesis [39]. This allowed them to incorporate a variety of conditions into the definition of the Shintani zeta functions, including the Davenport–Heilbronn maximality conditions (given in Proposition 5.7). A bogus proof of Roberts’ conjecture is as follows. For a set of primes P , define the P -maximal Shintani zeta function by the Dirichlet series (5.16), with the added condition that the only cubic forms counted are those in the set Up for each p 2 P . Therefore, letting P be the set of primes < X , it follows that Z 2Ci1 X0 1 Xs ˙ D (5.18)

P .s/ ds jAut.R/j s 2i1 ˙Disc.x/
course this begs the question of why the Shintani zeta functions have poles at s D 5=6. This can of course be explained by the various calculations in Shintani’s work, but for now a satisfying highbrow argument is unknown to the author.

Chapter 5 Secondary Terms in the Davenport-Heilbronn Theorems

65

therefore the implied constant depends on X (indeed, when standard techniques are used, exponentially so!). However, this flawed argument enabled Datskovsky and Wright [14] to give a rigorous analytic proof of the Davenport–Heilbronn theorem. The key step7 is to use (5.18), but with P equal to the set of all primes < Y , where Y tends to infinity very slowly with X . Given their adelic framework, their work yielded an asymptotic formula for the number of cubic extensions of any global field of characteristic not 2 or 3. Moreover, Roberts [32] computed the limit of the secondary terms in (5.18) and found an excellent match with numerical data. This led him to conjecture that (5.18) (equivalently, (5.1)) was correct with some undetermined error term smaller than X 5=6 . He closed his paper with the following (paraphrased very slightly): The pessimistic discussion in [14] suggests to us that the way may be difficult. However, one ˙ ingredient of a proof might be the functional equation of P .s/ with respect to s ! 1  s, studied in [13, 41]. Another ingredient might be [34, Theorem 3], which concerns growth of arithmetic functions whose associated Dirichlet series satisfy such a functional equation.

This is actually a good summary of our proofs of Theorems 5.1 and 5.2! To get the ball rolling, we needed to incorporate the sieve used by Belabas, Bhargava, and Pomerance described in Section 5.3. In place of the P -maximal zeta function we introduced the q-nonmaximal zeta function, which counts only cubic rings not maximal at each prime dividing q. This allowed us to count maximal cubic rings, and the irreducible such rings correspond to cubic fields. Remark 5.8. As described at the end of Section 5.2, we count 3-torsion elements of class groups by a variation of this argument. Such elements correspond to cubic fields not totally ramified at any prime, and so we expand the definition of the q-nonmaximal zeta function to count cubic rings either nonmaximal or totally ramified at every prime dividing q. Equation (5.18) holds for each q-nonmaximal zeta function, and the q-dependence of the error term can be bounded in terms of the exponential sum ˇ X X ˇˇ 1 X ˇ 2 ˇ bq .x/j WD jˆ ˆq .y/ exp.2 i Œx; y=q /ˇˇ; (5.19) ˇ q8 x2VZ=q 2 Z

x2VZ=q 2 Z

y2VZ=q 2 Z

where ˆq .x/ is the characteristic function of the Davenport-Heilbronn q-nonmaximality condition, and Œx; y D x4 y1  13 x3 y2 C 13 x2 y3  x1 y4 : Each inner sum appears naturally when we shift the contour in Equation (5.18) to the left of <.s/ D 0 and apply the functional equation. Trivially, the sum in Equation (5.19) is bounded 7 We

have adapted their argument somewhat to our point of view. Datskovsky and Wright also need, as did Davenport and Heilbronn, the bound (5.15) for the number of cubic rings nonmaximal at any prime > Y .

66

Frank Thorne

by q 8 ; this is enough to allow us to take Q D X 1=25 in (5.15) and obtain an error term of X 24=25C in the Davenport-Heilbronn theorems. We could improve this using the Cauchy–Schwarz inequality, but not below X 5=6 . In [37] we incorporated several improvements to lower the error terms in Theorems 5.1 and 5.2 to O.X 7=9C / and O.X 18=23C /, respectively. Among these is a careful analysis of the sum in Equation (5.19), carried out in [38]. For each x we obtained exact formulas for the inner sum in Equation (5.19), proving that the outer sum is q 1C .

5.5.1 Nonequidistribution in Arithmetic Progressions The Shintani zeta-function approach can also be used to study the distribution of cubic field discriminants in arithmetic progressions. To do this, one twists the Shintani zeta functions by Dirichlet characters, obtaining L-functions which are also proved to enjoy analytic continuation and a functional equation. The details are complicated, so here we simply present representative numerical data. The tables below list the number of cubic fields K with 0 < Disc.K/ < 2  106 and Disc.K/  a .mod m/, for m D 5 and m D 7, and each residue class a. Discriminant modulo 5 Actual count Theoretical result Difference

Discriminant modulo 7 Actual count Theoretical result Difference

0

1

2

3

4

21277 21307 30

22887 22757 130

22751 22757 6

22748 22757 9

22781 22757 24

0

1

2

3

4

5

6

15330 15316 14

17229 17209 20

14327 14277 50

15323 15316 7

17027 17024 3

18058 18063 5

15150 15131 19

The data modulo 5 is essentially equidistributed, apart from the a D 0 column. (The relative deficit of cubic fields divisible by 5, or by any other modulus, can be explained by a careful reading of Davenport and Heilbronn’s original paper.) However, the data modulo 7 shows a striking lack of equidistribution.8 The theoretical results above come from a generalization of Theorem 5.1 to arithmetic progressions; the secondary term now involves a sum of residues of twisted Shintani zeta functions. These residues are evaluated in [38], and for characters  for 8 The counting functions for cubic field discriminants 

0 .mod 7/ and  3 .mod 7/ are not the same. The main term for 0 .mod 7/ is smaller than that for 3 .mod 7/, the secondary term is larger, and the equality in the second row of the table is a coincidence of roundoff error.

Chapter 5 Secondary Terms in the Davenport-Heilbronn Theorems

67

which 6 ¤ 1, the associated residue at s D 5=6 is zero. However, when 6 D 1 this residue is often nonzero, and nonzero residues explain biases in arithmetic progressions such as the one seen above.

5.6

A Refined Geometric Approach

To give an overview of Bhargava, Shankar, and Tsimerman’s proof [7], we start with their treatment of the main term in Theorem 5.1. Later, we will describe how the secondary term appears in their work, but only in the context of irreducible cubic forms (corresponding to cubic orders). The argument in [7] largely follows Davenport and Heilbronn’s original work. Fix a sign, and let Virr˙ denote the irreducible points x 2 VZ with ˙Disc.x/ > 0. Write nC D 6, n D 2 for the order of the stabilizers of the action of GL2 .R/ on VR˙ . The equality nC D 3n is reflected in the fact that cubic fields of negative discriminant are three times as common as those of positive discriminant. Write F for a certain fundamental domain for GL2 .Z/nGL2 .R/ in GL2 .R/ (see Equation (12) of [7]), originally constructed by Gauss. For any v 2 V ˙ , consider the multiset F v, where the multiplicity of an element x 2 V ˙ is equal to the number of g 2 F for which gv D x. It is readily checked that each element x 2 GZ nVZ˙ is represented in this multiset n˙ =m.x/ times, where m.x/ denotes the size of the stabilizer of x in GL2 .Z/. We conclude that N.V ˙ I X / WD

X 0<˙Disc.x/<X x irred.

1 1 D ¹x 2 F v \ Virr˙ W jDisc.x/j < X º: jStab.x/j n˙

(5.20) The reducible points can be handled without too much difficulty: The number of reducible integral binary cubic forms au3 C bu2 v C cuv 2 C dv 3 in the multiset RX .v/ WD ¹w 2 F v W jDisc.w/j < X º, satisfying a ¤ 0, is X 3=4C . Conversely, all cubic forms with a D 0 are reducible. This condition a ¤ 0 meshes well with the 1 occurring in (5.20) geometric arguments that follow. In addition, the weight jStab.x/j can be neglected, as there are X 1=2 points x with jDisc.x/j < X with nontrivial stabilizer. The formula in Equation (5.20) does not depend on v, and one innovation of [7] (previously used by Bhargava in [3]) is to average over many v. In particular, define B WD ¹w D .a; b; c; d / 2 V W 3a2 C b 2 C c 2 C 3d 2  10; jDisc.w/j  1º, and we have R 1 ˙ 1 v2B\V ˙ n˙ ¹x 2 F v \ Virr W jDisc.x/j < X ºjDisc.v/j dv ˙ R : N.V I X / D 1 v2B\V ˙ jDisc.v/j dv (5.21)

68

Frank Thorne

The authors describe this step as “thickening the cusp”. As jDisc.v/j1 dv is GL2 .R/invariant, we may rewrite this as Z 1 #¹x 2 Virr˙ \ gB W jDisc.x/j < X ºdg; (5.22) N.V ˙ I X / D ˙ M g2F n˙ R 1 where M ˙ WD 2 v2B\V ˙ jDisc.v/j dv: This is rewritten as Z 1 #¹x 2 Virr˙ \ B.n; t; ; X /ºt 2 d nd  t d   (5.23) N.V ˙ ; X / D ˙ 0 0 M g2N .a/A ƒ using a standard decomposition of F ((12) of [7]), where B.n; t; ; X / is the region ¹x 2 gB W jDisc.x/j < X º. A proposition of Davenport establishes that in this situation, the count of lattice points above is well approximated by the volume of B.n; t; ; X /, provided that t is not too large. Conversely, when t is large all the lattice points are in the cusp a D 0, and hence reducible and not counted. Thus, the above is reduced to Z 1 ˙ Vol.B.n; t; ; X //t 2 d nd  t d   C O.   /: N.V ; X / D ˙ g2N 0 .a/A0 ƒ M t
1 2 Vol.RX .v// D X: n˙ 12n˙

(5.26)

5.6.1 Origin of the Secondary Term We now explain how Bhargava, Shankar, and Tsimerman refine these calculations to obtain a secondary term. Our brief explanation will necessarily be somewhat vague, which should encourage the reader to read [7]. We focus on their count of irreducible binary cubic forms; as in [2, 37], they also incorporate a sieve to count cubic fields. They begin by incorporating several tweaks to Equation (5.23). For technical reasons, they count discriminants in dyadic intervals ŒX=2; X . In addition, they introduce a smooth function ‰0 .t / on R0 , such that ‰0 .t / D 0 for t  2 and ‰0 .t / D 1 for t  3. They thus rewrite Equation (5.23) (restricted to a dyadic interval ŒX=2; X ) as      Z t t 1 ˙ N.V ; ŒX=2; X / D ˙ ‰0 1=3 C ‰ 1=3 M   g2N 0 .a/A0 ƒ #¹x 2 Virr˙ \ B.n; t; ; ŒX=2; X /t 2 d nd  t d  ;

(5.27)

Chapter 5 Secondary Terms in the Davenport-Heilbronn Theorems

69

where ‰ WD 1  ‰0 , and  is a parameter to be chosen later (to minimize error terms). The decomposition ‰ C ‰0 D 1 splits this integral into two, and we restrict our attention to the ‰0 part, which yields the secondary term. They now “slice” the count of binary cubic forms by the first coordinate. In particular, for a 2 Z, let Ba .n; t; ; ŒX=2; X / denote the set of binary cubic forms in Ba .n; t; ; ŒX=2; X / whose u3 coefficient is equal to a. Then we have ® ¯ # x 2 Virr˙ \ B.n; t; ; ŒX=2; X / D X ® ¯ # x 2 Virr˙ \ Ba .n; t; ; ŒX=2; X / : (5.28) a2Z a¤0

Skipping ahead in their work, the ‰0 -contribution to (5.27) is equal to  1=3  10=3 1=3 Z 1=4 1 Z u   2 X X u ˆ 0 p ˙ 1=3 1=3 3M aD1 D. 3=2/3 =C u>0 a a   Vol Bu .ŒX=.24 /; X=4 / d  ud   C O.   /; (5.29) where Ba .ŒY =2; Y / denotes the set of cubic forms in B with first coordinate a and discriminant in ŒY =2; Y . By Mellin inversion, we have  1=3  1 X u  1=3 e0 .2/. 3 u/2=3 C O.   /; D .1=3/ C 3ˆ a ˆ0 (5.30) 1=3 a aD1 and it is this (negative) .1=3/ which contributes the secondary term. In contrast, the second term of Equation (5.30) (which “looks larger”, especially in light of the eventual choice  D X 1=12 ) is reincorporated into Equation (5.29) and then combined with the ‰ term of Equation (5.27) to obtain the main term in Equation (5.26). When the .1=3/ term is plugged into Equation (5.29), the resulting integral can be evaluated without undue difficulty, and it has order X 5=6 . Remark 5.9. As Shankar has explained to the author, a more geometric argument can also be given which does not rely on the analytic continuation of the Riemann zeta function. However, it is unclear whether the resulting error term would be smaller than X 5=6 when combined with the sieve.

5.6.2

A Correspondence for Cubic Forms

To count cubic fields one must sieve for maximality, using the Davenport–Heilbronn conditions of Proposition 5.7. These could be treated in a naive manner, but the authors improve their error terms by introducing a useful correspondence for nonmaximal cubic forms.9 9 As Shankar explained to me, in the case of cubic fields it is this correspondence which allows them to cross the X 5=6 barrier.

70

Frank Thorne

We introduce some notation. Let N ˙ .VZ I X / count all cubic forms v with 0 < ˙Disc.v/ < X . For a prime p, let N ˙ .Upc I X / count those cubic forms which are nonmaximal at p, i.e., which do not lie in the set Up defined in Proposition 5.7. (In referring to cubic forms as “nonmaximal” we are implicitly appealing to the Delone– Faddeev correspondence.) Finally, for any ˛ 2 P 1 .Z=pZ/, let N ˙ .Vp;˛ I X / count those cubic forms v such that the reduction of v modulo p has ˛ as a root. With this notation, we have the following correspondence: Proposition 5.10 ([7]). We have the identity X X N ˙ .Vp;˛ I X=p 2 / N ˙ .Vp;˛ I X=p 4 /CN ˙ .VZ I X=p 4 /: N ˙ .Upc I X / D ˛2P 1 .Fp /

˛2P 1 .Fp /

(5.31)

This identity generalizes in a straightforward way from prime p to squarefree q, where we count cubic forms not in Up for any pjq. The identity amounts to a combinatorial argument, enumerating ways in which a nonmaximal cubic ring is contained in a larger ring (with smaller discriminant), and then counting these larger rings. This identity is crucial in [7], and it can also be translated into an identity for the q-nonmaximal Shintani zeta function! In work in progress by Bhargava, Taniguchi, and the author [8] we are developing this combined approach. We have proved an error term of O.X 2=3C / in Theorem 5.1 as well as a secondary term for relative cubic extensions of quadratic base fields, and we are currently working on further extensions and generalizations.

5.7 Equidistribution of Heegner Points Remarkably, Hough [24] demonstrated that the Delone–Faddeev correspondence is not the only path to the Davenport–Heilbronn theorem. Hough studied the distribution of Heegner points associated to 3-torsion elements of ideal class groups of quadratic fields, and he obtained a result related to Theorem 5.2 as a consequence. For simplicity, Hough worked only with imaginary discriminants D  2 .mod 4/; note pthat any such discriminant is not fundamental, and the discriminant of the field Q. D/ is 4D.10 He proves that X 4 #Cl3 .D/ D 2 X C O.X 19=20C /; (5.32)  0
10 It seems that the restrictions that D  2 .mod 4/ and D < 0 could both be lifted with some p effort. The restriction on the sign is the more serious of the two, as elements of class groups of Q. D/ for positive D correspond to closed geodesics rather than Heegner points. However, Duke’s result holds for either sign, so in [24] Hough expresses optimism that the positive discriminant case of Theorem 5.2 could be handled as well.

Chapter 5 Secondary Terms in the Davenport-Heilbronn Theorems

71

and for compactly supported, infinitely differentiable test functions , he proves that X

#Cl3 .D/.D=X / D

0
4 b .1/X C O.X 7=8C /: 2

(5.33)

The correct secondary term of order X 5=6 appears in both formulas, in spite of the larger error terms, and in follow-up work (in progress) he obtains an error less than X 5=6 in (5.33), thereby obtaining a proof of a related secondary term. Hough’s work has very different prospects for generalization from the approaches described previously. In particular, his approach can be used to study k-torsion in class groups for odd k > 3, and he makes the following conjecture: Conjecture 5.11 (Hough [24]). For good test functions  and odd k  3, we have X

#Clk .D/.D=X / D

0
  4b 1 1 b .1/X C C  C X 1=2C1=k C o.X 1=2C1=k /; (5.34) 1;k 2 2 k

where C1;k WD

 1 .1  k2 / . 12 /. 12  k1 /

1=k 11=k 1  2 C 2 6k .2/ .1  k1 /   Y  1  1=k 1C2=k 1C1=k 1 p p p p : (5.35) 1C pC1 p>2

Indeed he “proves” his conjecture, but with error terms larger than X . Even a proof of the main term for the single case k D 5 would be a major achievement. Indeed, the average size of Clp .D/ was conjectured by Cohen and Lenstra [11] to be 2 for each prime p  3, and Cohen and Lenstra further conjectured that the p-part of Clp .D/ is 1 isomorphic to a fixed p-group H with probability proportional to jAut.H . No case of /j these conjectures is currently known for any p  5, and essentially nothing is known for p D 3 beyond Theorem 5.2.

5.7.1

Heegner Points and Equidistribution

Hough’s main result concerns the equidistribution of Heegner points associated to the 3-part of the class group. We recall p the relevant background. If a D Œx1 ; x2  is an ideal of an imaginary quadratic field Q. D/, where by reordering if necessary we have x1 =x2 2 H WD ¹z 2 C W =.z/ > 0º, we call x1 =x2 the Heegner point za associated to the ideal a. A change of basis for a corresponds to a linear fractional transformation

72

Frank Thorne

aza Cb za ! cz with ad  bc D 1, and therefore the Heegner point za is well-defined in a Cd p the quotient SL2 .Z/nH. One checks thatpza D z˛a for any ˛ 2 Q. D/, and hence za depends only on the class ofpa in Cl. D/. One similarly checks that za ¤ zb if a and b are inequivalent in Cl. D/. p Therefore, there are h.D/ Heegner points associated to Q. D/ in SL2 .Z/nH, and in [17] Duke proves that they equidistribute with respect to the hyperbolic measure dx dy as D ! 1. Hough proves the same for the Heegner points associated to the y2 3-part of the ideal class group:

Theorem 5.12 (Hough [24]). Let B be a bounded Borel measurable subset of H having boundary of measure zero. Then as D ! 1, X

p ® ¯ 6X # a primitive, nonprincipal W Œa 2 C l3 .Q. D//; za 2 B 3 vol.B/:  0>D>X d 2 .mod 4/ (5.36) squarefree

Here we take B  H rather than B  SL2 .Z/nH, so that each Heegner point is counted once for each representative of its SL2 .Z/ orbit in B. (We recall that a primitive ideal is one without any rational integer divisors other than 1.) Straightaway one can see that this is an equidistribution statement. But it also implies the Davenport–Heilbronn theorem! Taking B to be the standard (or any other) fundamental domain for the action of SL2 .Z/pon H, one counts each Heegner point (and thus each nontrivial element of Cl3 .Q. D//) exactly once. The DavenportHeilbronn theorem follows from the classical computation vol.B/ D 3 . We will follow Hough’s approach sufficiently far as to see the secondary term. His work begins with the following parameterization of ideals (and therefore Heegner points), essentially due to Soundararajan [36]. Proposition 5.13 ([24, 36]). Let D  2 .mod 4/ be squarefree and let k  3 be odd. The set ¹.l; m; n; t / 2 .ZC /4 W lmk D l 2 n2 C t 2 D; ljD; .m; ntD/ D 1º

(5.37)

k is in p bijection with primitive ideal pairs .a; a/ with a ¤ 1 and a principal in Q. D/. Explicitly, the ideal a is given as a Z-module by p a D Œlm; lnt 1 C D

where Na D lm and t 1 is the inverse of t modulo m. k and coprime The proof is relatively p straightforward. For example, if a kis principal k to D, then a D .n C t D/ and taking norms yields .Na/ D n2 C t 2 D, a solution to Equation (5.37). Ideals a not coprime to D yield solutions to Equation (5.37) with l > 1.

Chapter 5 Secondary Terms in the Davenport-Heilbronn Theorems

73

As one of the two main steps in his proof, Hough now applies this parameterization to count Heegner points in the region ² ³ 1 1 1 RY WD z 2 H W  < <.z/ < ; =.z/ > : (5.38) 2 2 Y This establishes the vertical distribution of Heegner points, and he separately establishes that they are equidistributed horizontally. He proves that the number of Heegner points in RY is equal to 6 YX C C5=6 X 5=6 C O.   /; 3

(5.39)

for a fairly complicated error term depending on both X and Y . The secondary term appears here for the following p reason: By the parameterization above, we may write an ideal a as Œlm; lnt 1 C D. We have lm3 D l 2 np2 C t 2 D, so that lm  D 1=3 , implying that the Heegner point za has imaginary part lmD  D 1=6  X 1=6 . In other words, the Heegner points equidistribute as X ! 1, but for fixed X they do not appear high in the cusp. The hyperbolic volume of the subset of RY with =.z/ > X 1=6 is equal to Z

Z

1=2

xD1=2

1

yDX 1=6

dx dy D X 1=6 : y2

(5.40)

Accounting (much more carefully than done here!) for the fact that this subset contains no Heegner points yields the negative secondary term in Equation (5.39), and an analogous argument explains the second term in Equation (5.34). Remark 5.14. Hough’s methods also extend to arithmetic progressions, where he finds the same lack of equidistribution observed at the end of Section 5.5; some results along these lines are in preparation.

5.8

Hirzebruch Surfaces and the Maroni Invariant

Zhao’s work [45] is still in preparation, so we offer only a very brief overview here. Using algebraic geometry, Zhao estimates the number of cubic extensions of the rational function field Fq .t /. Fix a finite field Fq of characteristic not equal to 2 or 3, and let S.2N / be the number of isomorphism classes of cubic extensions K=Fq .t / of degree 3, such that jDisc.F /j D q 2N . In [14], Datskovsky and Wright proved that S.2N / D 2

RessD1 Fq .t/ .1/ Fq .t/ .3/

q 2N C o.q 2N /;

(5.41)

74

Frank Thorne

where Fq .t/ .s/ is the Dedekind zeta function of Fq .t /, or equivalently the zeta function of the algebraic curve PF1q . Datskovsky and Wright’s proof uses the adelic Shintani zeta function; in [45] Zhao obtains another proof of Equation (5.41) using algebraic geometry. A trigonal curve is a threefold cover of P 1 , and there is a bijection between isomorphism classes of smooth trigonal curves and cubic rational function fields. Already this perspective is enlightening; for example, the Riemann–Hurwitz formula implies that the discriminant has even degree, explaining why Equation (5.41) is an equation for S.2N / and not S.N /. The problem, then, is to count smooth trigonal curves. These may be counted by embedding the curves into Hirzebruch surfaces Fk ; for each trigonal curve C there is a unique nonnegative integer k, called the Maroni invariant of C , such that C embeds into Fk WD PP 1 .O ˚ O.k//: Zhao now counts curves on each surface Fk and sieves for smoothness, using a sieve method related to one developed by Poonen [31]. However, the main technique is different and new. The main term in Equation (5.41) comes from estimating the number of trigonal curves C with each Maroni invariant k and then summing the results over all integers k. However, the Maroni invariant of a smooth trigonal curve C is at most N=3, provided that the field corresponding to C has absolute discriminant q 2N . Limiting our sum to k  N=3 naturally introduces a secondary term of order q 5N=3 into Equation (5.41). Zhao’s method, like all other methods described in this paper, comes with error terms; at present he has obtained an error term of O.N 2 q 5N=3 / in Equation (5.41), narrowly missing a proof of the secondary term. He has some optimism that this error term can be reduced below q 5N=3 , and also that some of his methods may be adapted to say something interesting in the number field case.

5.9 Conclusion Cubic fields have seen a great deal of recent study recently, and we recommend the papers of Cohen–Morra [12] and Martin–Pollack [26] (among others) for recent studies of cubic fields from other perspectives. Many interesting open questions remain in the subject. For example, what is the “correct” error term in the Davenport–Heilbronn theorems? Even if we cannot prove it, it would still be interesting to determine the expected order of magnitude. To get the best error terms it seems that we should count fields K of degree at most 3, each 1 weighted by jAut.K/j . This is suggested both by numerical data and also by considering the space of binary cubic forms, where reducible maximal forms correspond to quadratic fields or to Q. (Note that this heuristic does not generalize directly to higher degree fields.) The data in [32] suggests that the true error may be smaller than X 1=2 , and a comparison with the divisor problem suggests that the error might be on the

Chapter 5 Secondary Terms in the Davenport-Heilbronn Theorems

75

order of X 3=8 . Nevertheless, the comparison with the divisor problem is not exact, and the numerical data is inconclusive. Perhaps the most compelling open question concerns quartic and quintic fields. Asymptotic formulas were proved by Bhargava [3,6]; should these formulas have secondary terms? It is generally believed that they should. However, so far we lack even a good conjecture. The zeta-function approach seems likely to work, and Yukie [44] has made some progress in this direction; the geometric approach seems likely to work as well. However, for now the difficulties with either approach appear rather severe. Another open question concerns the multiplicity of cubic fields. It is believed that there should be n cubic fields of discriminant ˙n, but the best bound known is O.n1=3C /, due to Ellenberg and Venkatesh [19]. Nontrivial bounds were also obtained by Helfgott and Venkatesh [22] and Pierce [30], using a variety of methods. Any of the methods described in the present paper could potentially yield improvements, but none of them have succeeded to date. (The present author has attempted improvements by means of Shintani zeta functions, which have led to a number of instructive and interesting failures.) Finally, this paper raises the question of whether or not unexpected connections might be found between the four perspectives presented here, and some promising preliminary results have been obtained in this direction. We anticipate that this and related questions will be addressed in the near future. Acknowledgments. I would like to thank the many people who have shared their insights on these secondary terms, especially Manjul Bhargava, Jordan Ellenberg, Bob Hough, Arul Shankar, Kannan Soundararajan, Takashi Taniguchi, Melanie Matchett Wood, and Yongqiang Zhao. I would further like to thank Bhargava, Shankar, Taniguchi, Zhao, and an anonymous referee for comments on this paper in particular. Most of all I would like to thank Akshay Venkatesh who originally brought this topic to my attention. In addition, I would like to thank the organizers of Integers 2011 for hosting an outstanding conference.

References [1] A. M. Baily, On the density of discriminants of quartic fields, J. Reine Angew. Math. 315 (1980), 190–210. [2] K. Belabas, M. Bhargava, and C. Pomerance, Error estimates in the Davenport-Heilbronn theorems, Duke Math. J. 153(1) (2010), 173–210. [3] M. Bhargava, The density of discriminants of quartic rings and fields, Ann. of Math. 162(2) (2005), 1031–1063. [4] M. Bhargava, Mass formulae for extensions of local fields, and conjectures on the density of number field discriminants, Int. Math. Res. Not. (2007), no. 17, Art. ID rnm052.

76

Frank Thorne

[5] M. Bhargava, Higher composition laws and applications, in: Proceedings of the International Congress of Mathematicians, vol. II, 271Ð294, Eur. Math. Soc., Zürich, 2006. [6] M. Bhargava, The density of discriminants of quintic rings and fields, Ann. of Math. 172(3) (2010), 1559-1591. [7] M. Bhargava, A. Shankar, and J. Tsimerman, On the Davenport-Heilbronn theorem and second order terms, preprint; available at http://arxiv.org/abs/1005.0672, last accessed 6/19/2013. [8] M. Bhargava, T. Taniguchi, and F. Thorne, work in preparation. [9] H. Cohen, F. Diaz y Diaz, and M. Olivier, Enumerating quartic dihedral extensions of Q, Compositio Math. 133 (2002), 65–93. [10] H. Cohen, F. Diaz y Diaz, and M. Olivier, Counting discriminants of number fields, J. Théor. Nombres Bordeaux 18(3) (2006), 573–593. [11] H. Cohen and H. Lenstra, Heuristics on class groups of number fields, in: Number theory, Noordwijkerhout 1983, pp. 33–62, Lecture Notes in Math. 1068, Springer, Berlin, 1984. [12] H. Cohen and A. Morra, Counting cubic extensions with given quadratic resolvent, J. Algebra 325 (2011), 461–478. [13] B. Datskovsky and D. Wright, The adelic zeta function associated to the space of binary cubic forms. II. Local theory, J. Reine Angew. Math. 367 (1986), 27–75. [14] B. Datskovsky and D. Wright, Density of discriminants of cubic extensions, J. Reine Angew. Math. 386 (1988), 116–138. [15] H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields. II, Proc. Roy. Soc. London Ser. A 322(1551) (1971), 405–420. [16] B. N. Delone and D. K. Faddeev, The theory of irrationalities of the third degree (in English translation), AMS, Providence, 1964. [17] W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math. 92(1) (1988), 73–90. [18] J. Ellenberg and A. Venkatesh, The number of extensions of a number field with fixed degree and bounded discriminant, Ann. of Math. 163(2) (2006), 723–741. [19] J. Ellenberg and A. Venkatesh, Reflection principles and bounds for class group torsion, Int. Math. Res. Not. (2007), no. 1, Art. ID rnm002. [20] W. T. Gan, B. Gross, and G. Savin, Fourier coefficients of modular forms on G2 , Duke Math. J. 115 (2002), 105–169. [21] E. Golod and I. Shafarevich, On the class field tower, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 261–272, in Russian. [22] H. Helfgott and A. Venkatesh, Integral points on elliptic curves and 3-torsion in class groups, J. Amer. Math. Soc. 19(3) (2006), 527–550. [23] C. Hermite, Extrait d’une lettre de M. C. Hermite à M. Borchardt sur le nombre limité d’irrationalités auxquelles se réduisent les racines des équations à coefficients entiers complexes d’un degré et d’un discriminant donnés, J. Reine Angew. Math. 53 (1857), 182–192. [24] B. Hough, Equidistribution of Heegner points associated to the 3-part of the class group, preprint, available at http://arxiv.org/abs/1005.1458, last accessed 6/19/2013.

Chapter 5 Secondary Terms in the Davenport-Heilbronn Theorems

77

[25] F. Lemmermeyer, Class field towers, http://www.rzuser.uni-heidelberg.de/˜hb3/publ/ pcft.pdf, last accessed 6/19/2013. [26] G. Martin and P. Pollack, The average least character nonresidue and further variations on a theme of Erd˝os, preprint, available at http://arxiv.org/abs/1112.1175, last accessed 6/19/2013. [27] J. Martinet, Petits discriminants, Ann. Inst. Fourier 29 (1979), 159–170. [28] J. Neukirch, Algebraic number theory, Springer-Verlag, Berlin, 1999. [29] A. Odlyzko, Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results, Sém. Théor. Nombres Bordeaux (2) 2(1) (1990), 119–141. [30] L. Pierce, The 3-part of class numbers of quadratic fields, J. London Math. Soc. (2) 71(3) (2005), 579–598. [31] B. Poonen, Bertini theorems over finite fields, Ann. of Math. (2) 160(3) (2004), 1099– 1127. [32] D. Roberts, Density of cubic field discriminants, Math. Comp. 70(236) (2001), 1699– 1705. [33] M. Sato and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants. Nagoya Math. J. 65 (1977), 1–155. [34] M. Sato and T. Shintani, On zeta functions associated with prehomogeneous vector spaces, Ann. of Math. (2) 100 (1974), 131–170. [35] T. Shintani, On Dirichlet series whose coefficients are class numbers of integral binary cubic forms, J. Math. Soc. Japan 24 (1972), 132–188. [36] K. Soundararajan, Divisibility of class numbers of imaginary quadratic fields, J. London Math. Soc. (2) 61 (2000), 681–690. [37] T. Taniguchi and F. Thorne, Secondary terms in counting functions for cubic fields, submitted, available at http://arxiv.org/abs/1102.2914, last accessed 6/19/2013. [38] T. Taniguchi and F. Thorne, Orbital L-functions for the space of binary cubic forms, submitted, available at http://arxiv.org/abs/1112.5030, last accessed 6/19/2013. [39] J. Tate, Fourier analysis in number fields and Hecke’s zeta-functions, thesis (Princeton, 1950), reprinted in J. W.S. Cassels and A. Fröhlich, eds., Algebraic number theory, Academic Press, London, 1986. [40] S. Wong, Automorphic forms on GL.2/ and the rank of class groups, J. Reine Angew. Math. 515 (1999), 125–153. [41] D. Wright, The adelic zeta function associated to the space of binary cubic forms. I. Global theory, Math. Ann. 270(4) (1985), 503–534. [42] D. Wright, Twists of the Iwasawa-Tate zeta function, Math. Z. 200 (1989), 209–231. [43] D. Wright and A. Yukie, Prehomogeneous vector spaces and field extensions, Invent. Math. 110(2) (1992), 283–314. [44] A. Yukie, Shintani zeta functions, London Mathematical Society Lecture Note Series 183, Cambridge University Press, Cambridge, 1993. [45] Y. Zhao, doctoral thesis, University of Wisconsin, in preparation.

78

Frank Thorne

Author information Frank Thorne, Department of Mathematics, University of South Carolina, Columbia, South Carolina, USA. Email: [email protected]

Combinatorial Number Theory, 79–89

© De Gruyter 2013

Chapter 6

Spotted Tilings and n-Color Compositions Brian Hopkins Abstract. Spotted tilings are presented as a new combinatorial interpretation of n-color compositions. Previous and new results are proven using this tool, an interpretation of the Fibonacci numbers, and a case of Terquem’s problem. The spotted tilings allow for MacMahon’s zig-zag graphs to be applied to n-color compositions, addressing a question of Agarwal in the article where these compositions were introduced in 2000. Keywords. Integer Compostions, Tilings, Zig-Zag Graph. Mathematics Subject Classification 2010. 05A17.

6.1

Background

For a given positive integer n, the compositions of n are ordered t -tuples .c1 ; : : : ; c t / of positive integers with c1 C    C c t D n. For instance, there are four compositions of 3, namely .3/; .2; 1/; .1; 2/; and .1; 1; 1/. The individual ci are called parts of the composition. Compositions are sometimes referred to as ordered partitions. Compositions of n may be represented graphically as tilings of a 1 n board, where a part k corresponds to a 1 k rectangle. This is essentially MacMahon’s construction using nodes on a line [7]. Figure 6.1 shows the tilings for the compositions of 3. Notice that each tiling has a vertical bar at the leftmost edge, possibly vertical bars separating the rectangles corresponding to parts, and a vertical bar at the rightmost edge. 3

21

12

111

Figure 6.1. The 4 tilings representing the compositions of 3.

Agarwal [1] introduced the concept of n-colored compositions, where a part k has one of k possible colors, denoted by a subscript 1; : : : ; k. There are eight n-colored compositions of 3, namely .31 /; .32 /; .33 /; .21 ; 11 /; .22 ; 11 /; .11 ; 21 /; .11 ; 22 /; .11 ; 11 ; 11 /: Let C C.n/ denote the number of n-color compositions of n, so C C.3/ D 8. These compositions, inspired by the analogous concept of n-color partitions, have been further considered in several articles and monographs.

80

Brian Hopkins

We introduce a combinatorial tool, spotted tilings, to provide a graphic representation of n-colored compositions. A part ki corresponds to a 1 k rectangle with a spot in position i . Spotted tilings for the n-color compositions of 3 are shown in Figure 6.2. Notice that the bars and spots alternate, since there is one spot per part. 31 u 32 33

21 11 u u

22 11

u u u

11 21 u u 11 22 u

11 11 11 u u u u

u

Figure 6.2. The 8 spotted tilings for C C.3/.

We will use this new tool to prove previous and new results about colored compositions (Section 6.2) and to develop an analog of MacMahon’s zig-zag graphs (Section 6.3), addressing a question of Agarwal [1]. There are two combinatorial results we will use in the sequel. The Fibonacci numbers are given by F0 D 0, F1 D 1 and the recurrence Fn D Fn1 C Fn2 for n  2. The following theorem is proven using generating functions in [6]. Theorem 6.1. Let C o.n/ denote the number of compositions of n whose parts are all odd. Then C o.n/ D Fn . Proof. Proceed by induction. Certainly C o.0/ D 0, as 0 cannot be written as a sum of positive odd integers. Also, C o.1/ D 1, since the single composition of 1 consists of an odd part. Assume the claim is true for n1 and n2. The compositions counted by C o.n/ include the odd-part compositions of n  1 with an additional part 1 included at the end, and the odd-part compositions of n2 with the last part increased by 2 (which results in another odd part). Since any odd-part composition of n ends in either 1 or a larger odd number, C o.n  1/ C C o.n  2/ counts all odd-part compositions of n by the induction hypothesis. Terquem [9] considered the problem of counting subsequences of ¹1; : : : ; nº where the terms alternate parity. For instance, such 3-term subsequences of ¹1; : : : ; 5º with first term odd are ¹1; 2; 3º, ¹1; 2; 5º, ¹1; 4; 5º, and ¹3; 4; 5º. Church and Gould [4] give a combinatorial proof of Terquem’s general problem. We give a different proof of the special case needed below. Theorem 6.2. The number of subsequences ¹x1 ; : : : ; x2j 1 º of ¹1; :::; 2k  1º where  Ck1 the parity of xi matches the parity of i is given by j2j 1 . Proof. Let T .j; k/ denote the number of desired subsequences. Since there are k odd numbers in ¹1; : : : ; 2k  1º, we have T .1; k/ D k. To determine T .j; k/ inductively, there are two cases to consider. If a subsequence counted by T .j; k/ does not include

Chapter 6 Spotted Tilings and n-Color Compositions

81

the number 2k  1, then it is among the subsequences counted by T .j; k  1/ D j Ck2 2j 1 . If a subsequence counted by T .j; k/ does end with the number 2k  1, then the second to last entry is one of the even numbers 2j  2; : : : ; 2k  2 and the number of possible ¹x1 ; : : : ; x2j 3 º are counted by T .j  1; j  1/; : : : ; T .j  1; k  1/, respectively. Together, this gives T .j; k/ D T .j; k  1/ C T .j  1; k  1/ C    C T .j  1; j  1/ ! ! ! j Ck2 j Ck3 2j  3 D C C C 2j  1 2j  3 2j  3 ! ! j Ck2 j Ck2 D C 2j  1 2j  2 ! j Ck1 D ; 2j  1 using the binomial coefficient identities known as the “hockey-stick theorem” and Pascal’s lemma (identities 135 and 127 of [3]).

6.2

n-Color Composition Enumerations

In this section, we use spotted tilings to give new proofs of enumeration results for C C.n/ and two special classes, C C.n; k/, the number of n-color compositions with k parts, and C C e.n/, the number of n-color compositions with each part even. We also prove a new enumeration result for C C o.n/, the number of n-color compositions with each part odd. Theorem 6.3 ([1, Theorem 1b, d]). The number of n-color compositions of n satisfies the formula C C.n/ D F2n and the recurrence relation C C.n/ D 3  C C.n  1/  q C C.n  2/. The corresponding generating function is 13qCq 2. Proof. We establish C C.n/ D C o.2n/ by a bijection between n-color compositions of n and (normal) compositions of 2n with all parts odd. Consider a spotted tiling for an n-color composition of n. Recall that the bars and spots alternate. Consider the spot signifying part ki to be positioned at i  12 within the 1 k rectangle. Then the distance between a spot and an adjacent bar has the form m C 12 for some integer m  0. Create a (normal) tiling of 2n by replacing spots with bars and doubling all distances, which are now of the form 2m C 1, odd numbers. Figure 6.3 shows the transition from spotted tilings counted by C C.3/ to tilings counted by C o.6/. The process is invertible because the bars in an all-odd composition of 2n alternate even/odd, considering the leftmost bar to be at 0. Because 2n is even, there are at least

82

Brian Hopkins u

u

u u u u u u u u u u u u

Figure 6.3. Row by row correspondence between C C.3/ and C o.6/.

bars at 0, some odd position, and 2n. To create the corresponding spotted tiling, halve all distances and replace the bars at any half-integer positions with spots. By Theorem 6.1, C C.n/ D F2n . The recurrence relation for F2n is known, but we demonstrate the recurrence in terms of spotted tilings to introduce ideas used in subsequent proofs. We establish a bijection between the n-color compositions counted by 3  C C.n  1/ and by C C.n/ C C C.n  2/. Given three copies of the spotted tilings counted by C C.n  1/, perform the following operations: (a) to the first set, add 11 at the end of each composition; (b) to the second set, for each composition, replace its last part kj with .k C 1/j ; (c) To the third set, for each composition, (c1) if the last part is k1 , replace it with .k C 1/kC1 ; (c2) if the last part is kj with j > 1, replace it with .k  1/j 1 . In terms of spotted tilings, operation (a) adds a box at the right hand side. Operations (b) and (c1) both extend the final rectangle by length one; (b) keeps the spots fixed, while (c1) moves the last spot from the first position of a 1 k rectangle to the last position of the new 1 .k C 1/ rectangle. The results of (a), (b), and (c1) are spotted tilings counted by C C.n/ with no repetition – notice that (b) produces tilings where the spot is not in the rightmost position, while tilings produced by (a) and (c1) all have a spot in the rightmost position. Operation (c2) moves the spot of the final rectangle one position to the left and decreases the rectangle by length one, producing a tiling counted by C C.n  2/. Both (c1) and (c2) can be considered to move the last spot one position to the left, allowing “wrap around” from the leftmost to rightmost position. We demonstrate this with 3  C C.3/ D C C.4/ C C C.2/. Since operations (a) and (b) are less complicated, Figure 6.4 shows them only applied to 32 , while all 8 spotted tilings counted by C C.3/ are shown under (c1) and (c2). For the reverse direction, there are three cases for n-color compositions counted by C C.n/. If the last part is 11 , remove it; this is the inverse of operation (a). If the last part is kj for k  2 and j < k, replace it with .k  1/j ; this is the inverse of operation (b). If the last part is kk for k  2, replace it with .k  1/1 ; this is the

83

Chapter 6 Spotted Tilings and n-Color Compositions

(a)

u

u

(b)

u

u

(c1) (c2) (c2) (c1) (c1) (c1) (c2) (c1)

u

u

u u u u u u u u u u u u

u u

u

u u u

u u u u u

u u u u

Figure 6.4. For three copies of C C.3/, examples of operations (a) and (b), and row by row details of (c1) and (c2).

inverse of operation (c1). For an n-color composition counted by C C.n  2/ with last part kj , replace it with .k C 1/j C1 ; this is the inverse operation of (c2). We have established C C.n/ D 3  C C.n  1/  C C.n  2/. The generating function follows from the recurrence (denominator) and initial values of the sequence (numerator); see [10] for more details. Theorem 6.4 ([1, Theorem 1a, c]). The number of n-color compositions of n with   m parts satisfies the formula C C.n; m/ D nCm1 2m1 . The corresponding generating function is

qm . .1q/2m

Proof. Consider the spotted tiling associated to an n-color composition of n with m parts. The spotted tiling will include m spots and m C 1 bars, more specifically bars at 0, 2n, and m  1 other positions. As in the previous proof, consider the spot signifying part ki to be positioned at i  12 within the 1 k rectangle. Doubling all distances places spots at odd positions and bars at even positions. Since spots and bars alternate, the positions of the m spots and m  1 internal bars create a subsequence ¹x1 ; : : : ; x2m1 º of ¹1; : : : ; 2n  1º where the parity of xi matches the parity of i . From Theorem 6.2,   we know there are nCm1 2m1 such subsequences. For generating functions associated with binomial coefficients, see [10]. Since C C.n/ D

P

m C C.n; m/, n X mD1

combining Theorems 6.3 and 6.4 gives ! nCm1 D F2n 2m  1

which is equivalent to the even index case of an identity connecting Fibonacci numbers to diagonals in Pascal’s triangle (identity 4 in [3]).

84

Brian Hopkins

Theorem 6.5 ([5, Theorems 1.2, 5.1]). The number of n-color compositions of n having only even parts satisfies the formula C C e.n/ D 4  C C e.n  2/  C C e.n  4/ with initial values C C e.0/ D 0 and C C e.2/ D 2. The corresponding generating function is

2q 2 . 14q 2 Cq 4

Proof. Note that C C e.n/ D 0 for all odd n. We establish a bijection between the n-color compositions counted by C C e.n/ C C C e.n  4/ and by 4  C C e.n  2/. Similar to the recurrence relation proof of Theorem 6.3, we perform the following operations on four copies of the spotted tilings counted by C C.n  2/: (a) to the first set, add 21 at the end; (b) to the second set, add 22 at the end; (c) for the third set, replace the final part kj with .k C 2/j ; (d) For the fourth set, (d1) if the last part is k1 , replace it with .k C 2/kC1 ; (d2) if the last part is k2 , replace it with .k C 2/kC2 ; (d3) if the last part is kj with j > 2, replace it with .k  2/j 2 . Notice that the three (d) operations each move the spot in the last rectangle two to the left, with the convention that the rightmost position is to the left of the leftmost position. Adding parts 2j and changing between part lengths k, k C 2, and k  2 maintains the parity restriction. Figure 6.5 shows the results on four copies of the spotted tilings counted by C C e.4/.

(a)

u

u

(b)

u

u

(c)

u

u

(d1) u u (d2) u (d3) u (d3) u (d1) u u (d2) u u u (d1) u u (d2)

u u

u u u u

u u u

u u

u

u u

Figure 6.5. For three copies of C C e.4/, examples of operations (a), (b), and (c), and row by row details of (d1), (d2), and (d3).

Chapter 6 Spotted Tilings and n-Color Compositions

85

As in the proof of Theorem 6.3, operations (a), (b), (c), (d1), and (d2) produce ncolor compositions counted by C C e.n/ with no repetition, by inspection of the last part. Operation (d3) produces n-color compositions counted by C C e.n  4/. For the inverse map, there are again several cases for an n-color composition counted by C C e.n/. If the last part is 21 or 22 , remove it. If the last part is kj with j  k  2, replace it with .k 2/j . If the last part is kk1 or kk , replace it with .k 2/1 or .k 2/2 , respectively. These are inverse operations for (a), (b), (c), (d1), and (d2), respectively. Given an n-color composition counted by C C e.n  4/, replace the last part kj with .k C 2/j C2 , the inverse of operation (d3). We have established C C e.n/ D 4C C e.n2/C C e.n4/. The initial conditions are easily checked. The sequence begins 0; 0; 2; 0; 8; 0; 30; 0; 112; 0; 418. Removing the zeros, this is double the sequence A001353 in [8]. The generating function follows from the recurrence relation and the initial values. Theorem 6.6. The number of n-color compositions of n having only odd parts satisfies the recurrence C C o.n/ D C C o.n  1/ C 2  C C o.n  2/ C C C o.n  3/  C C o.n  4/ with initial values 0; 1; 1; 4. The corresponding generating function is qCq 3 . 1q2q 2 q 3 Cq 4

Proof. We establish a bijection between the n-color compositions counted by C C o.n/ C C C o.n  4/ and by C C o.n  1/ C 2  C C o.n  2/ C C C o.n  3/. Similar to the previous proofs, we perform the following operations on spotted tilings counted by C C o.n  1/ C 2  C C o.n  2/ C C C o.n  3/: (a) given an n-color composition counted by C C o.n  1/, add 11 at the end; (b) for the first set of n-color composition counted by C C o.n  2/, replace the final part kj with .k C 2/j ; (c) for the second set of n-color composition counted by C C o.n  2/; (c1) if the last part is k1 , replace it with .k C 2/kC1 ; (c2) if the last part is k2 , replace it with .k C 2/kC2 ; (c3)] if the last part is kj with j  3, replace it with .k  2/j 2 ; (d) given an n-color composition counted by C C o.n  3/, add 33 at the end. As in the proof of Theorem 6.5, the three (c) operations move the spot in the last rectangle two to the left, with the wrap around convention. Adding parts 11 and 33 and changing between part lengths k, k C 2, and k  2 maintains the parity requirement. Figure 6.6 shows the results on C C o.5/ C 2  C C o.4/ C C C o.3/. Notice that neither operations (c1) nor (c2) produces an n-color composition of n C 2 with last part 33 . As in the previous proofs, operations (a), (b), (c1), (c2), and (d) produce n-color compositions counted by C C o.n/ with no repetition, by inspection of the last part. Operation (c3) produces n-color compositions counted by C C o.n  4/.

86

Brian Hopkins

(a)

u

(b)

u

u

u

u u u u

(c1) (c1) (c1) (c1) (c2) (c3) (c1) (d)

u

u

u u u u u u u u u u u

u u

u u

u

u u u u u u u u u

u u u u

u

u u

Figure 6.6. For n D 6, examples of operations (a), (b), and (d), and row by row details of (c1), (c2), and (c3).

For the inverse map, consider an n-color composition counted by C C o.n/. If the last part is 11 or 33 , remove it. If the past part is kj with j  k  2, replace it with .k  2/j . If the last part is kk1 or kk , replace it with .k  2/1 or .k  2/2 , respectively. There are inverse operations for (a), (d), (b), (c1), and (c2), respectively. Given an ncolor composition counted by C C o.n  4/, replace the last part kj with .k C 2/j C2 , the inverse operation of (c3). We have established C C o.n/ D C C o.n  1/ C 2  C C o.n  2/ C C C o.n  3/  C C o.n  4/. The initial conditions are easily checked. The sequence begins 0, 1, 1, 4, 7, 15, 32, 65, 137, 284, 591, which is A119749 in [8]. The generating function follows from the recurrence relation and the initial values.

6.3 Conjugable n-Color Compositions In his foundational work Combinatory Analysis, MacMahon introduced the zig-zag graph of a composition as a way of defining conjugacy. A composition is broken up with one rectangular part per row, such that the first segment of a part is directly beneath the last segment of its predecessor; see Figure 6.7 for an example. MacMahon explains, “Whereas the composition is read from left to right in successive rows from top to bottom, the conjugate is read from top to bottom in successive columns from left to right.” [7, Sect. IV, Chap. 1, § 129] 1 3 1 1 2 1 3 Figure 6.7. Mahonian zig-zag graphs showing that the conjugate of (2,1,3) is (1,3,1,1).

87

Chapter 6 Spotted Tilings and n-Color Compositions

In the paper where Agarwal introduced n-color compositions [1], he concluded with two questions, including “What will be the shape of MacMahon’s zig-zag graph in the case of n-colour compositions?” The notion of spotted tilings introduced here allows a possible answer to this question – simply carry the spots into the zig-zag graph. But for which n-color compositions does conjugation of the spotted zig-zag graph lead to a valid spotted tiling? Consider the composition in Figure 6.7. Can .2; 1; 3/ be colored so that it is conjugable? In order for the first column to include a spot, the 2 must become 21 . The second column automatically has a spot since 1 can only become 11 . To avoid another spot in the second column, the 3 must become 32 or 33 , but either possibility leaves some column without a spot. So no coloring of .2; 1; 3/ gives a conjugable n-color composition. In contrast, Figure 6.8 shows that .21 ; 11 ; 32 ; 11 / is conjugable. 21 11 32 11

u

11 32 11 22 u

u

u

u

u

u

u

Figure 6.8. Spotted zig-zag graphs showing that the conjugate of .21 ; 11 ; 32 ; 11 / is .11 , 32 , 11 , 22 /.

Definition. An n-color composition is conjugable if its zig-zag graph has exactly one spot per column. Our final theorem characterizes and counts conjugable n-color compositions. Theorem 6.7. (a) Conjugable n-color compositions have the form   a 21 ; 11 ; 2b2 ; .32 ; 2a1 ; 11 ; 2b2 /c where exponents denote repetition, a; b; c  0, and a; b may vary in each occurrence. (b) The number of conjugable n-color compositions of n is 0 if n is even, 2.n1/=2 if n is odd. Proof. The analysis of conjugable n-color compositions is most easily done in terms of the zig-zag graphs. By construction, there is one spot per row. By definition, there is one spot per column. Since a subsequent part is positioned with its first segment under the last segment of its predecessor, its spot must appear to the right of the spot above it. In order to have one spot per column, it is exactly one position to the right. The spots, therefore, lie on the diagonal. The parts 31 , 33 , and kj for any k  4 cannot occur in a conjugable n-color compositions, since each would leave at least one column without a spot. As spots must appear on the diagonal, the allowed parts 11 ; 21 ; 22 ; 32 may occur within the constraints shown in Figure 6.9. The description in part (a) follows.

88

Brian Hopkins u

u

u

u

u

u

u

u

Figure 6.9. Parts 11 and 22 can only be followed by some allowed k2 while parts 21 and 32 can only be followed by some allowed k1 .

The spotted zig-zag graph of a conjugable n-color composition of n with m parts fits in an m m square with m spots on the main diagonal and exactly m  1 more segments in the two adjacent diagonals to “connect the spots”. Thus n D 2m  1 and there are no conjugable n-color compositions of even n. To count the conjugable n-color compositions of odd n D 2m  1, consider the original representation as a tiling of a 1 n board. Spots on the main diagonal of the zig-zag graph here means that the first, third, : : : ; .2m  1/-st positions contain spots, leaving m  1 gaps where there are two choices for the position of a vertical bar; see Figure 6.10. Thus there are 2m1 D 2.n1/=2 such compositions. z u

u

u

u

z u

u

z u

u

u

u

u

u

Figure 6.10. Between adjacent spots, there are two choices for the position of a vertical bar, so there are 4 conjugable n-color compositions of 5.

In conclusion, spotted tilings are a combinatorial interpretation of n-color compositions in the spirit of MacMahon that allow for bijective proofs of various enumeration results and an analog of the zig-zag graph and conjugacy. The interested reader will want to compare the spotted tilings introduced here to the combinatorial interpretation of n-color compositions using lattice paths given in [2]. Acknowledgments. Thanks to the organizers for putting together this fifth Integers Conference and to the anonymous referee for helpful comments.

References [1] A.K. Agarwal, n-colour compositions, Indian J. Pure Appl. Math. 31(11) (2000) 1421– 1437. [2] A.K. Agarwal and G. Narang, Lattice paths and n-colour compositions, Discrete Math. 308(9) (2008) 1732–1740. [3] A. Benjamin and J. Quinn, Proofs that Really Count: The Art of Combinatorial Proof, Mathematical Association of America, 2003. [4] C. Church Jr. and H. Gould, Lattice point solution of the generalized problem of Terquem and an extension of Fibonacci numbers, Fibonacci Quart. 5(1) (1967) 59–68.

Chapter 6 Spotted Tilings and n-Color Compositions

89

[5] G. Yu-Hong, n-colour even self-inverse compositions, Proc. Indian Acad. Sci. (Math. Sci.) 120(1) (2010) 27–33. [6] V. Hoggatt Jr. and D. Lind, Fibonacci and binomial properties of weighted compositions, J. Combin. Theory 4(2) (1968) 121–124. [7] P. A. MacMahon, Combinatory Analysis, volume 1, Cambridge University Press, 1915. [8] The On-Line Encyclopedia of Integer Sequences, published electronically at http://oeis. org, 2012, last accessed 12/12/2012. [9] O. Terquem, Sur un symbole combinatoire d’Euler et son utilité dans L’Analyse, J. Math. Pures Appl. 4 (1839) 177–184. [10] H. Wilf, generatingfunctionology, Academic Press, 1990.

Author information Brian Hopkins, Department of Mathematics, Saint Peter’s University, Jersey City, New Jersey, USA. Email: [email protected]

Combinatorial Number Theory, 91–107

© De Gruyter 2013

Chapter 7

A Class of Wythoff-Like Games Aviezri S. Fraenkel and Yuval Tanny Abstract. We present a class of two-player Wythoff game variations we dub Wyt(f ) that depends on a given function f .k/. In this class a move consists of removing either a positive number of tokens from precisely one of two given piles, or k tokens from one pile and ` from the other, subject to the constraint 0 < k  ` < f .k/. We analyze three classes of integer-valued functions f .k/: constant, superadditive, and polynomial of degree > 1 with nonnegative integer coefficients. The nature of the winning positions in the games is essentially unique for each class. Keywords. Combinatorial Games, Wythoff Games. Mathematics Subject Classification 2010. 91A46.

7.1

Introduction

We propose and analyze a family of two-player combinatorial take-away games played on two piles, dubbed Wyt(f ). The two players move alternately by selecting one of the following moves: 

move of the first type: take any positive number of tokens from a single pile, possibly the entire pile;



move of the second type: take k > 0 tokens from one pile and ` > 0 tokens from the other. This move is restricted by the condition 0 < k  ` < f .k/;

(7.1)

where f .k/ W Z0 ! Z0 is a function that distinguishes the games from each other. A position in the game can be represented by a pair .a; b/ (a; b 2 Z0 ) which denotes the number of tokens in each pile. Without loss of generality we assume throughout 0  a  b. The games we consider here are played under the normal play convention, that is, the player making the last move wins the game and the opponent loses. In the misère play convention the player making the last move loses and the opponent wins.

92

Aviezri S. Fraenkel and Yuval Tanny

Note that these are impartial games – the set of possible moves for a player depends only on the game position, not on the player. They are deterministic (no chance moves) and acyclic (the number of tokens decreases at every move until it becomes 0). Therefore we can partition the positions of the game into Next player winning positions (denoted N -positions, or by the set N WD N .f /  Z20 ) and Previous player winning positions (denoted P -positions, or by the set P WD P .f /  Z20 ). A game G is tractable if 

for every position, its state (P - or N -position) can be decided in polynomial time; the next move from any N -position to a P -position can be computed in polynomial time;



the winner can consummate a win in at most an exponential number of moves.



To the “run of the mill algorithmicians” the last item dooms the game as intractable. It may be quite a surprise to them that this is not the case: whereas we dislike computing in exponential time, the human race relishes observing some of its members being tormented for an exponential length of time! In fact, the easy game Nim and similar take-away games do require that amount of time. For games there are also notions of polynomiality and efficiency; see [12], especially Section 4. In the next sections of this chapter, we define sequences An and Bn to analyze the P -positions. We use the notation 1 A D [1 iD0 ¹Ai º; B D [iD0 ¹Bi º:

In addition, when analyzing the P -positions we frequently use the mex function: Let S be a finite set of nonnegative integers. Then mex.S / is defined to be the least nonnegative integer not in S . Note that the mex of the empty set is 0. This chapter presents characterizations of the P -positions and tractable winning strategies for the following types of functions: 

f .k/ D t; t > 0 an integer;



f .k/ is a strictly increasing superadditive function; P f .k/ D niD0 ai k i is a polynomial of degree n > 1 with integer coefficients ai  0 and a0 > 0.



Note that a function f W Z0 ! Z0 is called superadditive if it satisfies: f .k/  k and f .k C `/  f .k/ C f .`/ for all k; ` 2 Z0 . Wyt(f ) is a general class of games that encapsulates a number of previously analyzed games and new games. In its simplest case, where f .k/ 2 ¹0; 1º, the move of the second type cannot be done and the game reduces to Nim on 2 piles (in normal play). For the case f .k/ D k C 1 the game reduces to the classical Wythoff game [24], where in a move of the second type the player has to take the same positive number of tokens from both piles. An analysis of the constant difference class f .k/ D k C t , t  1 integer and later for the linear class f .k/ D sk C t s; t  1

93

Chapter 7 A Class of Wythoff-Like Games

integers was done by Fraenkel [8, 11]. For other examples of variations of 2-piles Wythoff, see [3, 5, 10, 13, 15, 18, 20]. In Section 7.2 we deal with the case f .k/ D t , and in Section 7.3 with superadditivity. In Section 7.4 we handle polynomials, where we resort to real analysis for the proofs. Further possible work is indicated in the final Section 7.5.

7.2

Constant Function

Considering Wyt(f ) for f .k/ D t; t > 0 an integer, using the move of the second type, a player can take k tokens from one pile and ` tokens from the other as long as 0 < k  ` < t . As implied by the penultimate paragraph of the previous section, we may assume t  2. The class of games with the restriction k C ` < t for moves of the second type is called Cyclic Nimhoff and was settled by Fraenkel and Lorberbom [14] for a general n-piles game; see also [5]. Duchêne and Gravier [7] examined (among other geometrical extensions of Wythoff) a bounded Wythoff game in which it is only possible to take k < t tokens from one pile or k < t tokens from both piles. Given fixed t 2 Z2 , we define g W Z0 ! Z0 such that for all m 2 Z0 , g.m/ D t m  .t 2  1/bm=.t C 1/c and An D mex¹Ai ; Bi W 0  i < nº;

Bn D g.An /;

n  0:

[1 iD0 ¹.Ai ; Bi /º

In this section we show that are the P -positions of Wyt(f ) when f .k/ D t for any constant integer t  2. To prove this, we begin by showing the relation between the An and Bn sequences using a specific numeration system. This numeration system will also help us to derive a tractable strategy for winning the game. The first few P -positions for the games where t D 3 and t D 10 are shown in Table 7.1 and Table 7.2, respectively. After Theorem 7.13 (Section 7.2.2) we point out in Remark 7.14 that Table 7.1 exhibits certain periodicities which help in exhibiting the structure of the sequences A and B. We begin with an auxiliary result. Lemma 7.1. (i) Let t  2. For every m 2 Z0 , g.m/  m, with equality if and only if m  0 .mod .t C 1//. (ii) The function g is an injection. Proof. (i) Write m D s.t C 1/ C r, 0  r < t C 1, s; r 2 Z0 . Then g.m/  m D .t  1/.m  .t C 1/b.s.t C 1/ C r/=.t C 1/c/: Thus .g.m/  m/=.t  1/ D m  s.t C 1/  0 with equality if and only r D 0. (ii) An elementary algebra exercise.

94

Aviezri S. Fraenkel and Yuval Tanny

Table 7.1. t D 3: n

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

An Bn

0 0

1 3

2 6

4 4

5 7

8 8

9 11

10 14

12 12

13 15

16 16

17 19

18 22

20 20

21 23

24 24

25 27

Table 7.2. t D 10: n

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

An Bn

0 0

1 10

2 20

3 30

4 40

5 50

6 60

7 70

8 80

9 90

11 11

12 21

13 31

14 41

15 51

16 61

Lemma 7.1 immediately implies: Corollary 7.2. For every n  0, Bn D An if and only if An  0 .mod .t C 1//.

7.2.1 A Numeration System Lemma 7.3. Let t 2 Z2 . Every nonnegative integer m can be represented uniquely in the form (7.2) m D m2 .t 2  1/ C m1 t C m0 ; where the digits m0 ; m1 ; m2 are integers satisfying: 0  m 0 ; m1 < t I

m2  0I

m0 D m1 D t  1 is not permitted:

(7.3)

Proof. Theorem 1 of [9] states: let 1 D u0 < u1 < u2 <    be any finite or infinite sequence of integers. Every nonnegative integer m has precisely one representation in the system S D ¹u0 ; u1 ; u2 ; : : : º of the form m D †niD0 mi ui , where the mi are nonnegative integers satisfying mi ui C mi1 ui1 C    C m0 u0 < uiC1 for i  0: By this theorem, every nonnegative integer m can be represented uniquely in the form (7.2) where m0 < t and m0 C m1 t < t 2  1. It follows immediately that m1 < t and that m0 D m1 D t  1 is not permitted. Throughout this section, we use the convention that for every nonnegative number m we let m0 ; m1 ; m2 be the appropriate digits of m in the above numeration system. Corollary 7.4. With hypotheses as in Lemma 7.3, g.m/ D m if and only if m0 D m1 . Proof. m D m2 .t 2  1/ C m1 .t C 1/ C m0  m1  0 .mod .t C 1// if and only if m0 D m1 , since 0  m0 ; m1 < t . The proof is complete by Lemma 7.1.

Chapter 7 A Class of Wythoff-Like Games

95

Using our numeration system we can present the g function in a more intuitive way. Lemma 7.5. Let t 2 Z2 , and m 2 Z0 which is represented uniquely in the form (7.2) with digits m0 ; m1 ; m2 . If m1  m0 , then g.m/ D m2 .t 2  1/ C m0 t C m1 : Otherwise, g.m/ D .m2 C 1/.t 2  1/ C m0 t C m1 : Proof. We have, g.m/ D g.m2 .t 2  1/ C m1 t C m0 / D m2 .t 2  1/t C m1 t 2 C m0 t  .t 2  1/ .b.m1 t C m0 /=.t C 1/c C m2 .t  1// D m2 .t 2  1/ C m1 t 2 C m0 t  .t 2  1/b.m1 t C m0 /=.t C 1/c: If m1  m0 , then b.m1 t C m0 /=.t C 1/c D b.m1 .t C 1/ C .m0  m1 //=.t C 1/c D m1 C b.m0  m1 /=.t C 1/c D m1 since 0  m0 ; m1 < t . Therefore, the g function for this case becomes: g.m/ D m2 .t 2  1/ C m1 t 2 C m0 t  m1 .t 2  1/ D m2 .t 2  1/ C m0 t C m1 : Otherwise, b.m1 t C m0 /=.t C 1/c D m1  1 and the g function becomes: g.m/ D m2 .t 2  1/ C m1 t 2 C m0 t  .m1  1/.t 2  1/ D .m2 C 1/.t 2  1/ C m0 t C m1 : Remark 7.6. Notice that in the two displayed formulas of Lemma 7.5 the digits m0 , m1 of the unique representation (7.2) of m have been switched. Furthermore, m has a unique representation of the form (7.2) if and only if m0 WD m2 .t 2  1/ C m0 t C m1 has a unique representation. Their value difference is m  m0 D .m1  m0 /.t  1/. We presented the relation between An and Bn using the special numeration system. In the next couple of lemmas we show that an independent simple characterization of the ¹Ai º and ¹Bi º sets follows from the use of this numeration system. This characterization will enable us to prove that the An and Bn pairs constitute the P -positions of the game. Lemma 7.7. Let t 2 Z2 , and m 2 Z0 which is represented uniquely in the form (7.2) with digits m0 ; m1 ; m2 . Then m 2 A if and only if m1  m0 .

96

Aviezri S. Fraenkel and Yuval Tanny

Proof. Induction on m. Assume that the statement is true for all n in K D ¹0; 1; : : : ; m  1º. Suppose first that m1  m0 . Using the induction hypothesis, we show that then m … ¹Bn D g.n/ W n 2 K \ Aº. It then follows from the mex definition that m 2 A. Let n 2 K \ A. Then n1  n0 by the induction hypothesis. If n1 D n0 , then n D g.n/ < m by Corollary 7.4. So we may assume n1 < n0 . Let r D r2 .t 2 1/Cr1 t Cr0 be the unique representation of g.n/. Lemma 7.5 and Remark 7.6 then imply that r1 D n0 and r0 D n1 . Thus r0 < r1 . Assuming m D r, the uniqueness of the representation (Lemma 7.3) and Remark 7.6 then imply r1 D m1 and r0 D m0 , contradicting m1 < m0 . Therefore, m ¤ g.n/. Secondly, suppose that m 2 A. Let m0 WD m2 .t 2  1/ C m0 t C m1 . Assume that m1 > m0 . Then m  m0 D .m1  m0 /.t  1/ > 0, so m0 2 K \ A by the induction hypothesis. By Lemma 7.5, m D g.m0 / 2 B. In fact, if, say, m D An , then m0 D Ai for some i < n, and m D g.m0 / D Bi . But this contradicts the definition of the mex function. Hence m1  m0 . Consequently we have: Lemma 7.8. m D m2 .t 2  1/ C m1 t C m0 2 B if and only if m0  m1 : Proof. If m1 > m0 then m … A by Lemma 7.7 and therefore m 2 B by the mex property. If m1 D m0 then m 2 A by Lemma 7.7 and m D g.m/ 2 B by Corollary 7.4. Conversely, if m 2 B, then by definition there exists a 2 A such that m D g.a/. Let a0 ; a1 ; a2 be the digits of a represented in form (7.2). Then by Lemma 7.7, a1  a0 ; and by Lemma 7.5, m0 D a1 and m1 D a0 , so m0  m1 . Corollary 7.9. (i) If An D m2 .t 2  1/ C m1 t C m0 then Bn D m2 .t 2  1/ C m0 t C m1 . (ii) Bn An D .m0 m1 /.t 1/  0. In particular, Bn D An if and only if m0 D m1 . Proof. (i) By Lemma 7.7, m D An if and only if m1  m0 ; and if m1  m0 , then Lemma 7.5 implies that g.m/ D Bn D m2 .t 2  1/ C m0 t C m1 . (ii) This also follows from m1  m0 . Theorem 7.10. For f .k/ D t a constant, the set of P -positions of Wyt.f / is given by W WD [1 iD0 ¹.Ai ; Bi /º. Proof. Since Wyt(f ) is an acyclic game, it suffices to show two things: (1) every move from any position in W lands in a position outside W ; (2) for every position outside W , there is a move to some position in W . (1) A move of the first type from .An ; Bn / 2 W leads to a position not in W , because there are no repeating terms in A, nor in B: (i) The mex property implies that the sequence A is strictly increasing thus has no repeating terms. (ii) There are also no

97

Chapter 7 A Class of Wythoff-Like Games

repeating terms in B, since the function g is an injection (Corollary 7.1 (ii)) and there are no repeating terms in A. Suppose that a move of the second type from .An ; Bn / 2 W produces a position .Am ; Bm / 2 W . Then m < n. This move can be made either in the form (i) An ! Am ; Bn ! Bm I or in the form (ii) An ! Bm ; Bn ! Am : (i)

An ! Am , Bn ! Bm . Then

 2

Bn  Bm D t .An  Am /  .t  1/



 Am An  : t C1 t C1

If bAn =.t C1/c D bAm =.t C1/c then Bn Bm D t .An Am /  t , contradicting the move rule (7.1). Otherwise, bAn =.t C 1/c  bAm =.t C 1/c  1. Since An  Am D k < t , we get Bn  Bm  t .t  1/  .t 2  1/ D 1  t < 0; again contradicting (7.1). (ii) An ! Bm , Bn ! Am . Let r WD Am . Then r1  r0 by Lemma 7.7. We may assume r0 > r1 , because r0 D r1 implies Bm D Am (Corollary 7.9(ii)), whence case (ii) reverts back to case (i). The move (ii) can only be made if An > Bm . Then Bn Am D g.An /Am  An Am > Bm Am D .r0 r1 /.t 1/  t 1, contradicting move rule (7.1). (2) Let .x; y/ with x  y be any position not in W . The construction of An by the mex rule implies that the set [1 iD0 ¹Ai ; Bi º D Z0 . Therefore, x D Bn or x D An for some n  0. We consider several cases. (i) x D Bn . Then move y ! An using the first move rule. (ii) x D An and y > Bn , then move y ! Bn using the first move rule. (iii) x D An  y < Bn . Recall that we can represent x and y uniquely in our numeration system: x D x2 .t 2  1/ C x1 t C x0 and y D y2 .t 2  1/ C y1 t C y0 . Let x 0 WD x2 .t 2  1/ C x1 t C s; y 0 WD x2 .t 2  1/ C st C x1 ; where s WD y1 C b.y0  x1 /=t /c: We show the following: (a) (a1) If y0  x1 then s D y1 , x0 > y1 ; if y0 < x1 then s D y1  1, x0  y1 . (a2) 0  x1  s < x0 < t . (b) There exists m such that x 0 D Am and y 0 D Bm . (c) The move x ! x 0 D Am , y ! y 0 D Bm is a legal move, that is, 0 < x  x 0 < t , 0  y  y0 < t .

98

Aviezri S. Fraenkel and Yuval Tanny

(a) Notice that s D y1 C b.y0  x1 /=t /c 2 ¹y1  1; y1 º, since 0  x1 ; y0 ; y1 < t . By Corollary 7.9(i), An and Bn have the same coefficient (digit) multiplier of t 2  1, so a fortiori y2 D x2 . Hence x D An  y < Bn D g.An / implies: x1 t C x0  y1 t C y0 < x0 t C x1 :

(7.4)

The right-hand side of Equation (7.4) is equivalent to .x0 y1 /t > y0 x1 . If y0  x1 , then s D y1 and x0 > y1 , so s < x0 < t . If y0 < x1 , then s D y1  1, yet x0  y1 , and we have again s < x0 < t . The left-hand side of Equation (7.4) is equivalent to y0  x0  .x1  y1 /t . By Lemma 7.7, x1  x0 , since x D An . Hence s  y1 C b.y0  x0 /=t /c  x1  0: (b) By (a2), x1  s < t , so by Lemma 7.7, x 0 D Am for suitable m. Again by (a2), 0  s < t , so y 0 D Bm by Corollary 7.9(i). (c) From (a), 0 < x  x 0 D x0  s < t . Now y  y 0 D .y1  s/t C y0  x1 , since y2 D x2 . By (a1), if y0  x1 then s D y1 , so 0  y  y 0 D y0  x1 < t ; and if y0 < x1 then s D y1  1, so 0 < y  y 0 D t  .x1  y0 / < t .

7.2.2 Strategy Tractability and Structure of the P-Positions Theorem 7.11. Given game position .x; y/ with 0  x  y, there is a tractable linear-time algorithm to decide whether or not .x; y/ 2 P . Proof. We wish to compute whether or not .x; y/ 2 P in time linear in the input size ‚.log xy/. Expand x and y in our numeration system, which can be done linearly by using the simple greedy algorithm from [9]: x2 D bx=.t 2  1/c x1 D b.x  x2 .t 2  1//=t c x0 D x  x2 .t 2  1/  x1 t: Then, check whether x0  x1 . If negative, .x; y/ 2 N . If positive, check whether y2 D x2 and y1 D x0 and y0 D x1 . If positive, then .x; y/ 2 P (Corollary 7.9(i)); otherwise, .x; y/ 2 N . Corollary 7.12. Let t 2 Z2 and f .k/ D t a constant function. Then there is a tractable strategy for winning Wyt(f ). Proof. Using the instructions in the second part of the proof of Theorem 7.10, the greedy algorithm in Theorem 7.11, and Lemmas 7.7 and 7.8, it is clear how to construct a tractable strategy for winning Wyt(f ) for any given N -position. We now turn to the structure of the sequences A and B.

Chapter 7 A Class of Wythoff-Like Games

99

A sequence ¹sn ºn0 with limn!1 sn D 1 is called arithmetically periodic if there exist integers 1 < q  p (the periods), and distinct nonnegative integers r1 ; : : : ; rq 2 Œ0; p  1, such that si  ri .mod p/ whenever n  i .mod q/ .i D 1; : : : ; q/. This is a variation of the definitions of arithmetic (or additive) periodicity given or implied in [1,4,17]. It accommodates sequences that are not necessarily monotonically increasing. Theorem 7.13. Let t  2. (i) Each of the sequences A and B is arithmetically periodic with periods q D .t  1/.t C 2/=2 and p D t 2  1. (ii) Each of the sequences A and B contains .t 1/.t C2/=2 distinct residues mod p, t  1 of which are common to both sequences. (iii) A [ B contains all the t 2  1 residues mod p, t  1 of which appear in both sequences. (iv) For all n  0, 0  Bn  An  .t  1/2 . Proof. (i) Lemmas 7.7 and 7.8 and Corollary 7.9 imply that the sequences A and B have residues that are periodic mod p. Indeed, membership of x in A and B depends only on the relative size of x0 and x1 , not on the size of x2 . (ii) Using (i) we can assume x2 D 0. By the mex property, all the residues of A in Œ0; p  1 are increasing, so they are distinct. Their number is precisely the number of x0 , x1 satisfying 0  x1  x0  t  1. Since x0 D x1 D t  1 is forbidden, this number is precisely .t  1/.t C 2/=2, t  1 of which are common to both sequences, namely those for which 0  x0 D x1 < t  1. The proof for the residues of B is similar. (iii) Follows immediately from (ii). (iv) Follows immediately from Corollary 7.9(ii). Remark 7.14. Notice that the preceding propositions (i)-(iv) of Theorem 7.13 can be seen in Table 7.1. Almost all game positions .x; y/, 0  x  y are N -positions – see [22, 23]. Therefore .x; y/ … P with high probability. Theorem 7.13 can help to dispose of them quickly by computing the residues of x and y mod p and searching for them in a pre-constructed table of the p D t 2  1 residue pairs of .An ; Bn / mod p.

7.3

Superadditive Functions

In this section we examine Wyt(f ), when f is a strictly increasing superadditive function. We show that the P -positions of this game have a simple recursive formula. As

100

Aviezri S. Fraenkel and Yuval Tanny

an example, the first few P -positions for the game for which f .x/ D x 2 are shown in Table 7.3. Table 7.3. f .x/ D x 2 . n

0

1

2

3

4

5

6

7

8

9

10

11

12

13

An Bn

0 0

1 1

2 4

3 9

5 25

6 36

7 49

8 64

10 100

11 121

12 144

13 169

14 196

15 225

Remark 7.15. The function f in this example is a polynomial, and polynomials are considered in the next section. However, a polynomial of degree t  1 of the form f .x/ D

t X

ai x i ; ai 2 Z0 for 0  i  t; a t > 0;

(7.5)

iD0

is clearly superadditive if and only if a0 D 0. In the next section we consider the case a0 > 0. Remark 7.16. Notice that if f W Z0 ! Z0 is strictly increasing, then f .k/  k. Let An D mex¹Ai ; Bi W 0  i < nº;

Bn D f .An /;

n  0:

(7.6)

Theorem 7.17. Let f W Z0 ! Z0 be a strictly increasing superadditive function. Then the set of P -positions of Wyt.f / is given by W WD [1 iD0 ¹.Ai ; Bi /º. Proof. As in the proof of Theorem 7.10, it suffices to show: (1) Every move from any position in W lands in a position outside W . (2) For every position outside W , there is a move to some position in W: (1) The mex definition implies that An is a strictly increasing sequence. Thus, for every i > j we have Ai > Aj . Hence Bi D f .Ai / > f .Aj / D Bj since f is a strictly increasing function. Therefore, there are no repeating terms in An nor in Bn . It follows that a move of the first type from .An ; Bn / 2 W leads to a position … W . Suppose that a move of the second type from .An ; Bn / 2 W produces a position .Am ; Bm / 2 W . Then m < n. Let k WD An  Am and ` WD Bn  Bm . Since An is a strictly increasing sequence we have k > 0. The superadditivity of f then implies, ` D f .An /  f .Am /  f .An  Am / D f .k/: By Remark 7.16 we have `  k. Thus 0 < k  `  f .k/, contradicting move rule (7.1). The move .An ; Bn / ! .Bm ; Am / where m < n is impossible too: Let k 0 WD An  Bm and `0 WD Bn  Am . Since Bn D f .An /  An , we have: 0 < k 0  k  `  `0 .

Chapter 7 A Class of Wythoff-Like Games

101

As was shown above, `  f .k/, so `0  `  f .k/  f .k 0 / since f is a strictly increasing function. Hence 0  k 0  `0  f .k 0 /, again contradicting move rule (7.1). (2) Let .x; y/ with 0  x  y be any position not in W . It follows from the mex definition that [1 iD0 ¹Ai ; Bi º D Z0 . Therefore, x D Bn or x D An for some n  0. Case (i). x D Bn . Then move to the position .An ; Bn / by subtracting y  An tokens from one pile, using a move of the first type. Case (ii). x D An and y > Bn . Then subtract y  Bn tokens from one pile, using a move of the first type. Case (iii). x D An  y < f .An / D Bn . Then move to .0; 0/ using a move of the second type. The time complexity for deciding whether a given position is a P -position or N position depends on the time complexity of the function f . Assume f to be a polynomial time computable function. The naïve algorithm for deciding whether a position .x; y/ is a winning position consists of calculating An and Bn using their recursive definition (7.6) until it is known whether x 2 An or not. This method takes exponential time and space since the input position is given in succinct form. In the previous section and in other Wythoff-like games (e.g., [11]) constructing a special numeration system assisted in building a tractable strategy to win the game. We show here that this is not necessary for this case. Remark 7.18. For all superadditive functions f; f .0/ D 0. This follows from f .0/ D f .0 C 0/  f .0/ C f .0/. Lemma 7.19. Let f W Z0 ! Z0 be a strictly increasing superadditive function. For every x 2 Z0 , if f .x/ D x then x D Ax D Bx . Proof. Induction on x. Since A0 D 0 by the mex property, B0 D f .0/ D 0 (Remark 7.18). Assume that the statement is true for all m < x, m 2 Z0 . If f .x/ D x then since f is strictly increasing, for every m < x, f .m/ D m. The induction hypothesis then implies ¹A0 ; : : : Ax1 ; B0 ; : : : Bx1 º D ¹0; 1; : : : ; x  1º. Therefore, x D Ax by the mex property. If f .x/ D x then also Bx D f .x/ D x. Lemma 7.20. Let f W Z0 ! Z0 be a strictly increasing superadditive function. If f is not the identity function f .x/ D x, then there exist N 2 Z1 and a > 1 such that for every integer n  N , f .n/=n  a and for every integer n < N , f .n/ D n. Proof. Since f is not the identity function, there exists a minimum integer N 2 Z1 such that f .N / ¤ N . Thus for every integer n < N , f .n/ D n. Notice that N ¤ 0 because f .0/ D 0 (Remark 7.18). By Remark 7.16, we have f .N /  N C 1. Let a WD 1 C 1=.2N  1/. Since N  1 it is clear that a > 1. We can write every integer n  N in the form n D kN C r, where 0  r < N and k 2 Z1 . The

102

Aviezri S. Fraenkel and Yuval Tanny

superadditivity of f and f .N /  N C 1 then imply that f .n/ D f .kN C r/  f .kN / C f .r/  kf .N / C f .r/  k.N C 1/ C r: Since r < N and k  1, we have f .n/ k.N C 1/ C r k k 1  D1C 1C 1C  a: n kN C r kN C r kN C kr N Cr Theorem 7.21. Let f W Z0 ! Z0 be a polynomial time computable strictly increasing superadditive function. Let An , Bn be given by (7.6), and let .x; y/ with 0  x  y be any game position. Then there is a polynomial-time algorithm to decide whether or not x 2 A. Proof. We apply the following recursive Procedure 1. Input: x. Output: 1 if x 2 A; 0 otherwise. Find z such that f .z/ is as close as possible to x, beginning with z D x=2 and then proceeding in the form of a binary search on the interval Œ0; x (O.log x/ operations). There are three possible outcomes: (1) there is no z such that x D f .z/: then return 1 since x 2 A by the mex definition; (2) there exists z such that x D f .z/ and z D x: then return 1 since x 2 A by Lemma 7.19; (3) There exists z such that x D f .z/ and z < x: then x 2 A if and only if z … A. So apply Procedure 1 with z as input and return 1 minus the returned value from this new call of Procedure 1. Suppose the recursive search algorithm stops with x t , for which f t .x t / D x, where f t denotes f composed with itself t -times. We show that t D O.log x/. Consequently, the algorithm is polynomial because each recursive call is polynomial in log x. By Lemma 7.20 there exist a > 1 and minimal N 2 Z1 , such that for every integer n  N , f .n/=n  a. In particular, f .N /=N  a. Since f .N /  N it follows that f .f .N //=f .N /  a. Hence, f 2 .N / f .N / f 2 .N / D  a2 I N f .N / N and by induction we have

f t .N /  at : N Therefore f t .N /  at N . We assume f .xn / ¤ xn . Otherwise, f .x/ D x D f t .x t / D x t and then t D 0. By Lemma 7.20, x t  N . Since f is an increasing function, f .x t /  f .N /, and by simple induction, f t .x t /  f t .N /. Hence x D f t .x t /  f t .N /  at N . Then loga .x/  t loga .aN /  t , since a > 1 and N  1. Consequently, t D O.log x/.

103

Chapter 7 A Class of Wythoff-Like Games

Corollary 7.22. Let f be a polynomial time computable strictly increasing superadditive function. Then there is a tractable strategy for winning Wyt(f ). Proof. Using the algorithm in Theorem 7.21 and the instructions in the second part of the proof of Theorem 7.17, it is clear how to construct a tractable strategy for winning Wyt(f ) for any given N -position. Note that by using the binary search described in the beginning of Theorem 7.21, it is possible to calculate f 1 .x/, if it exists, in polynomial time.

7.4

Polynomial

In this section we consider the game Wyt(f ) for which f W R0 ! R0 is a polynomial of the form (7.5) with t  2 and a0 > 0. Thus f is convex although not superadditive (Remark 7.15). The analysis of this class of games is similar to that of the previous section, since the function which defines the B sequence is superadditive, as we show presently. We define here, as before, the sequences that compose the P -positions of this class of games. For x 2 R0 , we define ´ max¹f .x/; f .1/xº if x 2 Œ1; 1/ g.x/ D f .1/x if x 2 Œ0; 1/; and An D mex¹Ai ; Bi W 0  i < nº;

Bn D g.An /;

n  0:

(7.7)

The function g is defined on R0 in order to enable us to use basic calculus for proving that g is a superadditive function. Note that g.Z0 /  Z0 . As an example, the first few P -positions for the game for which f .x/ D x 2 C 9 are displayed in Table 7.4. Table 7.4. f .x/ D x 2 C 9: n

0

1

2

3

4

5

6

7

8

9

10

11

12

13

An Bn

0 0

1 10

2 20

3 30

4 40

5 50

6 60

7 70

8 80

9 90

11 130

12 153

13 178

14 205

Lemma 7.23. The function g.x/ is a strictly increasing superadditive function. Proof. If g.x/ is a nonnegative continuous convex function which vanishes at the origin, then g.x/ is superadditive – see [2, Theorem 5]. Since g.x/ is a nonnegative continuous function and g.0/ D 0, it suffices to show that g.x/ is a convex function 0 .x/ is nondecreasing in in Œ0; 1/. By [16, Corollary 1.1.9], if the right derivative gC

104

Aviezri S. Fraenkel and Yuval Tanny

0 .x/ is Œ0; 1/ then g.x/ is a convex function in this interval. We show here that gC 0 .x/ is nondecreasing in Œ0; 1/. (ii) g 0 .x/ is nondenondecreasing by showing: (i) gC C 0 0 .x/ for x 2 Œ0; 1/. creasing in .1; 1/. (iii) gC .1/  gC 0 .x/ D g 0 .x/ D f .1/. Hence, g 0 .x/ nondecreasing in Œ0; 1/. (i) In Œ0; 1/, gC C (ii) In Œ1; 1/, g.x/ is a convex function, since a maximum of convex functions is 0 .x/ is nondecreasconvex [16, Theorem 1.1.3]. Therefore by [16, Corollary 1.1.6], gC ing in Œ1; 1/. (iii) For h > 0,

f .1/.1 C h/  f .1/ g.1 C h/  g.1/  D f .1/: h h 0 0 Hence gC .1/  f .1/ D gC .x/ for x 2 Œ0; 1/. 0 .x/ is nondecreasing in Œ0; 1/ and thus g.x/ is superadditive there. Therefore, gC 0 .x/  f .1/ > 0 for all x 2 Œ0; 1/, g.x/ is strictly Since g.x/ is continuous and gC increasing there.

Let S D ¹n 2 Z0 W f .An /  f .1/An º: Remark 7.24. Since t  2, f .An / > f .1/An for all sufficiently large n, so S is finite, containing at most the first few nonnegative integers. The next lemma throws some light on the set S . Lemma 7.25. The set S is nonempty, and for every n 2 S we have An < f .1/ and An D n. Note 7.26. For the example f .x/ D x 2 C 9 we have An < f .1/ and f .An /  f .1/An for 0  n  9, but f .An / > f .1/An for all n > 9. Thus An D n for 0  n  9, but An > n for all n > 9 (see Table 7.4). Proof. By the mex property, A0 D 0. Hence B0 D g.0/ D f .1/0 D 0. Again by mex, A1 D 1. Thus f .A1 / D f .1/A1 , so S ¤ ;. Also, Bn D g.An / D max¹f .An /; f .1/An º for all n  1. In particular, B1 D f .1/  2. Next we show An < f .1/ for all n 2 S . Suppose An  f .1/ for some n 2 S . We note that n  1, since 0 … S : f .0/ > f .1/0. Let k WD An  f .1/  0. Since t  2 and a0 > 0 we have, f .An / D f .f .1/ C k/  a t .f .1/ C k/t C a0 > .f .1/ C k/2  f .1/.f .1/ C k/ D f .1/An : Thus, f .An / > f .1/An D Bn D g.An /, contradicting the definition of g. Therefore An < f .1/ D B1 for all n 2 S . We already verified that An D n for n 2 ¹0; 1º. Suppose that Am D m for all m < n for which An < f .1/ D B1 . Then An D mex ¹A0 ; : : : ; An1 ; B1 ; : : : ; Bn1 º D mex ¹0; : : : ; n  1º D n, since Bn1 >    > B1 D f .1/ > An .

Chapter 7 A Class of Wythoff-Like Games

105

Theorem 7.27. Let f W Z0 ! Z0 be a polynomial of the form .7:5/ with t  2 and a0 > 0. Then the set of P -positions of Wyt.f / is given by W WD [1 iD0 ¹.Ai ; Bi /º, where Ai , Bi are defined in .7:7/. Proof. As in the proof of Theorems 7.10 and 7.17, it suffices to show that (1) every move from any position in W lands in a position outside W ; (2) for every position outside W , there is a move to some position in W: (1) The proof is the same as in part 1 of Theorem 7.17, using g instead of f and using g.x/  f .x/. (2) The proof for this statement is as in part 2 of Theorem 7.17 up to and including Case (iii) x D An  y < f .An / D Bn . There are two additional cases: Case (iv) x D An , f .An /  y < Bn and there exists m for which y D Bm < Bn . Then move to the position .Am ; Bm / by subtracting An  Am tokens from the first pile, using a move of the first type. Case (v) x D An , f .An /  y < Bn and y D Bm for no m. Since Bn D g.An / > f .An / it follows from the definition of g and Remark 7.24 that: Bm D g.Am / D f .1/Am for all 0  m  n:

(7.8)

Further, by Lemma 7.25, Am D m < f .1/ for all 0  m  n:

(7.9)

Write y D f .1/s C r; 0  r < f .1/; s; r 2 Z0 ; and u WD An  As : We claim that the move .An ; y/ ! .As ; Bs / is a valid move of the second type. For proving this claim, it suffices to show: (a) 0  s < n. (b) y > Bs . (c) Either 0 < u  r < f .u/ or 0 < r  u < f .r/. Thus, condition (7.1) of the move of the second type is satisfied by subtracting u tokens from the first pile and r tokens from the second pile. (a) We have y D f .1/s C r < Bn D f .1/n by (7.9). Thus s < n, since r < f .1/. Clearly s  0. (b) By (7.8) and (a), r D y  f .1/s D y  Bs . Therefore r > 0 since y ¤ Bs . (c) By (7.9) and (a), u D An  As D n  s > 0 and n  s  n < f .1/. Thus if u  r, then 0 < u  r < f .1/  f .u/. Otherwise, if u > r, then 0 < r < u < f .1/  f .r/. Corollary 7.28. Let f be a polynomial of the form .7:5/ with t  2 and a0 > 0. Then there is a tractable strategy for winning Wyt.f /.

106

Aviezri S. Fraenkel and Yuval Tanny

Proof. By Lemma 7.23, g is a strictly increasing superadditive function, and clearly polynomial-time computable. Using the algorithm in Theorem 7.21 and the instructions how to move from any N -position to a P -position in the second part of the proof of Theorem 7.27, it is clear how to construct a tractable strategy for winning Wyt.f / for any given N -position.

7.5 Further Work In this chapter we studied the P -positions and winning strategies of the game Wyt.f / for three classes of functions: constant, superadditive, and polynomial with nonnegative coefficients. There are two main direction in extending this work. One is to examine other classes of functions and possibly to generalize the results presented here to other classes of functions. For example, Wyt.f / where f .k/ D sk C t , for s; t 2 R>0 (the case s; t 2 Z>0 was studied in [11]), or for an arbitrary polynomial function f . Another direction is to study deeper properties in each class. One of the important properties of impartial games is the Sprague–Grundy function, which enables playing sums of games. Some results on the Sprague–Grundy function for Wythoff’s game can be found in [4, 17, 21]. An additional direction is to study the sets of restrictions and extensions of the game, in the sense of a subset or superset of the set of possible moves, that preserve its P -positions. These directions were studied for Wythoff and other Wythoff-like games in [6, 14, 19].

References [1] E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways for your Mathematical Plays, vols. 1–4, A .K. Peters, Wellesley, MA, 2001–2004; 2nd edition: vol. 1 (2001), vols. 2, 3 (2003), vol. 4 (2004); translation by G. Seiffert of 1st edition (1982) into German: Gewinnen, Strategien für Mathematische Spiele, Foreword by K. Jacobs, M. Reményi and Seiffert, four volumes, Friedr. Vieweg & Sohn, Braunschweig, 1985. [2] A. M. Bruckner and E. Ostrow, Some function classes related to the class of convex functions, Pacific Journal of Mathematics 12(4) (1962), 1203–1215. [3] I. G. Connell, A generalization of Wythoff’s game, Canad. Math. Bull 2 (1959), 181–190. [4] A. Dress, A. Flammenkamp, and N. Pink, Additive periodicity of the sprague-grundy function of certain nim games, Adv. in Appl. Math. 22 (1999), 249–270. [5] E. Duchêne, A. Fraenkel, S. Gravier, R. J. Nowakowski, et al., Another bridge between Nim and Wythoff, Australasian Journal of Combinatorics 44 (2009), 43–56. [6] E. Duchêne, A. S. Fraenkel, R. J. Nowakowski, and M. Rigo.,Extensions and restrictions of Wythoff’s game preserving its P positions, Journal of Combinatorial Theory, Series A 117(5) (2010), 545–567. [7] E. Duchêne and S. Gravier, Geometrical extensions of Wythoff’s game, Discrete Mathematics 309(11) (2009), 3595–3608.

Chapter 7 A Class of Wythoff-Like Games

107

[8] A. S. Fraenkel, How to beat your Wythoff games’ opponent on three fronts, Amer. Math. Monthly 89 (1982), 353–361. [9] A. S. Fraenkel, Systems of numeration, Amer. Math. Monthly 92 (1985), 105–114. [10] A. S. Fraenkel, Wythoff games, continued fractions, cedar trees and Fibonacci searches, Theoretical Computer Science 29(1–2) (1984), 49–73. [11] A. S. Fraenkel, Heap games, numeration systems and sequences, Annals of Combinatorics 2(3) (1998), 197–210. [12] A. S. Fraenkel, Complexity, appeal and challenges of combinatorial games, Expanded version of a keynote address at Dagstuhl Seminar Feb. 17–22, 2002, Theoretical Computer Science 313 (2004), 393-415, special issue on Algorithmic Combinatorial Game Theory. [13] A. S. Fraenkel, New games related to old and new sequences, Integers, Electr. J of Combinat. Number Theory 4 (2004), #G6, 18 pp., Comb. Games Sect. [14] A. S. Fraenkel and M. Lorberbom, Nimhoff games. Journal of Combinatorial Theory, Series A 58(1) (1991), 1–25. [15] V. E. Hoggatt Jr, M. Bicknell-Johnson, and R. Sarsfield, A generalization of Wythoff’s game, Fibonacci Quart. 17 (1979), 198–211. [16] L. Hörmander. Notions of convexity, vol. 127. Birkhauser, 1994. [17] H. Landman, A simple FSM-based proof of the additive periodicity of the Sprague– Grundy function of Wythoff’s game, in: R. J. Nowakowski (ed.), More Games of No Chance, Proc. MSRI Workshop on Combinatorial Games, July, 2000, Berkeley, CA, MSRI Publ., vol. 42, pp. 383–386, Cambridge University Press, Cambridge, 2002. [18] U. Larsson, Blocking Wythoff Nim, Electronic J. of Combinatorics 18(P120) (2011), 1. [19] U. Larsson, A generalized diagonal Wythoff Nim, Integers 12 (2012), #G2, 24 pp., Comb. Games Sect. [20] W. A. Liu, H. Li, and B. Li, A restricted version of Wythoff’s game, Electronic J. of Combinatorics 18(P207) (2011), 1. [21] G. Nivasch, More on the Sprague–Grundy function for Wythoff’s game, Games of No Chance III 56 (2009), 377–410. [22] D. Singmaster, Almost all games are first person games, Eureka 41 (1981), 33–37. [23] D. Singmaster, Almost all partizan games are first person and almost all impartial games are maximal, J. Combin. Inform. System Sci. 7 (1982), 270–274. [24] W. A. Wythoff, A modification of the game of Nim, Nieuw Arch. Wisk 7 (1907), 199–202.

Author information Aviezri S. Fraenkel, Department of Applied Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel. Email: [email protected] Yuval Tanny, Department of Applied Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel. Email: [email protected]

Combinatorial Number Theory, 109–121

© De Gruyter 2013

Chapter 8

On the Multiplicative Order of FnC1=Fn Modulo Fm Takao Komatsu, Florian Luca, and Yohei Tachiya Abstract. Here, we show that if s 62 ¹1; 2; 4º is a fixed positive integer and m and n are coprime positive integers such that the multiplicative order of FnC1 =Fn modulo Fm is s, where Fk is the k-th Fibonacci number, then m < 500s 2 . Keywords. Fibonacci Sequence, Multiplicative Order. Mathematics Subject Classification 2010. 11A07, 11B39.

8.1

Introduction

Let ¹Fk ºk0 be the Fibonacci sequence given by F0 D 0; F1 D 1 and FkC2 D FkC1 C Fk

for all

k  0:

Let m  3 and n be positive integers such that Fm and Fn are coprime. Since gcd.Fm ; Fn / D Fgcd.m;n/ , this last property holds when gcd.m; n/ 2 ¹1; 2º. Then Fn is invertible modulo Fm . Assuming also that FnC1 is coprime to Fm , we can think of the rational number FnC1 =Fn as an invertible element modulo Fm . Here we look at its order denoted by s. Formally, s depends on both m and n, but we shall omit this dependence in what follows. It is quite possible that this order is s D 1. Indeed, this happens precisely when FnC1  Fn .mod Fm /, so Fm j FnC1  Fn D Fn1 , and this holds when m j n  1. Hence, when n  1 .mod m/. 2 It is also possible that s D 2. In this case, FnC1  Fn2 .mod Fm /, so 2  Fn2 D .FnC1  Fn /.FnC1 C Fn / D Fn1 FnC2 : Fm j FnC1

Assume that m > 12. Then, by Carmichael’s primitive divisor theorem (see [1]), Fm has a primitive prime factor p. This primitive prime has the property that p j Fm but p − F` for any positive integer 1  ` < m. Furthermore, p j F` if and only if m j `. From the above divisibilities, we see that either p j Fn1 , case in which m j n  1, or p j FnC2 , case in which m j n C 2. The situation when m j n  1 leads to s D 1 and this is not convenient, so we must have m j n C 2. Thus, n  2 .mod m/.

110

Takao Komatsu, Florian Luca, and Yohei Tachiya

4 It is also possible that s D 4. In this case, FnC1  Fn4 .mod Fm /, so 4 2 Fm j FnC1  Fn4 D .FnC1  Fn /.FnC1 C Fn /.FnC1 C Fn2 / D Fn1 FnC2 F2nC1 :

If m > 12, then Fm has a primitive prime factor p. Since p divides the right-hand side of the above divisibility relation, we get that m divides one of n  1; n C 2 or 2n C 1. The first two cases lead to s 2 ¹1; 2º. The third case is possible only when m is odd and n  .m  1/=2 .mod m/. From the above discussion, we see that for each of s 2 ¹1; 2; 4º there exist infinitely many positive integers m such that the set of invertible residue classes modulo Fm contains a class representable as FnC1 =Fn for some appropriate positive integer n whose multiplicative order is s. We asked ourselves if this property holds for some other positive integers s. Maybe quite surprisingly, the answer is no. Our main result is the following: Theorem 8.1. If s 62 ¹1; 2; 4º is a positive integer and m is such that there exists an invertible class modulo Fm of the form FnC1 =Fn of multiplicative order s, then m < 500s 2 . For an algebraic number field K we put OK for the ring of algebraic integers in K.

8.2 Preliminary Results We need the following four lemmas. Lemma 8.2. Let X  3 be a real number. Let a and b be positive integers p with max¹a; bº  X . Then there p exist integers u; v not both zero with max¹juj; jvjº  X such that jau C bvj  3 X. p Proof.pConsider the nonnegative numbers as Cbt p for s; t 2 ¹0; 1; : : : ; b Xcº. There are .b Xc C 1/2 > X such numbers all in Œ0; 2X X. By the pigeon hole principle, there exist .s1 ; t1 / ¤ .s2 ; t2 / such that p p 2X X  3 X: ja.s1  s2 / C b.t1  t2 /j D j.as1 C bt1 /  .as2 C bt2 /j  X 1 Putting u D s1  s2 and v D t1  t2 , we get the desired conclusion. p We put ˛ D .1 C 5/=2 and ˇ D ˛ 1 . Lemma 8.3. Let  D e 2 iu=v with coprime positive integers u and v be a primitive root of unity of order v. If v 62 ¹1; 2; 4º, then the two numbers ˛

and

˛ ˛C

111

Chapter 8 On the Multiplicative Order of FnC1 =Fn Modulo Fm

are multiplicatively independent. Proof. Assume on the contrary that there exist integers m and n not both zero such that   ˛ m D ˛n : (8.1) ˛C If m D 0, then ˛ n D 1, therefore n D 0, which is impossible. So, we assume that m ¤ 0. Up to replacing the pair .m; n/ by .m; n/, we may assume that m > 0. Assume first that v is coprime to 5. Then ˛ 2 K D Q.e 2 i=5 / and  2 L D Q.e 2 i=v / and K and L are both Galois extensions of Q whose intersection is trivial (i.e., equal to Q). Thus, every Galois automorphism  of G D Gal.L=Q/ can be extended to a Galois automorphism of the compositum M D KL D Q.e 2 i=5v / of K and L in such a way that  .˛/ D ˛. Applying an arbitrary such  2 G to (8.1), we deduce that Equation (8.1) holds when we replace  by any conjugate of it. In particular, given u1 ; u2 2 ¹1; : : : ; vº both coprime to v, we have ˛  e 2 iu1 =v ˛ C e 2 iu1 =v

!m D ˛n D

˛  2e 2 iu2 =v ˛ C e 2 iu2 =v

!m :

(8.2)

Taking absolute values in (8.2) and then extracting mth roots, we get 1 C

˛ 2  2˛ cos.2u1 =v/ C 1 2˛ 2 C 2 D ˛ 2 C 2˛ cos.2u1 =v/ C 1 ˛ 2 C 2˛ cos.2u1 =v/ C 1 ˇ ˇ ˇ ˇ ˇ ˛  e 2 iu1 =v ˇ2 ˇ ˛  e 2 iu2 =v ˇ2 ˇ ˇ ˇ ˇ Dˇ ˇ Dˇ ˇ ˇ ˛ C e 2 iu1 =v ˇ ˇ ˛ C e 2 iu2 =v ˇ D

˛ 2  2˛ cos.2u2 =v/ C 1 ˛ 2 C 2˛ cos.2u2 =v/ C 1

D 1 C

2˛ 2 C 2 ; ˛ 2 C 2˛ cos.2u2 =v/ C 1

giving cos.2u1 =v/ D cos.2u2 =v/: This gives q q sin.2u1 =v/ D ˙ 1  cos.2u1 =v/2 D ˙ 1  cos.2u2 =v/2 D ˙ sin.2u2 =v/: This argument shows that there exist at most 2 primitive roots of unity of order v, therefore .v/  2, and since v 62 ¹1; 2; 4º, we get that v 2 ¹3; 6º. Let us look at these

112

Takao Komatsu, Florian Luca, and Yohei Tachiya

p p cases. In this instance, M D Q. 5; i 3/ is of degree 4 over Q. We compute 8 9 p !˙1 p = ˛   < 2 C 5 C "i 3 2 p W " 2 ¹˙1º : p ; 5 C "i 3 ˛C : Since p tells us p that the principal ideals in OM given by p ˛ ispa unit, Equation (8.1) . 5 C "i 3/m OM and .2 C 5 C "i 3/m OM are equal for some " 2 ¹˙1º. By unique factorization of ideals in OM , we get that p p p p . 5 C "i 3/OM D .2 C 5 C "i 3/OM : p p In particular, we deduce that 5 C "i 3 j 2. Taking norms in this last divisibility relation, we get that p p 64 D jNM=Q . 5 C "i 3/j j jNM=Q .2/j D 16; which is false. A similar argument applies when 5 j v. In this case K D Q.e 2 i=5 /  L, and so M D L and G D Gal.M=Q/ is isomorphic with the group of invertible elements modulo v which has order .v/. Further, by Galois theory, there are exactly .v/=2 Galois automorphisms  such that  .˛/ D ˛. We deduce that there exists a subset U ¹1; 2; : : : ; vº of positive integers coprime to v having exactly .v/=2 elements, such that Equation (8.1) holds for all  D e 2 iu=v with all u 2 U. The preceding argument shows that cos.2u1 =v/ D cos.2u2 =v/

holds for all

u1 ; u2 2 U;

therefore sin.2u1 =v/ D ˙ sin.2u2 =v/ holds for all

u1 ; u2 2 U:

This shows that the number of elements in U is at most 2, so .v/  4. Since we already have that 5 j v, we get that v 2 ¹5; 10º: We calculated that all numbers of the form ˛ ; ˛C when  is a primitive root of unity of order v 2 ¹5; 10º, are algebraic numbers in M D Q.e 2 i=5 / of norm 11˙1 , and therefore Equation (8.1) does not hold in this instance either for any pair of integers m; n with not both zero. Lemma 8.4. Let  D e 2 iu=v , where v ¤ 4 is a positive integer and u 2 ¹1; 2; : : : ; vº is coprime to v. Then the divisibility relation 1 C  2 j 2v holds in OK , where K is any number field containing Q./.

113

Chapter 8 On the Multiplicative Order of FnC1 =Fn Modulo Fm

Proof. We distinguish four cases. For a positive integer m we put ˆm .X / for the mth cyclotomic polynomial. 

If v is odd, then  2 is also a primitive root of order v of unity, so ˇ ˇ 1 C  2 j ˆv .1/ j X v  1ˇ D 2: XD1



If 2 j v and v=2 is odd, then  2 is a primitive root of unity of order v=2 and ˇ ˇ 1 C  2 j ˆv=2 .1/ j X v=2  1ˇ D 2: XD1



If 4 j v and v=4 is odd, then, since v=4 > 1, it follows that .X v=4  1/ and .X C 1/ are proper divisors of X v=2  1 and they do not have any common roots. Thus, 1 C  2 j ˆv=2 .1/ j

ˇ X v=2  1 X v=4 C 1 ˇˇ ˇ D D ˇ ˇ X C 1 XD1 .X v=4  1/.X C 1/ XD1 ˇ ˇ D v=4: X v=41  X v=42 C    C 1ˇ XD1



If 8 j v, then 1 C  2 j ˆv=2 .1/ j

ˇ X v=2  1 ˇˇ ˇ v=4 D X C 1 D 2: ˇ ˇ XD1 X v=4  1 XD1

For a prime number p and a nonzero integer m, we put p .m/ for the exponent of the prime p in the factorization of m. For a finite set of primes S and a positive integer m, we put Y p p .m/ mS D p2S

for the largest divisor of m whose prime factors are in S. Lemma 8.5. If S is any finite set of primes and m is a positive integer, then Y .Fm /S  2m FpC1 : p2S

Proof. For a prime p, let fp be its order of appearance in the Fibonacci sequence, which is the minimal positive integer k such that p j Fk . It is well-known that 8 ˆ 0 if m 6 0 .mod fp /I ˆ ˆ ˆ < .F / C .m=f / if m  0 .mod f /; p is oddI p fp p p p p .Fm / D ˆ 1 if m  3 .mod 6/; p D 2I ˆ ˆ ˆ : if m  0 .mod 6/; p D 2: 2 C 2 .m/

114

Takao Komatsu, Florian Luca, and Yohei Tachiya

In particular, the inequality p .Fm /  p .Ffp / C p .m/ C ıp;2 always holds with ıp;2 being 0 if p is odd and 1 if p D 2. Since fp  p C 1 holds for all primes p, we get that ! ! Y Y p .Ffp / p .m/ p p 22 .m/C1 .Fm /S  p2S

 2m

Y

pjm p>2

Ffp  2m

p2S

Y

FpC1 ;

p2S

which is what we wanted to prove.

8.3 Proof of Theorem 8.1 We use the Binet formula Fn D

˛n  ˇn ˛ˇ

valid for all

n  0:

(8.3)

We also use the inequalities ˛ n2  Fn  ˛ n1

valid for all

n  1:

(8.4)

We also use the fact that if m  1, then the sequence ¹Fk ºk0 is periodic modulo Fm with period 4m. Assume now that s ¤ ¹1; 2; 4º is a positive integer and that m > 1000 is such that there exist n with Fn coprime to Fm and FnC1 =Fn is invertible modulo Fm of multiplicative order exactly s. From the periodicity of ¹Fk ºk0 modulo Fm , we may assume that n  4m, and since Fn FnC1 and Fm are coprime, we may assume that n  4m  2. We shall exploit the relation Y s Fm j FnC1  Fns D .FnC1  Fn /: (8.5) W s D1

We split Fm into various factors. Step 1. We put A D gcd.Fm ; FnC1  Fn /; B D gcd.Fm ; FnC1 C Fn /; 2 C Fn2 /; C D gcd.Fm ; FnC1

and we bound ABC .

115

Chapter 8 On the Multiplicative Order of FnC1 =Fn Modulo Fm

Then, A D gcd.Fm ; Fn1 / D Fd1 ; where d1 D gcd.m; n  1/I B D gcd.Fm ; FnC2 / D Fd2 ; where d2 D gcd.m; n C 2/I C D gcd.Fm ; F2nC1 / D Fd3 ; where d3 D gcd.m; 2n C 1/: The numbers d1 ; d2 ; d3 are divisors of m and they are proper, since if di D m for some i 2 ¹1; 2; 3º, then, from what we have seen in the Introduction, we would get that s 2 ¹1; 2; 4º, which is not the case. Observe that any two of d1 ; d2 ; d3 are coprime, or the greatest common divisors of any two of them is exactly 3. The second condition holds precisely when m  0 .mod 3/ and n  1 .mod 3/. Indeed, this holds because gcd.d1 ; d2 / D gcd.m; n  1; n C 2/ D gcd.m; n  1; 3/I gcd.d1 ; d3 / D gcd.m; n  1; 2n C 1/ D gcd.m; n  1; 3/I gcd.d2 ; d3 / D gcd.m; n C 2; 2n C 1/ D gcd.m; n C 2; 3/ D gcd.m; n  1; 3/: Let i 2 ¹1; 2; 3º be such that di D max¹d1 ; d2 ; d3 º and let j; k be indices such that ¹i; j; kº D ¹1; 2; 3º. Noting that since di is a proper divisor of m, we have di  m=2. When any two of d1 ; d2 ; d3 are coprime, we then have that d1 d2 d3  m;

therefore dj dk  m2=3 :

(8.6)

When the greatest common divisor of any two of the numbers d1 ; d2 ; d3 is exactly 3, we get         d2 d3 m dj dk m 2=3 d1  ; therefore  ; 3 3 3 3 3 3 3 leading to the slightly worse bound than Equation (8.6), namely dj dk  34=3 m2=3 :

(8.7)

Thus, using Equation (8.4), we get that 4=3 m2=3 2

ABC D Fd1 Fd2 Fd3  ˛ d1 Cd2 Cd3 3  ˛ m=2Cdj Cdk 3  ˛ m=2C3

; (8.8) where we used also the fact that the inequality a C b  ab C 1 is valid for all positive integers a and b with a D dj and b D dk . Step 2. We put S D ¹2º [ ¹p W p j sº and D D .Fm /S , and bound D. By Lemma 8.5 and inequalities (8.4), we have that P Y D  2mF3 FpC1 < 4m˛ pjs p < ˛ sClog.4m/= log ˛ ; pjs

(8.9)

116

Takao Komatsu, Florian Luca, and Yohei Tachiya

P where we used the fact that pjs p  s, which is easily proved by induction on the number of distinct prime factors of s. Step 3. We put ED

Fm ; gcd.ABCD; Fm /

and bound E. We shall estimate the number E by using the fact that E is coprime to 2s, as well as divisibility (8.5), which in particular tell us that Y Fm j ABC .FnC1  Fn /; W s D1 62¹˙1;˙iº

which shows that Ej

Y

.FnC1  Fn /:

(8.10)

W s D1

62¹˙1;˙iº

p

Let K D Q.e 2 i=s ; 5/, which is a number field of degree d equal to .s/ or to 2.s/, according to whether s is a multiple of 5 or not. Assume that there are ` roots of unity  participating in the product appearing in the right–hand side of (8.10) and label them 1 ; : : : ; ` . Clearly, ` 2 Œs  4; s  1. Write Ei D gcd.E; FnC1  i Fn / for all

i D 1; : : : ; `;

(8.11)

where Ei are ideals in OK . Then relations (8.10) and (8.11) tell us that EOK j

` Y

Ei :

(8.12)

iD1

Our next goal is to bound the norm NK=Q .Ei / of Ei for i D 1; : : : ; `: First of all, Fm 2 Ei . Thus, with Equation (8.3) and the fact that ˇ D ˛ 1 , we get ˛ m  .1/m ˛ m

.mod Ei /:

Multiplying the above congruence by ˛ m , we get ˛ 2m  .1/m

.mod Ei /:

(8.13)

We next use Equations (8.3) and (8.11) to deduce that .˛ nC1  .1/nC1 ˛ n1 /  .˛ n  .1/n ˛ n /  0 .mod Ei /;

. D i /:

Multiplying both sides above by ˛ n , we get ˛ 2n .˛  /  .1/nC1 .˛ 1 C /  0 .mod Ei /:

(8.14)

Chapter 8 On the Multiplicative Order of FnC1 =Fn Modulo Fm

117

Let us show that ˛   and Ei are coprime. Assume this is not so and let  be some prime ideal of OK dividing both ˛   and Ei . Then we get ˛   .mod / and so ˛ 1   .mod / by (8.14). Multiplying these two congruences we get 1   2 .mod /. Hence,  j 1 C  2 , and so by Lemma 8.4 we get that  j 2s. However, this contradicts the fact that  j Ei j E, with E an integer coprime to 2s. Thus, indeed ˛   and Ei are coprime, so ˛   is invertible modulo Ei . Now congruence (8.14) shows that ˛ 1 C  ˛ 2n  .1/nC1 .mod Ei /; ˛ therefore ! ˛C 2nC1 nC1 .mod Ei /: ˛  .1/  (8.15) ˛ We now apply Lemma 8.2 to a D 2m and b D 2n C 1  2.4m  2/ C 1 < 8m with the choice XpD 8m to deduce that there exist integers p u, v not both zero with max¹juj; jvjº  X such that j2mu C .2n C 1/vj  3 X : We raise congruence (8.13) to u and congruence (8.15) to v and multiply the resulting congruences getting !v 2muC.2nC1/v muC.nC1/v v ˛ C  ˛ D .1/  .mod Ei /: ˛ We record this as ˛Cı ˛  ˛ı a

!b .mod Ei /

(8.16)

for suitable roots of unity  and ı of order dividing 2s with ı not of order 1; 2 or 4, where a D 2mu C .2n C 1/v and b D v. We may assume that a  0, for if not, we replace the pair .u; v/ by the pair .u; v/, thus replacing .a; b/ by .a; b/ and  by 1 and leaving ı unaffected. We may additionally assume that b  0, for if not, we replace b by b and ı D  by ı D , again a root of unity of order dividing 2s but not of order 1; 2 or 4, and leave a and  unaffected. Thus, Ei divides the algebraic integer Ei D ˛ a .˛  ıi /b  i .˛ C ıi /b ; (8.17) where ıi 2 ¹i ; i º and i is some suitable root of unity of order dividing 2s. Let us show that Ei ¤ 0. If Ei D 0; we then get !b ˛ C ıi a ˛ D i ; ˛  ıi and after raising both sides of the above equality to the power 2s, we get, since 2s i D 1, that !2bs ˛ C ıi 2sa ˛ D : ˛  ıi

118

Takao Komatsu, Florian Luca, and Yohei Tachiya

By Lemma 8.3, we have that as D bs D 0, and so a D b D 0. Since b D 0, we get that v D 0, and later since 2mu C .2n C 1/v D a D 0 and v D 0, we get mu D 0, so u D 0, therefore u D v D 0, but this is not allowed. We now bound the absolute values of the conjugates of Ei . We find it more convenient to work with the associate of Ei given by Gi D ˛ ba=2c Ei D ˛ aba=2c .˛  ıi /b  ˛ ba=2c i .˛ C ıi /b : Note that

p p a  j2m C .2n C 1/vj  3 X D 6 2m;

and

b D jvj 

p p X D 2 2m:

Let  be an arbitrary element of G D Gal.K=Q/. We then have that  .i / D 0i ,  .ıi / D ıi0 , where 0i and ıi0 are roots of unity of order dividing 2s. Furthermore,  .˛/ 2 ¹˛; ˇº. If  .˛/ D ˛, we then get j .Gi /j D j˛ aba=2c .˛  ıi0 /b  0i ˛ ba=2c .˛ C ıi0 /b j  ˛ .aC1/=2 .˛ C 1/b C .˛ C 1/b p p p  2˛ .aC1/=2 .˛ C 1/b  .2 ˛/˛ 3 2m .˛ 2 /2 2m p p D .2 ˛/˛ 7 2m ;

(8.18)

while if  .˛/ D ˇ, we also get j .Gi /j D jˇ aba=2c .ˇ  ıi0 /b  ˇ ba=2c 0i .ˇ C ıi0 /b j  .˛ 1 C 1/b C ˛ a=2 .˛ 1 C 1/b D ˛ b C ˛ a=2Cb  2˛ 3 D 2˛

p 5 2m

p p 2m 2 2m

˛

: .1/

.d /

In conclusion, inequality (8.18) holds for all  2 G. Thus, if we write Gi ; : : : ; Gi for the d conjugates of Gi in K, we then get that p p jNK=Q .Ei /j  jNK=Q .Ei /j D jNK=Q .Gi /j  .2 ˛/d ˛ 7d 2m ;

where the first inequality above follows because Ei divides Ei ; hence Gi , and Ei ¤ 0. Multiplying the above inequalities for i D 1; : : : ; ` we get, also using Equation (8.12), ! that ` Y Ei E d D NK=Q .E/ D NK=Q .EOK /  N iD1



` Y

p p NK=Q .Gi /  .2 ˛/`d ˛ 7d ` 2m ;

iD1

and therefore

p p p p E  .2 ˛/` ˛ 7` 2m D ˛ 7` 2mC` log.2 ˛/= log ˛ :

Thus, we have bounded E.

(8.19)

119

Chapter 8 On the Multiplicative Order of FnC1 =Fn Modulo Fm

Step 4. The final inequality. We now use Equation (8.4) to bound Fm from below as Fm > ˛ m2 , and the fact that Fm  ABCDE and the estimates (8.8), (8.9), and (8.19), to bound Fm from above as p

4=3 m2=3 2CsClog.4m/= log ˛C7`

Fm  ˛ m=2C3

p 2mC` log.2 ˛/= log ˛

;

to conclude that p p m log.4m/ ` log.2 ˛/ 4=3 2=3 C3 m C 7` 2m C ; m2< 2CsC 2 log ˛ log ˛

(8.20)

where `  s  1. We look at p p m log.4m/ .s  1/ log.2 ˛/ 4=3 2=3 f .m; s/ D s 3 m  7.s  1/ 2m  : 2 log ˛ log ˛ Computing the partial derivative with respect to m, we get g.m; s/ D

1 2 31=3 7.s  1/ 1 @f .m; s/ D   p  : 1=3 @m 2 m log ˛ m 2m

(8.21)

The function g.m; s/ is positive when m  500s 2 and s  3, because in this range g.m; s/ 

2 31=3 7 1 1  p > 0:103:  2 .4500/1=3 4500 log ˛ 1000

Thus, in order to prove that m < 500s 2 , it suffices to prove f .500s 2 ; s/ > 0. We checked with Mathematica that this inequality holds for s  17. For the remaining values s 2 Œ3; 16, we checked individually by noticing that for each one of these values of s a slightly better inequality than (8.20) holds. For example, in the case when s 2 ¹3; 5; 7; 9; 11; 13; 15º, there is no need for d2 and d3 because s is odd. Thus, the analogue of inequality (8.20) for such values of s is simply p p log.4m/ .s  1/ log.2 ˛/ m 1CsC C 7.s  1/ 2m C : (8.22) m2< 2 log ˛ log ˛ Plugging in s D 3; 5; 7; 9; 11; 13, and 15 into (8.22), we got m bounded by 2000, 7000, 15000, 26000, 40000, 57000, and 77000, respectively, and so the inequality m < 500s 2 definitely holds for these values of s as well. When s D 6; 10; 14, we keep only two divisors in Case 1, namely d1 and d2 since there is no need for d3 . Putting i 2 ¹1; 2º such that di D max¹d1 ; d2 º and letting j be such that ¹i; j º D ¹1; 2º, the analog of inequality (8.7) is p dj  3m: Since `  s  2, when s D 6; 10; 14, the analog of inequality (8.20) in this case is p p log.4m/ .s  2/ log.2 ˛/ m p C 7.s  2/ 2m C ; (8.23) m  2 < C 3m  2 C s C 2 log ˛ log ˛

120

Takao Komatsu, Florian Luca, and Yohei Tachiya

giving for s D 6; 10, and 14 that m is bounded by 8000; 27000, and 60000; respectively. Thus, the inequality m < 500s 2 holds also for s D 6; 10; 14. Finally, for s D 8; 12; 16, we use the analog of inequality (8.20) with the value `  s  4, yielding p p m 4=3 2=3 log.4m/ .s  4/ log.2 ˛/ C7.s4/ 2mC ; (8.24) m2 < C3 m 2CsC 2 log ˛ log ˛ which at s D 8; 12, and 16 gives that m is bounded by 16000; 45000, and 88000, respectively, so the inequality m < 500s 2 holds in these last three cases as well. This completes the proof of the theorem.

8.4 Comments and Numerical Results Numerical results are few because the bounds of Theorem 8.1 are very weak. However, from what we have said above at Step 4 of the proof of Theorem 8.1, we have that m < 2000 when s D 3, and m < 7000 for s D 5. We ran a Mathematica code run for about a day and searched for all such m and for all n 2 Œ1; 4m for which FnC1 =Fn is indeed an element of order s modulo Fm . No example was found with s D 3, and the example F7 =F6 modulo F10 is the only example with s D 5. In [3], it was shown that the Diophantine equation x D Fm Fnx C FnC1

has no positive integer solutions with n  2 and x  3. The method there was based on linear forms in logarithms. Since for any potential solution of the above equation it is easy to check that FnC1 =Fn is an invertible element modulo Fm which is not of order 1; 2; or 4 but whose order divides 2x, we get right away from Theorem 8.1 that m < 2000x 2 . Next, by Equation (8.4), we get x x < Fnx C FnC1 D Fm < ˛ m1 < ˛ 2000x ˛ nx  FnC1

2 1

;

and we derive that n < 2000x. However, we could not find an elementary upper bound on x out of this equation (without appealing to linear forms in logarithms). We conclude by mentioning that all the solutions of the more general Diophantine equation y x Fnx C FnC1 D Fm in positive integers .n; m; x; y/ were found in [2]. Acknowledgments. T. K. was supported in part by the Grant-in-Aid for Scientific research (C) (No. 22540005), the Japan Society for the Promotion of Science. F. L. worked on this project during a visit to Hirosaki in January and February of 2012 with a JSPS Fellowship (No. S-11021). We thank the referee for comments which improved the quality of this paper.

Chapter 8 On the Multiplicative Order of FnC1 =Fn Modulo Fm

121

References [1] R. D. Carmichael, On the numerical factors of the arithmetic forms ˛ n ˙ ˇ n , Ann. Math. (2) 15 (1913), 30–70. x D Fmy , preprint, [2] N. Hirata-Kohno and F. Luca, On the Diophantine equation Fnx C FnC1 2012. [3] F. Luca and R. Oyono, On the sum of powers of two consecutive Fibonacci numbers, Proc. Japan Acad. Ser. A. 87 (2011), 45–50.

Author information Takao Komatsu, Graduate School of Science and Technology, Hirosaki University, Hirosaki, Japan. Email: [email protected] Florian Luca, Centro de Ciencias Matemáticas UNAM, Morelia, Michoacán, Mexico. Email: [email protected] Yohei Tachiya, Graduate School of Science and Technology, Hirosaki University, Hirosaki, Japan. Email: [email protected]

Combinatorial Number Theory, 123–136

© De Gruyter 2013

Chapter 9

Outcomes of Partizan Euclid Neil A. McKay and Richard J. Nowakowski Abstract. PARTIZAN EUCLID is a game based on the Euclidean algorithm. The outcome of any position .p; q/ is determined by a single path of the game tree; this path has connections to the furthest integer continued fraction of p=q. We convert the question of “Who wins?” to a word problem, then give a list of reductions that reduces the word/position to one of 9 positions. Keywords. Combinatorial Games, Partizan Euclid. Mathematics Subject Classification 2010. 91A46.

9.1

Introduction

Suggested by “Euclid” [3] and Richard K. Guy, the game of PARTIZAN EUCLID is played by two players, Left and Right, and starts with a pair of positive integers .p; q/ with p > q. Let p D kq C t where 0  t < q. If q j p (i.e., t D 0), then the game is over; otherwise, Left moves to .q; t / and Right moves to .q; q  t /. The game may seem trivial as there is only one move available for each player. However, as we shall show, answering the question “Who wins?” reveals some of the interesting structure of the game. We would like to answer the question of who wins in the disjunctive sum of this game, but this appears to be difficult. See the last section for a discussion of that problem. In the (impartial) game EUCLID, which is also played with .p; q/, a pair of positive integers, a player is allowed to remove any multiple of the smaller from the larger provided the remainder is positive. Lengyel [7] reports that Schwartz first found that EUCLID is the sequential sum [10] of nim-heaps: given .p; q/, suppose the normal continued fraction of pq is Œa1 ; a2 ; : : : ; an  (an > 1, except if Fibonacci numbers are involved); then the EUCLID position .p; q/ corresponds to playing the sequential sum of NIM with nim-heaps a1 ; a2 ; : : : ; an . EUCLID has attracted much attention and has been generalized (see, e.g., [4, 5]). PARTIZAN EUCLID is related to nearest and farthest integer continued fractions (NICF and FICF) (see [8]). In the case of both NICFs and FICFs we write rational numbers as a sum or difference of an integer and a rational less than 1. For example, the FICF for 11 8 is obtained by rewriting, noting that 

11 8

D2

1 , 8=5

since 2 is further away from

11 8

than 1;

124

Neil A. McKay and Richard J. Nowakowski



8 5

D1C

1 , 5=3

since 1 is further away than 2;



5 3

D1C

1 , 3=2

since 1 is further away than 2;



3 2

D2

1 2=1

D1C

1 , 2=1

since 1 and 2 are equally distant.

We are not interested in the continued fraction itself but in noting that during the calculation (i) “integer subtract fraction” corresponds to a move by Right and (ii) “fraction subtract integer” corresponds to a Left move. We will use the word rl le to represent this where e is the common move to .2; 1/. Section 9.2 reports on the structure of the game tree and shows there is one path, the path obtained from the FICF algorithm, that determines the whole game tree. For example, in Figure 9.1 the path formed by the moves (edges) Right

Left

Left

.11; 8/ ! .8; 5/ ! .5; 3/ ! .3; 2/ is the important path. Why is it important? Nontrivial parts of the tree that are not on that path are isomorphic to parts rooted on the path; “.8; 3/0 ’ is isomorphic to “.5; 3/0 ’, “.5; 2/0 ’ is isomorphic to “.3; 2/”, and “.3; 1/” is isomorphic to “.2; 1/”. All the information needed to determine the outcome and value of .11; 8/ is found on this path. A game tree (position) is represented as a word from the alphabet r; l ending in e. In Lemma 9.13, we find reduction rules that preserve the outcome class of the word, moreover, any word reduces to one of just 9 words each with length at most 4. This can be accomplished in time linear in the length of the corresponding FICF. Unfortunately, these reductions most of the time do not preserve the value.

Figure 9.1. Some of the game tree of .11; 8/:

Chapter 9 Outcomes of Partizan Euclid

125

We will denote a game position as E.p; q/. Also, we will use p%q for p mod q. We try to present a sufficient amount of game theory to make the paper self-contained (with the exception of the last two sections). For terms not defined in the paper we follow [1]. A position, say h, is defined in terms of its options as follows: h D ¹hL jhR º. For example, where t D p%q, E.p; q/ D ¹E.q; t /jE.q; q  t /º. The outcome of a position is Left, Right, Next or Previous depending on (under perfect play), respectively, whether Left can win going first and second, Right can win going first and second, the next player to move wins regardless if this Left or Right, the next player cannot win regardless if this Left or Right. We phrase this more formally. Lemma 9.1. Let h be a position. The outcome of h is determined by the outcomes of its options. Specifically, 

o.h/ D L iff 9hL , o.hL / 2 ¹L; P º and 8hR , o.hR / 2 ¹L; N º;



o.h/ D P iff 8hL , o.hL / 2 ¹N ; Rº and 8hR , o.hR / 2 ¹L; N º;



o.h/ D N iff 9hL , o.hL / 2 ¹L; P º and 9hR , o.hr / 2 ¹P ; Rº;



o.h/ D R iff 8hL , o.hL / 2 ¹N ; Rº and 9hR , o.hr / 2 ¹P ; Rº.

Let g D E.p; q/. Since there is at most one option for each player we will abuse notation and write o.g/ D o.¹g L jg R º/ as ¹o.gL /jo.gR /º. For example, ¹N jP º D R.

9.2

Game Tree Structure

Lemmas 9.2 and 9.3 each show that for every position there are infinitely many positions with the same game tree. We call positions equivalent if they have the same game tree. Lemma 9.2. For all k, E.kp; kq/ D E.p; q/. Proof. Recall that q j p if and only if kq j kp. Thus, E.p; q/ D ¹jº if and only if E.kp; kq/ D ¹jº. Let t D p%q, then by induction E.p; q/ D ¹E.q; t /jE.q; q  t /º D ¹E.kq; k t /jE.kq; kq  k t /º D E.kp; kq/. Note that if h D E.n; m/ is a follower of a position g D E.p; q/ with gcd.p; q/ D 1, then gcd.n; m/ D 1. In the rest of the chapter we will assume that every position has gcd.p; q/ D 1 and thus p%q D 0 if and only if q D 1. Lemma 9.3. If p > 2q, then E.p; q/ D E.p  q; q/. Proof. Let p D kq C t , 0  t < q and k  2. Then p  q D .k  1/q C t , 0  t < q and k  1  1. Consider the options of both positions: E.p; q/ D ¹E.q; t /jE.q; q  t /º E.p  q; q/ D ¹E.q; t /jE.q; q  t /º: Since they have identical options the two positions are equivalent.

126

Neil A. McKay and Richard J. Nowakowski

A position E.p; q/ will be called standard if q < p < 2q. All positions E.p; q/ with q > 2 are equivalent to some standard position (which is reachable with repeated applications of Lemma 9.3). Notably E.2; 1/ is not standard, and positions of the form E.k; 1/ are neither standard nor equivalent to some standard position. A follower of a standard position may not be standard. For example, E.3; 2/ has only one proper follower, E.2; 1/, which is not standard. Lemma 9.4. Let g D E.p; q/ and t D p%q. If t 6D 0 then g has exactly one standard option except when q D 2t (i.e. E.3; 2/). Moreover; 

if 2t > q, then exactly g L is standard;



if 0 < 2t < q; then exactly gR is standard.

Proof. As t > 0, g L D E.q; t / and g R D E.q; q  t /. Recall by the definition of standard; that t > q2 if and only if g L is standard and t < q2 if and only if g R is standard. Otherwise, 2t D q, implying that p D 3t and subsequently that g D E.3; 2/; from E.3; 2/ both players have the option E.2; 1/, which is not standard. There is a unique standard position with a given left or right option. Lemma 9.5. Let g D E.p; q/ and 0 < t < q. If g is standard, then 

if g L D E.q; t /; then g D E.q C t; q/;



if g R D E.q; q  t /; then g D E.q C t; q/.

Proof. If g L D E.q; t / then p D kq C t ; as g is standard, k D 1. If gR D E.q; q  t / then p D kq C .t  q/ D .k  1/q C t ; as g is standard, k D 2. The essential game tree structure is given by Theorem 9.6. Theorem 9.6. Let g D E.p; q/ and t D p%q: 

if t D 0 then g D ¹jº;



if 2t D q then g L D g R ;



if 2t > q then gLL D gR ;



if 0 < 2t < q then g L D g RL .

Proof. If t D 0 then the game is over and g D ¹jº. Thus we suppose t > 0 and hence g L D E.q; t / and g R D E.q; q  t /. Suppose 2t D q; then g L D E.q; t / D E.q; q  2t C t / D E.q; q  t / D gR . Moreover, q  t j q, so E.q; t / D E.q; q  t / D 0. (For the purists, g D ¹0j0º D .) Suppose 2t > q; then q > 2.q  t / and t > q  t so gR D E.q; q  t / D E.q .q t /; q t / D E.t; q t / by Lemma 9.3, and g LL D E.q; t /L D E.t; q t /, giving g R D E.t; q  t / D g LL :

Chapter 9 Outcomes of Partizan Euclid

127

Suppose 2t < q; then g L D E.q; t / D E.q  t; t / by Lemma 9.3. Since gR D E.q; q  t / we have that g RL D E.q  t; t /, giving g RL D g L . Corollary 9.7. Let p > q with p D kq C t , 0 < t < q and let g D E.p; q/: 

if 2t > q; then g D ¹g L jgLL º;



if 2t < q; then g D ¹g RL jgR º.

Proof. This is a simplification of Theorem 9.6. We note the similarity of Lemma 9.4 and Corollary 9.7. In Corollary 9.7, the two options are the standard option and its left option. The case 2t > q is when the lower integer is the “farthest” integer when calculating the FICF and 2t < q is when the higher integer is the “farthest”. This motivates our next definition, the signature of a position, in which we highlight the important option at each stage. Recall that when we refer to E.p; q/ we are assuming that gcd.p; q/ D 1. Definition 9.8. Let g D E.p; q/. The signature of g, denoted Sg , is defined as follows. If q D 1 then Sg D , the empty word. If q D 2 then Sg D e. Otherwise, let h be the standard option from g. If gL D h then Sg D lSh . If gR D h then Sg D rSh . The position g and the standard positions that are successively the standard option (as per Lemma 9.4) starting from g are the spine of g. For example, if g D E.12; 7/ then the signature of g is lrle and the spine of g is ¹E.12; 7/; E.7; 5/; E.5; 3/; E.3; 2/º. Often, we will write the signature with superscripts; for example, S D l l lrrl l lre is the same as S D l 3 r 2 l 3 e. If two positions have the same signature then they have the same game tree. We use signatures liberally to represent positions. Furthermore, we use ˛f to denote the position g where Sg D ˛Sf . The position E.3; 2/ is the unique standard position with signature e. This position is at the bottom of every spine for every position other than E.k; 1/ and E.2k C 1; 2/ for k  2. Theorem 9.9. Let g be a PARTIZAN EUCLID position. Every follower of g not of the form E.k; 1/ is equivalent to some position on the spine of g. Proof. A position is on its spine, so we only need consider proper followers. If the length of the signature is 0 then there are no proper followers. If the length of the signature is 1; then Sg D e and g D E.3; 2/; where the only proper follower is E.2; 1/. We proceed by induction on the length of signature. If the length of the signature is at least 2, then the standard option is on the spine; the nonstandard option is (by Theorem 9.6) either gL D g RL or g R D gLL , which is the left option of the standard

128

Neil A. McKay and Richard J. Nowakowski

option and is by induction on the spine of the standard option or of the form E.k; 1/. As the spine of the standard option is part of the spine of g, this completes the proof. Corollary 9.10. Consider the position g, and let k represent an unfixed nonnegative integer. We can write Sg as either r k l˛e or r k e; Left’s move has signature ˛e or , respectively. We can also write Sg as either r˛e, lr k l˛e, or lr k e; Right’s move has signature ˛e, ˛e, or , respectively. Proof. If Sg is r k l˛e or r k e, then Sg L is ˛e or , respectively, because gL D gRL D k

g RRL D g RRRL D    D g R L . If Sg D r˛e then Sg R D ˛e. Otherwise, gR D g LL so Sg R is the signature of the position resulting from two Left moves namely ˛e or , as seen by the first part. In the examples below, we repeatedly use Theorem 9.6, but using Corollary 9.10 one can easily jump from the leftmost term in a line of equalities to the rightmost. Let g D l lrle then e D ¹e L je R º D ¹jº; le D ¹le L jle R º D ¹ejle LL º D ¹eje L º D ¹ejº; rle D ¹rle L jrle R º D ¹rle RL jleº D ¹le L jleº D ¹ejleº; lrle D ¹lrle L jlrle R º D ¹rlejlrle LL º D ¹rlejrle L º D ¹rlejrle RL º D ¹le L jeº D ¹rlejeº; l lrle D ¹l lrle L jl lrle R º D ¹lrlejl lrle LL º D ¹lrlejlrle L º D ¹lrlejrleº:

9.3 Reducing the Signature The paired outcome of a position g (or signature Sg ) is the pair .o.g L /; o.g//, denoted by po.g/ or po.Sg /. For example, if Sg D e, then po.g/ D po.e/ D .P ; N /. Note that po./ is not defined. The paired outcome of a position depends upon which option is standard and the paired outcome of that option. Lemma 9.11. If Sg D lSh then po.g/ D .o.h/; ¹o.h/jo.hL /º/. If Sg D rSh then po.g/ D .o.hL /; ¹o.hL /jo.h/º/. Proof. Follows immediately from Theorem 9.6.

Chapter 9 Outcomes of Partizan Euclid

129

That is, the paired outcome of a position with a standard option is determined by the paired outcome of the standard option. We use l and r to denote functions on paired outcomes; we write l ıpo.Sg / to mean po.l ıSg / and r ıpo.Sg / to mean po.r ıSg /. There are 4 4 D 16 ordered pairs of outcome classes. However, as stated in Lemma 9.1, there are relationships between the outcome of a position and the outcome of its options; there are only eight ordered pairs that are paired outcomes of positions. Figure 9.2 has a vertex for each of the eight paired outcomes. The directed edges labelled l and r from a paired outcome, say x, lead to l ı x and r ı x, respectively. The outcome of a position is given by the paired outcome of the signature. Provided we know the paired outcome of some suffix of the signature, we can find the paired outcome of the next larger suffix using Figure 9.2 and eventually the desired paired outcome. As e is the suffix of every nonempty signature, we only need to know that po.e/ D po.E.3; 2// D .P ; N / is where we start and to read the signature from the right starting after e. We have now described a relatively efficient method to determine the outcome of a position given its signature, but we give a better way to determine the outcome than a walk through the graph for each letter in the signature. Just as l and r are functions on paired outcomes, we use words (denoted by Greek letters) from ¹l; rº such as ˛ D lrr as functions on paired outcomes in the natural way where ˛ ı x D l ı r ı r ı x. For any such ˇ, ˇ ı po.Sh / D po.˛Sh /. We give reduction rules by which we can simplify a word (signature) that preserves outcome, in the sense that if two positions have the same reduced signature then they have the same outcome. The main goal of this section is to prove Theorem 9.15, in which we give a short list of words to which any signature will reduce. Lemma 9.12. If ˛ 2 ¹l; rº then ˛ ı .L; L/ D .L; L/ and ˛ ı .R; R/ D .R; R/. Proof. Immediate from Figure 9.2.

Figure 9.2. Paired outcome of signatures.

130

Neil A. McKay and Richard J. Nowakowski

Lemma 9.13. Let x be a paired outcome: 1. l 3 ı x D x; 2. r 2 ı x D r ı x; 3. ˛rlr ı x D rlr ı x; 4. .rl 2 /2 r ı x D r ı x; 5. rl lr ı .P ; N / D l ı .P ; N /; 6. ˛rl l ı .P ; N / D rl l ı .P ; N /; 7. rl ı .P ; N / D l ı .P ; N /. Proof. In this proof we make extensive use of Figure 9.2. The most common use is to find a vertex with the desired paired outcome then look up the paired outcomes reached by the directed edges. For the first four rules, we need to show that these equation holds for all paired outcomes x. Rule 1: we apply l to each x three times to observe that we return to the x with which we started. Rule 2: following two edges marked r and, regardless, the second edge is a loop. Rule 3: following three edges marked r, l and r results in .L; L/ or .R; R/ so the first part of the signature it is irrelevant. Rule 4: following the walk with edges marked rl lrl lr either (i) goes once around the 6-cycle in Figure 9.2 ending at the starting vertex; (ii) or if the starting vertex is .L; L/ or .R; R/ then remains at .L; L/ or .R; R/ respectively; (iii) starts at .L; N / or .R; P / then the first r-edge goes to .L; L/, .R; R/ respectively. In all cases the effect of following the last r edge in the walk is the same as following the first. Rule 5: obvious from the figure. Rule 6: from .P ; N / the walk l, l, r ends at .R; R/ and any further edges do not change the paired outcome. Rule 7: obvious from the figure. Lemma 9.14. If x is a paired outcome and  ı x D ı ı x, then po.˛ˇe/ D po.˛ıˇe/. Proof. po.˛ˇe/ D ˛ ı po.ˇe/ D ˛ı ı po.ˇe/ D po.˛ıˇe/. In applying Lemma 9.14 to reduce a word, we make reference to particular rules from Lemma 9.13. We call a word irreducible if none of the reduction rules are applicable. Reduction rules may be applied in any order to the signature of a position, say g, to derive an irreducible word; the irreducible word corresponds to some other position, say h. The outcome of g is the same as the outcome of h, as they have the same paired outcome,

131

Chapter 9 Outcomes of Partizan Euclid Table 9.1. Irreducible signatures with corresponding positions and outcomes. Sg  e re le l le lre rlre rl le l lre

o.g/

g

P N L R P N L R P

E.2; 1/ E.3; 2/ E.4; 3/ E.5; 3/ E.8; 5/ E.7; 4/ E.10; 7/ E.11; 8/ E.11; 7/

which is a stronger condition. As there are a finite number of irreducible words, we will be able to compute and store the outcomes of those positions. Theorem 9.15. There are nine irreducible words: , e, re, le, l le, lre, rlre, l lre, and rl le. For the proof of Theorem 9.15 we need the following Lemma: Lemma 9.16. A word containing 4 rs is reducible. Proof. Suppose ˛ is an irreducible word containing 4 rs. By Rule 2, each pair of consecutive rs is separated by at least one l. By Rule 3, each pair of rs except possibly the leftmost, is separated by more than one l. By Rule 1, each pair of rs is separated by at most 2 ls. That is, the rightmost 3 rs form the pattern rl lrl lr, which contradicts the assumption that ˛ is irreducible by Rule 4. Proof of Theorem 9.15. The irreducible words that do not end in e are easy to list and count with the help of Lemma 9.16; such words have at most 3 r’s. In what follows, ˛ and ˇ are one of either , l, or l l. Words with 3 rs are of the form rlrl lrˇ, of which there are 3. Words with 2 rs are of the form ˛rl lrˇ or rlrˇ, of which there are 12. Words with 1 r are of the form ˛rˇ, of which there are 9. Words with no r are of the form ˛, of which there are 3. There are a total of 27 such words. The only irreducible signatures are among the set containing these strings but with a trailing e appended, and the empty word. We show that 19 of the 27 strings reduce to the remaining 8: e, le, l le, re, lre, l lre, rl le and rlre.

132

Neil A. McKay and Richard J. Nowakowski



The 7 strings of the form  rl le where  is nonempty reduce by Rule 6 to rl le.



The 4 words of the form  rl lrle reduce to  rl l le and then to  re by Rules 7 and 1, respectively, leaving re, lre, l lre, and rlre.



The 4 words of the form  rl lre reduce to le by Rule 5, leaving the 3 strings with no r (note l l le D e) and rl le.



The 4 words of the form  rle reduce to le by Rule 7, leaving the 3 strings with no r (note l l le D e) and rl le.

We note that of the none of the reductions apply to the 9 claimed irreducible words (8 from above and ).

9.3.1 Algorithm We present an algorithm that efficiently determines the outcome of a PARTIZAN EUCLID position. Step 0. Let S be the signature of E.p; q/. Let S 0 be the empty string. Step 1. If S is nonempty, remove the first letter of S and add it to the end of S 0 ; go to Step 2. Otherwise, go to Step 3. Step 2. – If you added l to S 0 , use Rule 1 on the suffix of S 0 if applicable. Go to Step 1. – If you added r to S 0 , use Rule 2, 3, or 4 on the suffix of S 0 if applicable; at most one will apply. Go to Step 1. – If you added e to S 0 , use Rule 5, 6, or 7 on the suffix of S 0 if applicable, at most one will apply. If you applied Rule 5 or 7, then use Rule 1 if applicable. Go to Step 3. Step 3. The outcome of E.p; q/ is the outcome of S 0 given in Table 9.1. Reductions occur at the end of the word and the application of a reduction does not cause another reduction, except possibly in Step 2: part 3, with an l l l reduction. As such, Step 2 finishes in constant time (as do Steps 1 and 3). Step 1 takes about as long as the Euclidean algorithm. Steps 1 and 2 have to be performed at most p times; Steps 0 and 3 are each performed once. By Lemma 9.16, S 0 reaches a length of at most 8, as demonstrated by l lrl lrl l. That is, if at any point the length of S 0 is 9, then it will be irreducible in the next step. In the algorithm as given above, S is computed in full at the beginning, for ease of description. However, we can easily modify our algorithm to be an on-line algorithm by computing the next letter of S as we need it to add to S 0 . In that case, to run the algorithm we store at most 3 integers no larger than p and a string of length at most 9.

Chapter 9 Outcomes of Partizan Euclid

9.4

133

Outcome Observations

There are several interesting observations that can be made about the outcomes which may be useful in actual play. Observation 9.17. If Sg D rˇe then o.g/ 2 ¹L; Rº. All signatures in Table 9.1 starting with r are in L or R. The reductions (from Lemma 9.13) change signatures starting with r to shorter signatures starting with r or to le, which is in R. Observation 9.18. Let g D E.p; q/ be a standard position. If o.g/ 2 ¹N ; P º, then 2p p 3  q > 2. If o.g/ 2 ¹N ; P º, then Sg is  (in which case g is not standard), e (in which case D q), or starts with l. As g is standard, if t D p%q, then t D p  q and q > p2 . For the signature to start with l, we need 2t > q, which is 2.p  q/ > q or 2p 3 > q; combining this with q > p2 gives the result. Lemma 9.13 can be restated in terms of positions in the game. 2p 3

Observation 9.19. If a and b are integers with a > b > 0; then (1) o.E.a C b; a// D o.E.5a C 3b; 3a C 2b//; (2) if o.E.2a C b; a C b// D P and a > 2b, then E.2a C 3b; a C 2b/ D P . For (1). Let E.a Cb; a/ D ˛e and consider E.5a C3b; 3a C2b/. First suppose a > b; the consecutive left options E.3aC2b; 2aCb/, E.2aCb; aCb/, and E.aCb; a/ are standard and are on the spine so E.5a C 3b; 3a C 2b/ D l l l˛e. By Rule 1, o.E.5a C 3b; 3a C 2b// D o.E.a C b; a//. Now suppose a < b; from E.5a C 3b; 3a C 2b/, both E.3a C 2b; 2a C b/, E.2a C b; a C b/ are still on the spine, and now E.a C b; b/ is, too. Since E.a C b; a/ D P , then by Theorem 9.6 o.E.a C b; b/L / D P , and thus o.E.2aCb; aCb// D L. Now o.E.3aC2b; 2aCb/L / D o.E.2aCb; aCb// D L, and again by Theorem 9.6 o.E.3a C 2b; 2a C b/R / D o.E.a C b; a// D P , thus o.E.3aC2b; 2aCb// D N . Finally, o.E.5aC3b; 3aC2b/L / D o.E.3aC2b; 2aC b// D N , and again by Theorem 9.6, o.E.5a C 3b; 3a C 2b/R / D o.E.2a C b; a C b// D L, and thus o.E.5a C 3b; 3a C 2b// D P . For (2): Since a > 2b then E.2a C b; a C b/ D lr˛e, where the left option is E.a C b; a/ and its right option is E.a; a  b/, they are both standard and on the spine. The left option of E.2a C 3b; a C 2b/ is E.a C 2b; a C b/ and its right option is E.a C b; a/; again both on the spine. Thus E.2a C 3b; a C 2b/ D lrr˛e. Therefore, from Lemma 9.13, o.E.2a C 3b; a C 2b// D o.E.2a C b; a C b//.

134

Neil A. McKay and Richard J. Nowakowski

9.5 Open Questions Our main work is describing the structure of postitions of PARTIZAN EUCLID and giving an efficient algorithm for determining the outcome. Thus we arrive at one main open question. Question 9.20. Is there an efficient method to play disjunctive sums of EUCLID positions?

PARTIZAN

For some families of positions (signatures) we can give the value easily. Observations 9.21 and 9.22 happen to correspond to the extreme cases of the Euclidean algorithm. Observation 9.21. Positions of the form E.k C 1; k/ have value  C .k  2/" for k  2. The signature r k e corresponds to E.k C 1; k/. When k  2 the Left option is to E.k; 1/ which is equal to 0, and the Right option is to E.k; k  1/. Observation 9.22. Let fn be the n-th Fibonacci number, where f0 D 0 and f1 D 1. The position E.fk ; fk1 / has the signature l k e, and the value is periodic in k; E.f3k ; f3k1 / D", E.f3kC1 ; f3k / D , and E.f3kC2 ; f3kC1 / D 0. Starting with E.2; 1/ D 0, E.3; 2/ D , and E.5; 3/ D", an easy induction gives the result. It seems unlikely that we would find a heuristic method to play a sum; we expect a solution would require first computing the values of the summands and a method to play on a sum of such values (see [1] for more on values). Values are harder to calculate and there are few easy reductions of the signature that allow short cuts. However, we present a general rule that we have found. Note from Corollary 9.10 that a Left move from a position whose signature has at least one l in it removes exactly one l, and a Right move from a position whose signature has at least two ls in it removes exactly one r or exactly two ls. Observation 9.23. Let two positions g and h have signatures ˛lr a lˇe and ˛lr b le; respectively. If r a lˇe D r b le and ˇe D e; then g D h. To see this, if ˛ D , then gL D r a lˇe D r b le D hL and g R D g LL D ˇe D e D hLL D hR . If ˛ D l, then g L D lr a lˇe D lr b le D hL and g R D ˇe D e D hR . If ˛ D r, then gL D r a lˇe D r b le D hL and g R D lr a lˇe D lr b le D hR . As there are many nontrivial P positions in PARTIZAN EUCLID, and all P positions have value 0, we think it is reasonable to expect many other values to occur repeatedly, perhaps with similar patterns to that of the P positions.

135

Chapter 9 Outcomes of Partizan Euclid Table 9.2. E.q; p/, q D 11; : : : ; 20, p D 10; : : : ; 15.

p D 10 11 12 13 14 15

q D 11

12

13

14

15

16

17

18

19

20

¹6j2º

3=2 ¹7j2º

2 0 ¹8j2º

0 3 ¹2j2º ¹9j2º

1 0 1 0 ¹10j2º

0 4 0 2 ¹3j2º ¹11j2º

2 3 3 3 3 0

0 0 1 0 0 3=2

0 4 1 5 0 0

0 0 0 4 2 0

Question 9.24. Which signature reductions preserve value in which instances? Canonical forms become messy with even small values of p and q. Atomic weights (see [1]) are an approximation to the value of a position. The atomic weights of Table 9.2 were generated by CGSuite [9]. (In version 1.0 of CGSuite, PARTIZAN EUCLID is used as an example, and tables of both canonical forms and atomic weights can be easily generated.)

Figure 9.3. Graph showing mean atomic weights of positions against ratio of p to q.

136

Neil A. McKay and Richard J. Nowakowski

The mean values of atomic weights show some regularity on a large scale; see Figure 9.3, where the figure is cut off above 25, removing the points corresponding to E.k C 1; k/ for k > 27. In addition to the variation of the means of atomic weights of positions there is great complexity among the atomic weights, which include positions such as ¹6j9 12 º, z3 , 8*, and ¹7jj6j5º. Question 9.25. Is there an efficient way to determine the atomic weight of a position from its signature? Acknowledgments. The authors wish to thank the NSERC for financial support.

References [1] M. H. Albert, R. J. Nowakowski, and D. Wolfe, Lessons in Play, A. K. Peters, Ltd., Nattick, MA., 2007. [2] E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning ways for your mathematical plays, vol. 1, A. K. Peters, Ltd., 2001. [3] A. J. Cole and A. J. T. Davie, A game based on the Euclidean algorithm and a winning strategy for it, Math. Gaz. 53 (1969), 354–357. [4] D. Collins, Variations on a theme of Euclid, Integers, 5 (2005), #G3. [5] D. Collins and T. Lengyel, The game of 3-Euclid, Discrete Mathematics 308 (2008), 1130–1136. [6] J. H. Conway, On Numbers and Games, 2nd edn., A. K. Peters, Ltd., 2001. [7] T. Lengyel, A Nim-type game and continued fractions, Fibonacci Quart. 41 (2003), 310– 320. [8] O. Perron, Die Lehre von den Kettenbrüchen, B.G. Teubner, Leipzig and Berlin, 1913. [9] A. N. Siegel, Combinatorial Game Suite, http://cgsuite.sourceforge.net/, 2000, A software tool for investigating games, last accessed 6/2013. [10] W. Stromquist and D. Ullman, Sequential compounds of combinatorial games, Theoret. Comput. Sci. 119 (1993), 311–321.

Author information Neil A. McKay, Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada. Email: [email protected] Richard J. Nowakowski, Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada. Email: [email protected]

Combinatorial Number Theory, 137–154

© De Gruyter 2013

Chapter 10

Lecture Hall Partitions and the Wreath Products Ck o Sn Thomas W. Pensyl and Carla D. Savage Abstract. It is shown that statistics on the wreath product groups, Ck oSn , can be interpreted in terms of natural statistics on lecture hall partitions. Lecture hall theory is applied to prove distribution results for statistics on Ck o Sn . Finally, some new statistics on Ck o Sn are introduced, inspired by lecture hall theory, and their distributions are derived. Keywords. Partitions, Lecture Hall, Wreath Product Group. Mathematics Subject Classification 2010. 05A17, 52B11.

10.1

Introduction

The purpose of this note is to show that statistics on the wreath product Ck o Sn of a cyclic group Ck , of order k, and the symmetric group Sn , can be interpreted in terms of natural statistics on lecture hall partitions. We demonstrate that lecture hall theory can be used to prove results about the distribution of statistics on Ck oSn . We introduce some new statistics on Ck oSn , inspired by lecture hall partitions, including a quadratic version of “flag-major index”, and prove distribution results for these statistics. This chapter is organized as follows. In Section 10.2, we define the s-lecture hall partitions and state a few useful results. Section 10.3 is devoted to statistics of interest on the wreath product groups and a very brief discussion of what is known. Section 10.4 introduces s-inversion sequences, which will be used to relate statistics on Ck o Sn to statistics on lecture hall partitions. Section 10.5 describes a bijection between .k; 2k; : : : ; nk/-inversion sequences and Ck o Sn that allows statistics to be translated from one domain to another. Section 10.6 reviews recent work of Savage and Schuster [13] relating inversion sequences to lecture hall partitions. This work was developed with the intention of extending work on permutation statistics to a more general setting. Section 10.7 is the heart of the chapter. We prove there a theorem which allows us to apply the tools of Section 10.6 to Ck o Sn . This contains our main results relating statistics such as descent, flag-major index and flag-inversion number to statistics on lecture hall partitions, also proving an Euler–Mahonian distribution result.

138

Thomas W. Pensyl and Carla D. Savage

In Section 10.8 we define a new statistic “lhall ” on Ck oSn and derive its surprisingly nice distribution. In Section 10.9, we are led to define a distorted version of the descent statistic on Ck o Sn , that reveals an even closer connection to lecture hall partitions. A few words about notation: Z is the set of integers, R the set of real numbers, Sn the set of permutations of n elements; Œ j  D ¹1; 2; : : : ; j º, where Œ 0  D ;; Œ n q D .1  q n /=.1  q/; and for x D .x1 ; x2 ; : : : ; xn /, jxj D x1 C x2 C    C xn .

10.2 Lecture Hall Partitions For a sequence s D ¹si ºi1 of positive integers, the s-lecture hall partitions are the elements of the set ² ³ ˇ 2 n 1 .s/ n ˇ     : Ln D  2 Z ˇ 0  s1 s2 sn .1;2;3;4/ , but .0; 1; 3; 4/ 62 L.1;3;5;7/ , since 3=5 > 4=7. For example, .0; 1; 3; 4/ 2 Ln n .1;2;:::;n/ The original lecture hall partitions Ln D Ln were introduced by BousquetMélou and Eriksson in [3], where they showed that

X

y jj D

n Y iD1

2Ln

1 : 1  y 2i1

(10.1)

In [4] they proved the following refinement, which will be useful in the present work. Theorem 10.1 (Refined lecture hall theorem [4]). For any nonnegative integer n, X

q

jdej jj

y

D

n Y iD1

2Ln

1 C qy i ; 1  q 2 y nCi

(10.2)

where de D .d1 =1e ; d2 =2e ; : : : ; dn =ne/. If the largest part in a lecture hall partition in Ln is constrained, we have the following: Theorem 10.2 ([8, 13]). For integers n  1 and t  0, X 2Ln I n tn

q jdej D Πt C 1 nq :

(10.3)

Chapter 10 Lecture Hall Partitions and the Wreath Products Ck o Sn

139

For example, when n D 3 and t D 1, the set ¹ 2 L3 j 3  3º has the eight elements ® ¯ .0; 0; 0/; .0; 0; 1/; .0; 0; 2/; .0; 0; 3/; .0; 1; 2/; .0; 1; 3/; .0; 2; 3/; .1; 2; 3/ and

X

q d1 =1eCd2 =2eCd3 =3e D 1 C 3q C 3q 2 C q 3 D Π2 3q :

2L3 I 3 3

10.3

Statistics on Ck o Sn

An element  2 Sn is a bijection  W Œ n  ! Œ n , and we write  D .1 ; : : : ; n /, to mean that .i / D i . A descent in  2 Sn is a position i 2 Œ n  1  such that i > iC1 . The set of all descents of  is Des  and des  D jDes j. The inversion number of  is ˇ ˇ inv  D ˇ¹.i; j / j 1  i < j  n and i > j ºˇ: For example, if  D .5; 4; 1; 3; 2/, then Des  D ¹1; 2; 4º, des  D 3 and inv  D 8. For positive integers k and n, we view Ck o Sn combinatorially as a set of pairs .; /: ® ¯ Ck o Sn D .; / j  2 Sn ;  2 ¹0; 1; : : : ; k  1ºn : We use the notation  to denote .; / and write  D .1 1 ; 2 2 ; : : : ; n n / D ..1 ; : : : ; n /; .1 ; : : : ; n // D .; /: Statistics on Ck o Sn (or k-colored permutations or k-indexed permutations) have been studied by many, starting with Reiner’s work on signed permutations [12], followed by independent work of Brenti [5] and Steingrímsson [14] on the more general wreath products. Pairs of (“descent, major index”) statistics have been found, satisfying relations like Carlitz’s q-Eulerian polynomials, starting with work of Adin, Brenti, and Roichman [1]. There have very recently been many new and exciting discoveries, including [2, 7, 9, 10]. It is remarkable how many variations in the definitions of the statistics there are, even when they give the same distribution. We start with a fairly standard definition of descent. The descent set of  2 Ck oSn is ® ¯ Des  D i 2 ¹0; 1; : : : ; n  1º j i < iC1 ; or i D iC1 and i > iC1 ; (10.4) with the convention that 0 D 0 D 0.

140

Thomas W. Pensyl and Carla D. Savage

We will consider the following statistics defined on Ck o Sn . des  D jDes  j X .n  i / comaj  D i2Des  

fmaj  D k comaj   finv  D inv  C

n X

n X

i

iD1

i i :

iD1

As an example, for  D .51 ; 41 ; 10 ; 30 ; 22 / 2 C3 o S5 , we have Des  D ¹0; 1; 4º; des  D 3; comaj  D 10; fmaj  D 26; and finv  D 21. Note that this definition of fmaj differs a bit from those appearing elsewhere, even among those who define the descent set as in Equation(10.4) [1, 7]. Using lecture hall theory, we will show, among other things, X X   q fmaj  D q finv  ; (10.5)   2Ck oSn

X

Œkt C

  2Ck oSn

P

1 nq x t

D

t0

X

q

P

jdej dn =.k n/e

x

D

2Ln

fmaj   x des     2Ck oSn q Qn ki iD0 .1  xq /

;

(10.6)

fmaj   x des     2Ck oSn q Qn ki iD1 .1  xq /

:

(10.7)

Relations of the form (10.6), for general k, have been found only recently, starting with Chow and Mansour [7] and Hyatt [10], sometimes with slightly different definitions of Des or fmaj . Our intention here is to highlight our methods, which are quite novel, and which allow us to prove new results like Equation (10.7).

10.4 Statistics on s-Inversion Sequences The connection between statistics on Ck o Sn and statistics on lecture hall partitions will be made via statistics on inversion sequences. Given a sequence s D ¹si ºi1 of positive integers, and positive integer n, the set .s/

In of s-inversion sequences is defined by D I.s/ n

®

¯ .e1 ; : : : ; en / 2 Zn j 0  ei < si for 1  i  n :

The familiar “inversion sequences” associated with permutations are the elements of .s/ In for s D .1; 2; : : : ; n/.

Chapter 10 Lecture Hall Partitions and the Wreath Products Ck o Sn

141

The ascent set of an inversion sequence e 2 I.s/ n is the set ³ ² ˇe eiC1 ˇ i ; < Asc e D i 2 ¹0; 1; : : : ; n  1º ˇ si siC1 with the convention that e0 D 0 and s0 D 1. For example, as an element of .3;6;9;12;15/ I5 , the inversion sequence e D .1; 3; 2; 2; 13/ has the ascent set Asc e D ¹0; 1; 4º. .s/ The following statistics on In were defined in [13]: ˇ ˇ asc e D ˇAsc e ˇ; X .n  i /; amaj e D jej D

i2Asc e n X

ei ;

iD1

lhp e D jej C

X

.siC1 C ::: C sn / :

i2Asc e .3;6;9;12;15/ , we have asc e D 3, amaj e D 10, jej D 21, For e D .1; 3; 2; 2; 13/ 2 I5 and lhp e D 81. In this chapter, our focus is the sequence s D .k; 2k; : : : ; nk/, where k is a positive .k;2k;:::;nk/ . We will require two new statistics on In;k : integer. Let In;k D In

N.e/ D

n X ej j D1

j

;

Ifmaj e D k amaj e  N.e/: .3;6;9;12;15/

For e D .1; 3; 2; 2; 13/ 2 I5

10.5

, N.e/ D 4 and Ifmaj e D 26.

From Statistics on Ck o Sn to Statistics on In;k

We will make use of the following bijection between Sn and In;1 which was proved in [13] to have the required properties. Lemma 10.3. For positive integer n, the mapping  W Sn ! In;1 defined by ./ D t D .t1 ; t2 ; : : : ; tn /, where ˇ ˇ ti D ˇ¹j 2 Œi  1 j j > i ºˇ is a bijection satisfying both Des  D Asc t and inv  D jt j.

142

Thomas W. Pensyl and Carla D. Savage

For example, if  D .5; 4; 1; 3; 2/, then t D ./ D .0; 1; 2; 2; 3/ 2 I5;1 . Checking the statistics, Des  D ¹1; 2; 4º D Asc t and inv  D 8 D jt j. Noting that, as sets, In;k and Ck o Sn have the same cardinality, we set up a bijection which translates statistics from one domain to the other in a useful way. Theorem 10.4. For each pair of integers .n; k/ with n  1, k  1, there is a bijection ‚ W Ck o Sn ! In;k with the following properties. If ‚. / D e D .e1 ; : : : ; en / then Asc e D Des  ; N.e/ D

n X

(10.8)

i ;

(10.9)

iD1

Ifmaj e D fmaj  ;

(10.10)

en D n.n C 1/  n ; jej D inv  C

n X

i i D finv  :

(10.11) (10.12)

iD1

Proof. Define ‚ by e D ‚.1 1 ; 2 2 ; : : : ; n n / D .1 C t1 ; 22 C t2 ; : : : ; nn C tn /; where .t1 ; t2 ; : : : ; tn / D ./, as in Lemma 10.3. For example, for  D .51 ; 41 ; 10 ; 30 ; 22 / 2 C3 o S5 , t D .5; 4; 1; 3; 2/ D .0; 1; 2; 2; 3/, so we get e D ‚. / D .1; 3; 2; 2; 13/. Note that properties (10.8) through (10.12) hold for this example: Asc e D ¹0; 1; 4º D Des  ; N.e/ D 4 D 1 C 1 C 0 C 0 C 4 D jj; Ifmaj e D 26 D fmaj  ; e5 D 13 D 5.5 C 1/  5 ; jej D 21 D finv  : Clearly, ‚. / 2 In;k . Since Ck o Sn and In;k have the same cardinality, to show that ‚ is a bijection, it suffices to show that ‚ is onto. Let e D .e1 ; : : : ; en / 2 In;k . Define  D .1 ; : : : n / by i D bei =i c. Then  2 ¹0; 1; : : : ; k  1ºn . Define t D .t1 ; : : : tn / by ti D ei  i i . Then t 2 In;1 . Finally, let  D  1 .t / 2 Sn . Then  2 Ck o Sn and ‚1 .e/ D  . To prove properties (10.8) through (10.12), observe first that tn D n  n , so property (10.11) holds. It is clear from the definition of ‚ that Equation (10.12) is true.

Chapter 10 Lecture Hall Partitions and the Wreath Products Ck o Sn

143

Also, note that bei =i c D i since 0  ti < i and property (10.9) holds. So property (10.10) will follow once we prove Equation (10.8). By Lemma 10.3, since t D ./, we know that Asc t D Des , so it remains to show Asc e D Des  . Note first that e1 D 1 C t1 D 1 , since t1 D 0. Thus, 0 2 Des  ” 1 > 0 ” e1 > 0 ” 0 2 Asc e: For 1  i  n, i 2 Asc e if and only if 0<

ei .i C 1/iC1 C tiC1 i i C ti eiC1  D  k.i C 1/ ki k.i C 1/ ki i.i C 1/.iC1  i / C i tiC1  .i C 1/ti ki.i C 1/ i ; D ki.i C 1/

D

where i D i.i C 1/.iC1  i / C i tiC1  .i C 1/ti : Therefore, i 2 Asc e if and only if i > 0. If i D iC1 , then i > 0 ” i tiC1  .i C 1/ti > 0 ” i 2 Asc t ” i 2 Des  ” i 2 Des  : For the remaining cases, note that since 0  tiC1  i and 0  ti  i  1, i.i C 1/.iC1  i /  i 2 C 1  i  i.i C 1/.iC1  i / C i 2 : If i 6D iC1 , then i 2 Des  if and only if i < iC1 . But if i < iC1 , then i  i.i C 1/  i 2 C 1 D i C 1 > 0; and so i 2 Asc e. And, if i > iC1 , then i  i.i C 1/ C i 2 D i  0 and i 62 Asc e. This completes the proof.

10.6

Lecture Hall Polytopes and s-Inversion Sequences

The s-lecture hall polytope was introduced in [13], for an arbitrary sequence s D ¹si ºi1 of positive integers, as ² ³ ˇ 2 n 1 .s/ n ˇ Pn D  2 R ˇ 0      1 : s1 s2 sn

144

Thomas W. Pensyl and Carla D. Savage

P.s/ n is a convex, simplicial polytope with the n C 1 vertices: .0; 0; : : : ; 0/; .s1 ; s2 ; : : : ; sn /; .0; s2 ; : : : ; sn /; .0; 0; s3 ; : : : ; sn /; : : : ; .0; 0; : : : ; 0; sn /; .s/

all with integer coordinates. The t -th dilation of Pn is the polytope ² ³ ˇ 2 n 1 n ˇ D  2 R        t : t P.s/ 0  ˇ n s1 s2 sn A multivariate function, fn.s/ .t I q; y; z/, was used in [13] to enumerate lattice points in .s/ t Pn according to statistics significant in the theory of lecture hall partitions: X C q jdes j y jj z j  ./ j ; fn.s/ .t I q; y; z/ D n 2tP.s/ n \Z

where



     1 2 n ; ;:::; ; des D s1 s2 sn         1 2 n  1 ; s2  2 ; : : : ; sn  n :  C ./ D s1 s1 s2 sn

(10.13) (10.14)

The following theorems show the connection between statistics on s-inversion sequences and statistics on s-lecture hall partitions. Theorem 10.5 ([13]). For any sequence s of positive integers, and any positive integer n, P x asc e q amaj e y lhp e z j e j X e2I.s/ .s/ t n fn .t I q; y; z/ x D Qn : (10.15) ni y si C1 CCsn / iD0 .1  xq t0 Theorem 10.6 ([13]). For any sequence s of positive integers, and any positive integer n, P x asc e q amaj e y lhp e z j e j X e2I.s/ jdejs jj j  C ./ j dn =sn e q y z x D Qn1n : (10.16) ni y si C1 CCsn / .s/ iD0 .1  xq 2Ln

145

Chapter 10 Lecture Hall Partitions and the Wreath Products Ck o Sn

10.7

Lecture Hall Partitions and the Inversion Sequences In;k

In order to apply the results of the previous section to the problem of interest, we need an analog of Ifmaj on In;k for lecture hall partitions. First, observe that the following sets of lecture hall partitions are all the same: D L.2;4;:::;2n/ D L.3;6;:::;3n/ D  : Ln D L.1;2;:::;n/ n n n However, the lecture hall polytopes Pn;k defined by ² ³ ˇ 1 2 n ˇ     1 Pn;k D  2 Rn ˇ 0  k 2k nk are different for different k. On the other hand, the following dilations are the same: t Pn;k D k t Pn;1 ;

(10.17)

a fact which we will exploit. Furthermore, k t Pn;1 \ Zn D ¹ 2 Ln j n  k t nº: Since the definitions (10.13) and (10.14) depend on the sequence s D .k; 2k; : : : ; nk/, we will make the dependence explicit in the notation. For  2 Ln and k  1, let       1 2 n ; ;:::; ; (10.18) dek D k 2k nk         1 2 n  1 ; 2k  2 ; : : : ; nk  n ; (10.19) kC ./ D k k 2k nk k ./ D k dek  de :

(10.20)

Note that for  2 Ln , de1 D de ; where de was defined in Theorem 10.1. We now show that the new statistic k on Ln corresponds to the statistic N on In;k . Theorem 10.7. For positive integers n; k, let X C q jdek j y jj z j k ./j w j k ./ j : fn;k .t I q; y; z; w/ D

(10.21)

2tPn;k \ Zn

Then, X t0

P t

e2In;k

fn;k .t I q; y; z; w/ x D Qn

iD0 .1

x asc e q amaj e y lhp e z j e j w N.e/  xq ni y k.n.nC1/i.iC1//=2 /

:

(10.22)

146

Thomas W. Pensyl and Carla D. Savage

Proof. If w D 1, this is just the case s D .k; 2k; : : : ; nk/ of Theorem 10.5. To include w, we appeal to the combinatorial proof of (10.15) in Theorem 10.5 which was presented in [13]. In that proof,  2 .t Pn;k \Zn / is associated with the inversion sequence kC ./, which, by definition, is in In;k . It suffices to check that jk ./j D N.kC .//:

n X i k di =.i k/e  i C N.k .// D i D D

iD1 n X iD1 n X

bk di =.i k/e  i =i c .k di =.i k/e  di =i e/

iD1

D jk dek  de1 j D jk ./j: The Ifmaj statistic is obtained by setting q D q k and w D q 1 in Theorem 10.7. Corollary 10.8. For positive integers n; k, X t0

X

C

q jdej y jj z j k ./j x t D P 2ktPn;1 \ Zn Qn

e2In;k

iD0 .1

x asc e q Ifmaj e y lhp e z j e j

 xq k.ni/ y k.n.nC1/i.iC1//=2 /

: (10.23)

Proof. With q D q k and w D q 1 , the numerator in the right-hand side of Equation (10.22) becomes x asc e q k amaj eN.e/ y lhp e z j e j D x asc e q Ifmaj e y lhp e z j e j : From (10.21), the left-hand side summand of Equation (10.22) becomes X C q kjdek jj k ./ j y jj z j k ./j fn;k .t I q k ; y; z; q 1 / D 2tPn;k \ Zn

X

D

C

q jdej y jj z j k ./j

2ktPn;1 \ Zn

by (10.18)–(10.20) and by (10.17). Corollary 10.9. For positive integers n; k, X 2Ln

q

jdej jj j kC ./j

y

z

x

dn =.nk/e

P

D Qn1

e2In;k

iD0 .1

x asc e q Ifmaj e y lhp e z j e j

 xq k.ni/ y k.n.nC1/i.iC1//=2 /

:

(10.24)

147

Chapter 10 Lecture Hall Partitions and the Wreath Products Ck o Sn

P C jde1 j y jj z j k ./j from Equation Proof. For t > 0, let H.t / D 2ktPn;1 \ Zn q (10.23), with H.0/ D 1. Then for t > 0, since       ³ ² ³ ² ³ ² n n n D t D  2 Ln I  t   2 Ln I t 1  2 Ln I nk nk nk     D k t Pn;1 \ Zn  k.t  1/Pn;1 \ Zn ; we have X

C

q jdej y jj z j k ./j x dn =.nk/e D

X

X

xt

l

t0

2Ln

X

D1C

2Ln I

C

n nk

q jdej y jj z j k ./j

m Dt

.H.t /  H.t  1//x t

t1

X

D1C D

X

H.t /x t 

t1

D .1  x/

H.t  1/x t

t1

t

H.t /x  x

t0

X

X

X

H.t /x t

t0

H.t /x t :

t0

P

But t0 H.t /x t is the left-hand side of Equation (10.23), so we simply multiply the right-hand side of Equation (10.23) by .1  x/ to complete the proof. We can now apply these results to the wreath product groups. First, we have the expected result that the pair .des ; fmaj / is Euler–Mahonian. Theorem 10.10. For positive integers n; k, P fmaj   x des   X   2Ck oSn q n t Œ k t C 1 q x D : Qn ki iD0 .1  xq / t0 Proof. Set y D z D 1 in Equation (10.23). On the left-hand side, in the summand, we get X q jdej : 2ktPn;1 \ Zn

Since k t Pn;1 \ Zn D ¹ 2 Ln j n  k t nº, by Theorem 10.2, X q jdej D Œ k t C 1 nq : 2ktPn;1 \ Zn

For the right-hand side, we get

P e2In;k

Qn

iD0 .1

x asc e q Ifmaj e  xq k.ni/ /

:

148

Thomas W. Pensyl and Carla D. Savage

Reindex the product in the denominator and for the numerator, using the fact that by Theorem 10.4, the distribution of .des ; fmaj / on Ck oSn is the same as the distribution of .asc ; Ifmaj / on In;k . Now, to interpret the distribution .des ; fmaj ; finv / on Ck o Sn in terms of lecture hall partitions, set y D 1 in Equation (10.24) and use Theorem 10.4. Theorem 10.11. For positive integers n; k, P fmaj   x des   z finv   X   2Ck oSn q jdej j kC ./j dn =.nk/e q z x D : Qn ki iD1 .1  xq / 2Ln

The implication of Theorem 10.11 for z D 1 is quite interesting. We have P fmaj   x des   X   2Ck oSn q jdej dn =.nk/e q x D : (10.25) Qn ki iD1 .1  xq / 2Ln

In the left-hand side of equaqtion (10.25), the only dependence on k is in the exponent of x, in a statistic involving only the last part of . We take this further in Section 10.9.

10.8 A Lecture Hall Statistic on Ck o Sn From the point of view of partition theory, the most important statistic for a lecture hall partition  is the number jj D 1 C    C n being partitioned. So, what does jj correspond to on Ck o Sn ? In [6], a quadratic version of the major index was defined on Sn by bin  D iC1 P i2Des  2 : In that spirit, we define “cobin ” on Ck o Sn by ! !! X nC1 i C1 cobin  D  : 2 2  i2Des 

Now define the statistic “lhall ” on Ck o Sn by lhall  D k cobin   finv  : Observe that under the bijection ‚ of Theorem 10.4, if e D ‚. / then lhall  D lhp e. This can be seen as follows, since jej D finv  and Asc e D Des e: X lhp e D jej C .k.i C 1/ C    C k n/ i2Asc e

D jej C k

X i2Asc e

! !! nC1 i C1  2 2

149

Chapter 10 Lecture Hall Partitions and the Wreath Products Ck o Sn



D finv  C k

! !! nC1 i C1  2 2

X i2Des e



D finv  C kcobin  D lhall  : The joint distribution of .lhall ; fmaj / on Ck o Sn has the following form. Theorem 10.12. For positive integers n; k, X

y

lhall   fmaj  

q

  2Ck oSn

n Y .1 C qy i /.1  q k.nC1i/ y k.iCCn/ D 1  q 2 y nCi iD1

D

dn=2e Y

Πk.2i  1/ qy nC1i

iD1

bn=2c Y

.Π2 qy i Πki q 2 y 2.ni /C1 /

iD1

Proof. Under the bijection ‚ of Theorem 10.4, if e D ‚. /, then lhall  D lhp e and fmaj  D Ifmaj e. Therefore, X X   y lhall  q fmaj  D y lhp e q Ifmaj e :   2Ck oSn

e2In;k

So, by Corollary 10.9 with x D z D 1, P P lhp e q Ifmaj e lhall   q fmaj   e2In;k y   2Ck oSn y D Qn1 Qn1 k.ni/ y k.n.nC1/i.iC1//=2 / k.ni/ y k.n.nC1/i.iC1//=2 / iD0 .1  q iD0 .1  q X D y jj q jdej : 2Ln

Now apply Theorem 10.1 to get P n lhall   q fmaj   Y 1 C qy i   2Ck oSn y D : Qn1 k.ni/ y k.n.nC1/i.iC1//=2 / 1  q 2 y nCi iD0 .1  q iD1 So, X

y

  2Ck oSn

lhall   fmaj  

q

D

n Y

.1q

k.niC1/ k.n.nC1/i.iC1//=2

y

iD1

iD1

which, after simplification, gives the theorem. Setting y D 1 in Theorem 10.12 and simplifying, we get X   2Ck oSn

/

n Y

q

fmaj  

D

n Y iD1

Πki q ;

1 C qy i ; 1  q 2 y nCi

150

Thomas W. Pensyl and Carla D. Savage

the same distribution as finv , Ifmaj , and jej, as expected. But the statistic lhall itself also has a surprisingly simple distribution: Theorem 10.13. For positive integers n; k, X

q

  2Ck oSn

lhall  

D

n Y

Πki q 2.ni /C1 :

iD1

Proof. Set q D 1 and y D q in the proof of the Theorem 10.12, but apply Equation (10.1) instead of (10.2) to get X

q lhp e D

.1;2;:::;n/

n Y 1  q k.iCCn/ 1  q 2i1

iD1

e2In

D

bn=2c Y iD1

D

n Y

dn=2e 1  q k.2i1/.niC1/ Y 1  q ki.2.ni/C1/ 1  q 2i1 1  q 2.ni/C1 iD1

Πki q 2.ni /C1 :

iD1

10.9 Inflated Eulerian Polynomials for Ck o Sn We showed in [11] how to obtain more refined information about the s-lecture hall .s/ partitions by considering the rational lecture hall polytope Rn : ² ³ ˇ 2 n 1 .s/ n ˇ     and n  1 : Rn D  2 R ˇ 0  s1 s2 sn R.s/ n is a convex simplicial polytope, whose vertices are    s2 s1 s2 sn sn ; 0; ; : : : ; ; .0; 0; : : : ; 0/ ; ; ;:::; sn sn sn sn sn     sn s3 sn 0; 0; ; : : : ; ; : : : ; 0; 0; : : : ; 0; ; sn sn sn 

with rational (but not necessarily integer) coordinates. Let X C q jdejk y jj z j k ./ j : gn.s/ .t I q; y; z/ D

(10.26)

n 2tR.s/ n \Z

The following theorems were proved in [11]. These are analogs of Theorems 10.5 and 10.6.

Chapter 10 Lecture Hall Partitions and the Wreath Products Ck o Sn

151

Theorem 10.14 ([11]). For any sequence s of positive integers, and positive integer n, P q amaj e y lhp e z j e j x sn asc een X e2I.s/ .s/ t n gn .t I q; y; z/x D : Q sn ni y si C1 CCsn / .1  x/ n1 iD0 .1  x q t0

Theorem 10.15 ([11]). For any sequence s of positive integers, and positive integer n, P q amaj e y lhp e z j e j x sn asc een X e2I.s/ jdejs jj j sC ./ j n n q y z x D Qn1 : sn ni y si C1 CCsn / .s/ iD0 .1  x q 2Ln

We can specialize Theorems 10.14 and 10.15 to s D .k; 2k; : : : ; nk/ and modify to track Ifmaj as in Theorem 10.7 and its corollaries. We should expect something interesting because D R.2;4;:::;2n/ D R.3;6;:::;3n/  : R.1;2;:::;n/ n n n We get the following theorem, which is an analog of Theorem 10.7. The proof, which is analog to that of Theorem 10.6, is omitted. Theorem 10.16. For positive integers n; k, let X C gn;k .t I q; y; z; w/ D q jdek j y jj z j k ./j w j k ./ j : 2tRn

Then X

(10.27)

\ Zn

P t

gn;k .t I q; y; z; w/ x D

t0

.1 

k nasc een q amaj e y lhp e z j e j w N.e/ e2In;k x : Qn1 x/ iD0 .1  x k n q ni y k.n.nC1/i.iC1//=2 /

(10.28)

The following corollaries of Theorem 10.16 are analogs of Corollaries 10.8 and 10.9 with y D z D 1. Note that in the left-hand sides of the equations there is no dependence on k. Corollary 10.17. For positive integers n; k, P k nasc een q Ifmaj e X X e2In;k x jdej t : q x D Q k n k.ni/ / .1  x/ n1 iD0 .1  x q t0 2tR \ Zn n

152

Thomas W. Pensyl and Carla D. Savage

Corollary 10.18. For positive integers n; k, X

P q

jdej n

x

2Ln

D

k nasc een q Ifmaj e e2In;k x : Qn1 k n k.ni/ / iD0 .1  x q

Making use of Theorem 10.4 giving the correspondence between statistics on In;k and on Ck o Sn , we have the following analogs of Theorems 10.10 and 10.11. First, We need a result from [8]. Lemma 10.19 ([8]). For integers t  0 and n > 0, let j and i be the unique integers satisfying t D j n C i , where j  0 and 0  i < n. Then, X q jdej D Πj C 1 ni Πj C 2 iq : q 2tRn \Zn

Theorem 10.20. For positive integers n; k, P fmaj   x n.k des   1 n /Cn X X n1   2Ck oSn q ni i nj Ci : Q Πj C 1 q Πj C 2 q x D .1  x/ niD1 .1  x k n q ki / j 0 iD0 Proof. By Lemma 10.19, X n1 X j 0 iD0

Πj C 1 ni Πj C 2 iq x nj Ci D q

X X n1

X

q jdej x j nCi :

j 0 iD0 2.j nCi/Rn \ Zn

Since every t  0 can be written uniquely as t D j n C i for nonnegative integers j and i with i < n, the last expression can be rewritten as X X q jdej x t ; t0 2tRn \ Zn

which, by Corollary 10.17, is equal to P k nasc een q Ifmaj e e2In;k x Qn : k n k.ni/ / iD0 .1  x q Under the bijection ‚ of Theorem 10.4, if e D ‚. /, then Ifmaj e D fmaj  , asc e D des  , and en D n.n C 1/  n . The result follows then, since k n asc e  en D k n des   n.n C 1/ C n :

153

Chapter 10 Lecture Hall Partitions and the Wreath Products Ck o Sn

Theorem 10.21. For any positive integers n; k, P fmaj   x n.k des   1 n /Cn X   2Ck oSn q jdej n q x D Qn : k n ki iD1 .1  x q / 2Ln

Proof. Start from Corollary 10.18 and apply Theorem 10.4. (Note: there is no dependence on k in the left-hand side). Let Qn;k .x/ be the q D 1 specialization: Qn;k .x/ D

X   2C

x n.k des 

 1

n /Cn

:

k oSn

The Qn;k .x/ are referred to as inflated Eulerian polynomials in [11]. To contrast the usual, Eulerian polynomials for Ck o Sn are X

En;k .x/ D



x des  :

  2Ck oSn

It is interesting that Qn;k .x/ is self-reciprocal, but in general En;k .x/ is not when k > 2.

10.10

Concluding Remarks

It is interesting from the results in Sections 10.7– 10.9 that for fixed n, statistics on Ck o Sn such as descent, flag-major index, and flag-inversion number appear naturally in the geometry of the same simplicial cone, Rn , independent of k. It would be interesting to see to what extent other statistics on Ck o Sn can be interpreted in terms of lecture hall partitions. Different orderings on Ck o Sn and different bijections Ck o Sn ! In;k would give different results. Lecture hall partitions were discovered in the setting of affine Coxeter groups, and Theorem 10.1 was inspired by Bott’s formula. It should be possible to trace through backwards to discover the algebraic significance of the statistic lhall , at least in the Coxeter groups An D C1 o Sn or Bn D C2 o Sn but we have not seen how to do this. Acknowledgments. The second author would like to thank Matthias Beck, Benjamin Braun, and Mattias Koeppe for discussions on Ehrhart theory and signed permutations. Thanks also to the American Institute of Mathematics, where some of those discussions were hosted. We are grateful to the referee for a careful reading of the manuscript and for many helpful suggestions to improve the presentation.

154

Thomas W. Pensyl and Carla D. Savage

References [1] Ron M. Adin, F. Brenti, and Y. Roichman, Descent numbers and major indices for the hyperoctahedral group, Adv. in Appl. Math. 27(2–3) (2001) 210–224, special issue in honor of Dominique Foata’s 65th birthday (Philadelphia, PA, 2000). [2] R. Biagioli and J. Zeng, Enumerating wreath products via Garsia–Gessel bijections, European J. Combin. 32(4) (2011) 538–553. [3] M. Bousquet-Mélou and K. Eriksson, Lecture hall partitions. Ramanujan J. 1(1) (1997), 101–111. [4] M. Bousquet-Mélou and K. Eriksson, A refinement of the lecture hall theorem, J. Combin. Theory Ser. A 86(1) (1999), 63–84. [5] F. Brenti. q-Eulerian polynomials arising from Coxeter groups, Eur. J. Comb. 15 (1994), 417–441. [6] K. L. Bright and C. D. Savage, The geometry of lecture hall partitions and quadratic permutation statistics, in: DMTCS Proceedings, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), San Francisco, (AN), pp. 569, 580, 2010. [7] C.-O. Chow and T.Mansour, A Carlitz identity for the wreath product Cr o Sn , Adv. in Appl. Math. 47(2) (2011), 199–215. [8] S. Corteel, S. Lee, and C. D. Savage, Enumeration of sequences constrained by the ratio of consecutive parts, Sém. Lothar. Combin. 54A (2005/07), Art. B54Aa, (electronic). [9] H. L. M. Faliharimalala and A. Randrianarivony, Flag-major index and flag-inversion number on colored words and wreath product, Sém. Lothar. Combin. 62 (2009/10), Art. B62c, 10. [10] M. Hyatt, Quasisymmetric functions and permutation statistics for Coxeter groups and wreath product groups, . Ph.D. Thesis, University of Miami, 2011. [11] T. W. Pensyl and C. D. Savage, Rational lecture hall polytopes and inflated Eulerian polynomials, Ramanujan J. (2012), DOI 10.1007/s11139-012-9393-7, to appear. [12] V. Reiner, Signed permutation statistics, European J. Combin. 14(6) (1993), 553–567. [13] C. D. Savage and M. J. Schuster, Ehrhart series of lecture hall polytopes and Eulerian polynomials for inversion sequences, J. Combin. Theory Ser. A 119 (2012), 850–870. [14] E. Steingrímsson, Permutation statistics of indexed permutations, European J. Combin. 15(2) (1994), 187–205.

Author information Thomas W. Pensyl, Department of Computer Science, North Carolina State University, Raleigh, North Carolina, USA. Email: [email protected] Carla D. Savage, Department of Computer Science, North Carolina State University, Raleigh, North Carolina, USA. Email: [email protected]

Index

admissible set, 39 affine Coxeter Groups, 153 alternating game, 1 AP -set, 19 atomic weight, 135 Bhargava, M. 58–62, 67–70 Bott’s formula, 153 bounded Wythoff game, 93 Bush, Albert, 52 C -set, 18 C  -set, 18 Central Sets Theorem, 17 central, 16 central*, 16 circle group, 25 compositions, 79 Conjecture H, 38, 39, 42 conjugate, 3 cubic forms, 60–61 -set, 16  -set, 16 Diananda, P. H., 53 difference set, 48 dilation, 144 distinguishable, 3 Erd˝os, P., 53 Euler-Mahonian, 147 Eulerian polynomials, 153 f˛ , 25 farthest continued fraction, 123 Fibonacci numbers, 80, 109, 123 function, constant, 91, 93, 98, 106 convex, 103, 104 mex, 92, 96 polynomial 91, 100, 103, 106

polynomial time computable, 101 Sprague-Grundy, 106, 107 superadditive, 91, 92, 99, 100, 101, 102, 103, 106 function fields, 73–74 g˛; , 15 game tree, 124, 125 geometry of numbers, 67–70 Green, B., 53, h˛ , 19 Heegner points, 70–73 Hermite’s theorem, 56–57 Hirzebruch surfaces, 73–74 Hough, B., 70–73 ideal class groups, 59–60, 70–73 ideals, 15 preserved, 33 if and only if, 28 impartial games, 92, 106, 107 inclusion-exclusion, 37, 39, 40 indistinguishable, 3 inflated Eulerian polynomials, 150, 153 intractable, 92 IP , 15 irreducible signature, 131 word, 130 J -set, 18 preserved, 27 J  -set, 18 J.S /, 18 ideal, 28 jumping champions, 38

156 k-colored permutations, 139 k-indexed permutations, 139 Kedlaya, K. S., 53 lattice points, 144 lecture hall polytopes, 143 statistics on wreath products (cobin, lhall), 148 left-handed game, 2, 5 MacMahon 79, 86 Maroni invariant, 73–74 Mertens’ theorem, 52 Minkowski’s theorem, 56–57 misère equivalence, 3 inequality, 3 play, 91 multiplicative independence, 111 order, 109 n-colored compositions, 79 N -position, 92, 98, 99, 103, 106 nearest continued fraction, 123 nonhomogeneous, 15 normal play, 91, 92 number fields, counting, 56–59 numeration system, 93, 94, 95, 97, 98, 101, 107 one-handed game, 2, 6, 7 outcome classes, 2, 129 outcomes, 10 paired outcomes, 129 P -position, 92, 93, 95, 96, 98, 99, 100, 101, 103, 105, 106 PENNY NIM, 2, 5 periodic, periodicity, 93, 99 additive, 99, 106, 107 arithmetic, 99 piecewise syndetic, 18 Poisson density, 38 distribution, 37 prehomogeneous vector space, 58–59 preserved, 27, 32

Index

prime k-tuple conjecture, 37, 38, 39 prime number theorem, 37, 39 product-free, 45 Pryby, Chris 52 rational lecture hall polytope, 150 really complicated proof for simple facts, 58 reduced signature, 128 right-handed game, 2, 5 Ruzsa, I. Z., 52, 53 s-inversion sequences, 140 s-lecture hall partitions, 138 S C -set, 20 Schinzel, A., 53 self-reciprocal, 153 Shankar, A., 67–70 Shintani zeta functions, 63–67 signature, 127, 128 singular series, 38 average, 41 Solymosi, J., 53 spectrum, 15 spine, 127 spotted tilings, 80 statistics on lecture hall partitions, 145 on permutations (Des, des, inv), 139 on s-inversion sequences (Asc, asc, amaj, lhp, N, fmaj), 141 on wreath products (Des), 139 on wreath products (des, comaj, fmaj, finv), 140 Stirling’s formula, 42 strongly central, 26 succinct, 101 sum-free, 45 sum-product problem, 45 syndetic, 18, 23 Szemerédi, E., 53 take-away games, 91, 92 Taniguchi, T., 63–67 Taylor’s theorem, 41 Terquem’s problem, 80 thick, 20, 23 time complexity, 101

157

Index

tractable, 92, 93, 98, 101, 103, 105, 106 Tsimerman, J., 67–70 two-handed game, 2 upper asymptotic density, 46 Vandehey, Joseph, 52 wreath product Ck o Sn , 137

Wythoff’s game, 91, 92, 93, 101, 106, 107 Yap, H. P., 53 Z˛ , 25 Zhao, Y., 73–74 zig-zag graph, 86