Lattice Points (Pitman Monographs & Surveys in Pure & Applied Mathematics.)

Lattice Points

Main Editors

H. Brezis, Universite de Paris R. G. Douglas, State University of New York at Stony Brook A. Jeffrey, University of Newcastle-upon-Tyne (Founding Editor) Editorial Board

R. Aris, University of Minnesota A. Bensoussan, INRIA, France S. Bloch, University of Chicago B. Bollobas, University of Cambridge W. Burger, Universitat Karlsruhe S. Donaldson, University of Oxford J. Douglas Jr, University of Chicago R. J. Elliott, University of Alberta G. Fichera, Universite di Roma R. P. Gilbert, University of Delaware R. Glowinski, Universite de Paris K. P. Hadeler, Universitat Tiibingen K. Kirchgassner, Universitat Stuttgart B. Lawson, State University of New York at Stony Brook W. F. Lucas, Claremont Graduate School R. E. Meyer, University of Wisconsin-Madison J. Nitsche, Universitat Freiburg L. E. Payne, Cornell University G. F. Roach, University of Strathclyde J. H. Seinfeld, California Institute of Technology B. Simon, California Institute of Technology I. N. Stewart, University of Warwick S. J. Taylor, University of Virginia

I

Pitman Monographs and Surveys in Pure and Applied Mathematics 39

Lattice Points P. Erdos, P. M. Gruber & J. Hammer Hungarian Academy of Sciences/Technical University of Vienna/University of Sydney

Longman

NN Scientific& now Technical Copublished in the United States with ,lohn Wiley & Sons, Inc, New York

Longman Scientific & Technical Longman Group UK Limited Longman House, Burnt Mill, Harlow Essex CM20 2JE, England and Associated Companies throughout the world.

Copublished in the United States with John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158

© P. Erdos, P. M. Gruber and J. Hammer 1989 All rights reserved; no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise without either the prior written permission of the Publishers or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, 33-34 Alfred Place, London, WC1E 7DP. First published 1989 ISSN 0269-3666

AMS Subject Classifications. (Main) 11HXX, 52-XX, 05CXX, (Subsidiary) 82A60, 53C65, 65030 British Library Cataloguing in Publication Data

Erdos, Paul, 1913Lattice points. 1. Lattice point geometry I. Title II. Gruber, P. M.

III. Hammer, J.

516.3'5

ISBN 0-582-01478-6 Library of Congress Cataloging-in-Publication Data Erdos, Paul, 1913-

Lattice points/P. Erdos, P. Gruber & J. Hammer. p. cm. - (Pitman monographs and surveys in pure and applied mathematics, ISSN 0269-3666; 39) Bibliography: p. Includes index. ISBN 0-470-21154-7

1. Lattice theory. 2. Geometry of numbers. I. Gruber, Peter M., 1941- . II. Hammer, J. (Joseph) III. Title. IV Series. QA171.5.E73 1989 511.3'3- dcl9

Typeset in 10/12 Times New Roman Printed and Bound in Great Britain at The Bath Press, Avon.

Contents

Preface

List of symbols

1 Equidissectable polytopes 2 Lattice polytopes, lattice point enumerators and a glimpse of algebraic geometry 3 Minkowski's fundamental theorem and some of its relatives 4 Blichfeldt's theorem 5 Successive minima 6 The Minkowski-Hlawka theorem 7 Mahler's selection theorem 8 Packing and covering 9 Packing and covering with balls 10 Crystallography, tiling and Hilbert's 18th problem 11 Geometry of positive quadratic forms: reduction, packing and covering with balls 12 Selected problems of number theory

vi

viii 1

6 14 25 28

34 37 40 55 67 81

13 Visibility 14 Lattice point problems of integral geometry 15 Applications to numerical analysis 16 Lattice graphs 17 Extremal combinatorial problems

96 107 111 116 123 133

References

139

Subject index

176

Author index

180

V

Preface

In this book we have tried to collect geometric, number-theoretic and also combinatorial and analytical results, theories and problems related to lattice points. It is clear that problems of the geometry of numbers comprise a sizeable part of this book, but we have tried to cover more topics dealing with dissection problems, lattice polytopes, packing, covering and tiling problems, mathematical crystallography, visibility, integral geometry, applications to numerical integration, combinatorics, graph theory and several others.

We hope that the book will convince the reader of the many interesting relations of the concept of lattice points to other areas of mathematics and that the great number of implicitly or explicitly stated classical and new problems will induce further research. Since it was our intention that the book should act as an `appetizer' we have included

only a small number of proofs, but we have not hesitated to state heuristic arguments and to give intuitive descriptions. We also make many comments on the results presented, and, in some instances our personal opinion, is made clear. The references are selective, and we have always tried to include more recent ones. If by chance we have omitted some prominent paper in the field of lattice points we ask to be pardoned. In any case readers interested in geometry of numbers may consult the comprehensive volume on Geometry of numbers, the second

edition of which was prepared by G. Lekkerkerker and one of the present authors. Many references are to surveys and monographs on various topics dealing with lattice points which the reader might wish to consult.

We gratefully acknowledge many helpful hints and discussions with colleagues and friends working in the area of lattice points, in particular Professors Coxeter, Danzer, Ewald, Gabor Fejes TOth, Groemer, Hlaw-

ka, Mack, McMullen, Ryskov, Seidel, Shephard, Uhrin and Wills. Professor Ewald helped us in the preparation of section 2.5 and

PREFACE

vii

Professor Shephard made many helpful comments on Penrose tilings. The figures were drawn by Hartwig Sorger. Christian Buchta, Gerhard Ramharter, Dinesh Sarvate and Esther Szekeres assisted us with checking the manuscript. The typing was done by Yit-Sin Choo. P. Erdos P. M. Gruber J. Hammer

To Laszlo Fejes Toth

List of symbols

A 23

A()7 a( 19 a( ) 2 Bd 3

B( ) 8 bd 7

d( ,

L L(

,)10

)j,A,(, )28 J1189

m()89 µ(

C* 30 C( , ) 10 D 23 D( )74,88

d(

L( ),) °( ), L°( ) 6, 125

14

) 18

A( ) 16 b( ) 16 OL( ) , O (), bc() 41 bi( ), oT( ), bc() 42 det 16

) 54

123

1

j

11 37

p81

P()8 PA( ) 21 7!( ) 92 pos 11 R 23 iP 83

P()89 S

11

[d, k, m] 60 (d, M, m) 60

S(

Ed 1

-T 82

E(

7, 123

F() 32

0L(

id 68

V(

<,>11 xd 17

'cO18,21,99

X()6 -e 37

L* 32

15

a()93 BT( ), BcO 41

BL( ), BT( ), e k( ) 42

3, 123

V( ,..., ,) 10 W 23

W;( ) 23

Z2 x()34

x(,)116

1

Equidissectable polytopes

1.1 The investigation of equidissectability of polytopes was greatly stimulated by Hilbert's third problem. First we shall consider a modern

variant of Hilbert's problem in the context of lattices and then a problem due to Hadwiger will be treated. The original problem of Hilbert, which essentially goes back to Gauss, is

specify two tetrahedra of equal bases and equal altitudes which can in no way be split up into congruent tetrahedra, and which cannot be combined with congruent tetrahedra to form two polyhedra which themselves could be split into congruent tetrahedra. (Hilbert [1900])

In order to formulate a general version of Hilbert's problem some precise definitions must be given. Let a class S of subsets of d-dimensional euclidean space E d be given. In particular we shall consider classes of proper polytopes in Ed. These

are finite unions of (compact) convex polytopes with non-empty interiors. A dissection in S of a set S E S is a finite family (Si, . ., S having pairwise disjoint interiors and such that S equals the union of S1, . . ., S,,. In addition to S let a group G of rigid motions in .

Ed be given. Two sets, S, T E S are called equidissectable in S with respect to G or G-equidissectable in S if there are dissections {S1, ..., S}, { T1, . . ., T,) in S of S and T respectively such that Ti = m,(S,) for i c- {1, ., n} for suitable rigid motions m1, ..., Mn E G. The sets S and T are .

.

G-equicomplementable in S if there are G-equidissectable sets U, V E S such that S and U have disjoint interiors and correspondingly for T and V and such that S u U and T u V are G-equidissectable. Papers related to Hilbert's problem typically deal with the following question: Given a class (S of subsets and a group G of rigid motions of 1

LATTICE POINTS

2

Ed, specify necessary and/or sufficient conditions for sets S, T E S to be G-equidissectable or G-equicomplementable.

An old result for the plane due to Farkas Bolyai (the father of the famous Janos Bolyai) and P. Gerwien says the following. Two proper polygons, i.e. proper polytopes in E2, are equidissectable with respect to the group of all rigid motions in the class of all polygons if and only if they have the same area. Unfortunately the corresponding result does not hold for dimensions > 3 as was shown by Dehn, then assistant to Hilbert. In particular, Dehn's result gives a positive answer to Hilbert's third problem. For exhaustive information on results in the context of

Hilbert's third problem we refer the reader to Hadwiger (1957), Boltyanskii (1978), Sah (1979) and the survey of McMullen and Schneider (1983). For results of a related character in the context of the famous

Banach-Tarski paradox, but based on a concept of dissection into disjoint sets, see Wagon (1985).

Let Z' denote the set of all points of Ed with integer coordinates. Z' is called the integer or fundamental lattice in Ed. See figure 1.1. (For the more general concept of a lattice see section 3.1.)

0

0

0

0

0

0

0

0

0

0

0

0

0

0

00

0

0

0

0

0

0

0

0

0

0

0

Figure 1.1 The integer lattice L2

Since we do not distinguish between points and vectors, Z' clearly forms a group. Obviously it can be interpreted as a group of translations: To each u E Z d corresponds the translation

x -x+uforx E Ed. 1.2 Hadwiger (1953, 1957) gave necessary and sufficient conditions for two proper polytopes in Ed to be Zd-equidissectable. In order to formulate his result we need some more definitions.

Let P be a proper polytope and p E P. Then the normalized internal angle a(p, P) of P at p is defined by

EQUIDISSECTABLE POLYTOPES

3

V(P n (pBd + p)) (1.1) V(pBd + p) Here V( ) denotes Lebesgue measure, Bd is the solid euclidean unit ball

a(p, P ) =

li m

P-+0

in Ed and pBd + p = {px + p: x E Bd}. Since the quotient in (1.1) is constant for sufficiently small p > 0, the existence of the limit is obvious. Next define a real functional L on the class of all proper polytopes P in Ed by

L-(P) = E{a(u, P): U E P n Zd}.

(1.2)

L-(P) is the weighted number of points of Z d contained in P and L- is called the weighted lattice point enumerator. For a slightly more general definition and the definition of three more lattice point enumerators see section 2.1. Before giving Hadwiger's criterion for Zd-equidissectability we quote

some of his remarks (1953). Let P be a proper polytope in Ed such that there are translates of P by suitable vectors of Z d which tile Ed, i.e. the translates have pairwise disjoint interiors and their union equals Ed (for more information on tiling see sections 10.4-8). Then

V(P) = L (P).

(1.3)

Thus it is possible to compute the volume of P by `counting' the number of points of Z d contained in P. Assume now that P is a parallelotope all

vertices of which belong to Zd. Then (1.3) holds. We may express L(P) in the form L (P) = E(Ecr(u, P)),

(1.4)

where the inner sum is extended over the points u of Z d contained in

the relative interiors of the i-dimensional faces of P. Denote the number of these points by Ni. It follows from (1.4) that L (P) is simply the sum extended over the values of 2'-dN1. This gives the formula of Hofreiter (1933a) for the volume of P: d

V(P) = E2i-dN. ;=o

Hadwiger's criterion for Z d-equidissectability says that two proper polytopes P, Q in E d are Z d-equidissectable in the class of all proper

polytopes in Ed if and only if L-(P + x) = L-(Q + x) for all x E Ed with 0 < x; < 1 for i e { 1, ., d}. (The coordinates of x E E d are denoted x 1, ... , Xd. Although x is always considered as the column with entries x1, ..., Xd, we shall write (X1, ..., Xd) for x in some .

.

LATTICE POINTS

4

instances.) For an example of a parallelogram equidissectable to a square, see figure 1.2. 0

0

0

which

is

ZZ-

0

0

Fi$ure 1.2 Z2-equidissectability (After Hadwiger (1957), p. 73)

Hadwiger (1953) showed that any two proper polytopes which are

Zd-equicomplementable are also Zd-equidecomposable in the class of all

proper polytopes in E'. Thus for the class of proper polytopes in E d the concept of Z d-equidissectability and Z d-equicomplementability coincide. Coincidence Of these two concepts holds for many classes of polytopes and many groups (see the references cited in section 1.1). For a case where this does not hold see the following subsection.

1.3

Let U denote the group of transformations of the form

x --*Ax+ a

for xEEd,

where A is a d x d matrix with integer elements and determinant ±1 and u E Zd. Each such transformation maps Zd onto itself. By a proper convex lattice polytope in E d we understand a convex polytope with nonempty interior all vertices of which are contained in Zd.

Betke and Kneser (1985) proved that two proper convex lattice polytopes are U-equicomplementable in the class of all convex lattice polytopes if and only if they have equal volume. Unfortunately the concepts of U-equidissectability and Uequicomplementability do not coincide for the class of proper convex

EQUIDISSECTABLE POLYTOPES

5

lattice polytopes. The following is an example of Betke (1985): The simplices in E3 with vertices o = (0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 3) and

o,

(1, 0, 0),

(0, 1, 0),

(0, 0, 3)

respectively

are

U-equi-

complementable but not U-equidissectable in the class of all proper convex lattice polytopes in E3.

2

Lattice polytopes, lattice point enumerators and a glimpse of algebraic geometry 2.1 In this section we consider relations between the number of points of the integer lattice contained in a lattice polytope and its volume and surface area. We also exhibit the connection of lattice polytopes and toric varieties.

A finite family of simplices in Ed is a (finite) simplicial complex if each face of a simplex of the family also belongs to the family and if the intersection of any two simplices of the family is a face of both of them. The 0-dimensional simplices are called vertices. A polytope in E' (or a polygon in case d = 2) is the union of a simplicial complex. The latter forms a simplicial decomposition of the polytope. Equivalently one may define a polytope as a finite union of convex polytopes of dimensions d. A polytope is proper if it is the closure of its interior or if it can be

represented as a finite union of convex polytopes of dimension d, cf. section 1.1. Call a polytope P a lattice polytope if for a suitable simplicial decomposition of P the set of vertices consists precisely of the points of Z d contained in P.

The Euler characteristic of a simplicial complex is the number of its 0-dimensional simplices (= vertices)

minus the number of

its 1-

dimensional simplices (= edges) plus the number of its 2-dimensional simplices, minus etc. For a polytope P the Euler characteristic x(P) can be uniquely defined as the Euler characteristic of a simplicial decomposition of P. For example the three polygons in figure 2.1 have Euler characteristics 1, 2 and 3 respectively.

For a polytope P let L(P), L'(P) and L°(P) denote the number of points of Z d contained in P, on the boundary of P and in the interior of

P, respectively. Further let L_(P) be defined by equations (1.1) and (1.2).

This defines functionals L, L*, L°, L_ on the class of all

polytopes in Ed, so-called lattice point enumerators. 6

LATTICE POLYTOPES AND LATTICE POINT ENUMERATORS

7

2.2 The values of the lattice point enumerators of a lattice polytope and its volume and surface area are closely related.

We consider the case d = 2 first. As may be expected, the results in this case contain dimensional results.

much more information than related higher-

Let P be a lattice polygon of area A(P). Consider a simplicial decomposition of P whose set of vertices is Z2 n P. Let E(P) denote the number of edges of the simplicial decomposition on the boundary bd P of P, where an edge which is contained in the closure of the interior of P is counted once and all other edges twice. A general result of Hadwiger and Wills (1976) says that

L(P) = A(P) + 'E(P) + X(P).

(2.1)

The reader may verify this for the lattice polygons in figure 2.1.

0

(a) polygon but not a lattice polygon

0

0

0

0

(b) lattice polygon

0

(c) proper lattice polygon

Figure 2.1 Lattice and non-lattice polygons

Several particular cases of (2.1) deserve mention. If P is a proper lattice polygon then (2.1) implies the case d = 2 of a formula of Reeve (1957):

L(P) = A(P) + 1L'(P) + x(P) - ix(bdP).

(2.2)

(Note that in Reeve's notation the Euler characteristic is -X(P).) If the boundary of a proper lattice polygon P is the disjoint union of closed Jordan polygons, then X(bd P) = 0 and (2) implies

L(P) = A(P) + zL'(P) + x(P).

(2.3)

A particular case of (2.3) arises when the boundary of a proper lattice polygon P consists of a single closed Jordan polygon. Then we obtain the well-known formula of Pick (1899)

LATTICE POINTS

8

L(P) = A(P) + ZL'(P) + 1.

(2.4)

Among the numerous proofs of Pick's theorem we just mention Coxeter

(1969) and Liu (1979). An application of Pick's theorem to lattice simplices is due to Reznick (1986).

Since each line segment connecting two points of Z2 has length at least 1, proposition (2.1) yields for a lattice polygon P the inequality

L(P) , A(P) + 1B(P) + X(P).

(2.5)

Here the `perimeter' B(P) of P is determined such that each edge on the boundary of P which is contained in the closure of the interior of P is counted twice. To obtain a corollary of (2.5) consider a plane compact convex set C. Let L(C) denote the number of points of Z2 contained in C. If L(C) = 0, inequality (2.5) trivially holds. Otherwise consider the convex hull of the points of Z2 in C. Then (2.5) implies that

L(C) , A(C) + ' P(C) + 1, a result first proved by Nosarzewska (1948). Here P(C) is the perimeter of C. Let P be a lattice polygon and let t E E 2, t f Z2 be chosen. Hadwiger and Wills (1976) give the following somewhat surprising upper bound for the number of lattice points in the translate P + t of P:

L(P + t) , L(P) - X(P). For any value the Euler characteristic can assume there are lattice polygons for which equality holds.

Let us consider the case d , 2. The example of the tetrahedra S with vertices o, (1, 0, 0), (0, 1, 0), (1, 1, m), where m is a positive integer, shows that it is not possible to express V(S) as a linear combination of L(Sm), L*(S,,,) and X(S,,,) (= 1) with real coefficients.

Let P be a proper lattice polytope in Ed. In order to determine V(P) Reeve (1957, 1959) had the idea of considering besides P the lattice polytopes 2P (= {2x: x E P}), 3P, .... The results of Reeve concern the cases d = 2, 3 only. Macdonald (1963) extended and proved them for all d: For d = 2 we have the formulae

A(P) = (L(P) - zL (P)) - (X(P) -'X(bdP)) 2A(P) = L(2P) - 2L(P) + X(P),

A(P) = L'(P), and for d = 3 we obtain

6V(P) _ (L(2P) - ' L'(2P)) - 2(L(P) - ' L'(P)) 2

+ (x(P) - 'X(bd P)),

2

(cf. (2.2)),

LATTICE POLYTOPES AND LATTICE POINT ENUMERATORS

9

6V(P) = L(3P) - 3L(2P) + 3L(P) - X(P), 6V(P) = L(2P) - 2L'(P). For general d the formulae are (d - 1)d! 2

V(P) = (L((d - 1)P) - 'L ((d - 1)P))

(d +

1

1)(L((d - 2)P) - 1L'((d - 2)P)) + . (-1)d-2(

d

d - 12 )((L(P)

- L '(P))

+ (-1)d-'(X(P) - 'X(bd P)),

d!V(P) = L(dP) - d)L((d - 1)P + L(P) + (-1)dX(P),

+

21)d!V(P) (d

= L-((d - 1)P) - (dl + (-1)d-2(d

1 ) L-((d

- 2)P) + .. .

- 2)L (P).

An inequality of Betke and McMullen (1985) relates V(P) and L(nP) for a convex lattice polytope P with non-empty interior:

(n + d

+

n+

d-1 2

(d!V(P) - 1)

(d odd)

d

(n+d)+1{(n+)(n_1+f)}(d!V(P)_1) d

L(nP)n+d-1)d!V(P)+(n+d

d

(d even)

d

d-1

1).

For a large body of further geometric properties of lattice polygons and lattice polytopes, we refer to the thesis of Rabinowitz (1986).

2.3 The functionals L, L', L°, L^ and related other functionals have many interesting properties, for which the reader is referred to McMullen (1975), Ehrhart (1977), Wills (1978, 1980, 1982), Betke and Wills (1979), McMullen and Schneider (1983) and Gruber and Lekkerkerker

LATTICE POINTS

10

(1987). Here we shall state only one particularly appealing result. In

order to appreciate this result we first state a classical theorem of Minkowski (1903); see also Bonnesen and Fenchel (1934), p. 40, or Leichtweiss (1979), p.162. For subsets X, Y of E I and A real define

X+Y={x+y:xEX,yEY}, AX

= {Ax: X E X}.

Let us give an example: assume that C is a convex body in Ed, that is a compact convex subset of Ed with non-empty interior. Then for A > 0 the set C + ABd is called the parallel body of C at distance A. It is easy to see that C + ABd is the union of all balls of radius A and centres in C.

Minkowski's theorem is the following: Let C1, ..., C" be n compact convex sets in Ed. Then there are coefficients V(Ci ..., C,d), i1i ... id E {1, ..., n}, which are symmetric in the indices i1, ..., id, such that for Al, ..., An , 0 the following equality holds:

+.."C') =

+ V(A1C1

n

E

. . .

V(C,,,

.

.

I

it. .id=1

Cid))'iI

.

.

. Aid.

The coefficient V(Ci,, ... , Cid) is called the mixed volume of C,, , ... , Cid .

By analogy with this a result of McMullen (1977) which in essence was proved also by Bernstein (1976) shows that for convex lattice polytopes P1, ., P. in Ed there are coefficients L(P1, i1, ..., P", in), il, ..., in E {0, ..., d} such that for k1i ..., kn c {0, 1, 2, ...} we .

.

have

L(k1p1 + ... + knPn) _

L(P1, il.... , Pn,

in)kl'...kn

.

Similar results hold with L replaced by L', L°, L_.

2.4 For P a lattice polytope Betke and McMullen (1985) consider the power series

L(P, t) = 1 + 7L(nP)t".

,1

If P is a proper lattice polytope, then L(P t) = C(P, t) (1 - t)d+i

LATTICE POLYTOPES AND LATTICE POINT ENUMERATORS

11

where C(P, t) is a suitable polynomial in t of degree at most d + 1 with integer coefficients; see Ehrhart (1967). Betke and McMullen give more precise information on C(P, t) which yields refinements and generalizations of earlier results of Macdonald (1971), Wills (1978) and Stanley (1980).

2.5 Since about 1970 it has become clear that there exists a close relationship between the theory of lattice polytopes and algebraic geometry, centred around the concept of a `toric variety'. In the following we will describe this relationship without giving too many details or exact definitions.

Let 01), . . ., u(k) be primitive points of Zd, i.e. o and u(') are the only points of Zd on the line segment with end points o, u(') for i E {1, ,

k}. Consider the cone

S = pos{u(1),

. .

., u(k)}

(= (Alu(1) + . . + Xku(k): Al, . . ., Ak -- 0})

and the `dual' cone S V of S,

Sv _ {x: (x, y) % 0 for all y E S}. (Here (,) denotes the inner product in Ed.) The set of all Laurent polynomials in d variables z 1, ... , zd , an,...naZ1

nd

a,

where the coefficients an,- nd are in an algebraically closed field K and the summation is over finitely many (n1, . . ., nd) E Sv n Zd, forms a ring Rs". The maximal (prime) ideals of Rs" represent points (irreduci-

ble curves, surfaces, etc.) of an `affine variety' Xs". If S is a cell complex of cones S, i.e. a fan, the affine varieties Xs,, Xs;' of any two cones S1, S2 E A are `glued together' by using the `natural injections' Rs," - R(s,r,s,)". The result is called a toric variety, X,r. If, for example, S1 = pos({(1, 0), (0, 1))),

S2 = pos({(0, 1), (-1, -1)1),

S3 = pos({(1, 0), (-1, -1)}), the affine varieties Xs; , Xsz , Xs, , are affine planes which are `charts' of the projective plane P2 = X,, (where d = {S1, S2, S3, pos{(1, 0)}, pos{(0, 1)}, pos{(-1, -1)}, o}). Toric varieties were introduced by Demazure (1970), Kempf et al.

LATTICE POINTS

12

(1973) and Oda (1978). (See also the survey of Danilov (1978).) There are many properties of the complex S and the toric variety X,, related to each other. Here are some examples:

b X, is compact (if K = C); 1. U{S: S E S} = Ed 2. S is strongly polytopal, that is, there is a lattice polytope P such that

S _ {S = pos{u('), . . , u(k)}: {u(), . ., u(k)} are the ver-

C=> Xy is projective;

.

tices of a face of P}

3. S is regular, that is each d-dimensional a X y has no singularities. cone of S is represented in the form pos{u(1),

...,

u(d)} where det{u(i>,

...,

u(d)} = ±1.

Toric varieties have been used by Stanley (1980) in the solution of

McMullen's conjecture on f-vectors of polytopes and by Teissier (1980/81) for a new proof of the Fenchel-Alexandrov inequalities on mixed volumes. On the other hand results on lattice polytopes were used for problems of algebraic geometry dealing with toric varieties, see for example Oda (1978), Danilov (1978), Batyrev (1982), Voskresenskii and Klyachko (1985), Ewald (1986) and the comprehensive monograph of Oda (1988). Apparently these connections are of increasing importance in contemporary convexity theory.

2.6

We close this section with some remarks on embedding of regular polytopes into V.

Apparently Lucas (1878) was the first to note that an equilateral triangle cannot be embedded into Z2 (here embedding means that all its vertices are in Z2). Scherrer (1946) showed that the square is the only regular polygon that can be embedded into Z2. More generally it was proved by Schoenberg (1937) that the regular triangle, the square and the regular pentagon are the only regular polygons that can be embed-

ded in a suitable Zd. He also showed that a regular d-simplex can be embedded in a suitable Z d' if

1. d is even and d + 1 is a perfect square;

2. d=3mod4;or

3. d is of the form (2h +1)2 + (2k + 1)2 - 1, h, k E {0, 1, 2, ...}. Finally Patruno (1983) completely determined which regular polytopes can be embedded in a suitable Zd. It remains an open question which

LATTICE POLYTOPES AND LATTICE POINT ENUMERATORS

13

semi-regular polytopes can be embedded in a suitable Z1. (For definition of semi-regular figures see Fejes Toth (1964).) For further embedding theorems, see Rabinowitz (1986).

3

Minkowski's fundamental theorem and some of its relatives

3.1 The geometry of numbers has a long history. The first sporadic results appeared in the work of Kepler (1611) and Lagrange (1773). Later contributors who emphasized the geometric aspect are Gauss (1831), Dirichlet (1850) and Klein (1895/96), whereas the numbertheoretic and arithmetic aspect is predominant in the papers of Hermite (1850) and Korkin and Zolotarev (1872, 1873, 1877). Systematic study of the subject started in the decades after 1880 with Fedorov (1885), the basic work of Minkowski (1896, 1907, 1911) and the investigations of Voronoi (1908(a), 1908(b), 1909, 1952). Many results and concepts of the modern geometry of numbers have their origin in the contributions

of Minkowski. It was Minkowski who coined the name geometry of numbers for this new branch of mathematics linking number theory and geometry. Important later contributions are due to Siegel, Furtwangler (mainly unpublished), Davenport, Venkov, Delone, Hajos, their students and a number of other living mathematicians. In this and in some of the following sections we will present basic ideas, results and open

problems of the geometry of numbers. In general, we will present geometric versions of the results chosen. This section deals with a simple but fundamental theorem of Minkowski, which is generally considered

as the central result of the geometry of numbers. Its importance lies in the wide range of interesting applications.

A lattice L in Ed is the set of all linear combinations with integer coefficients of d linearly independent vectors (figure 3.1 shows an example for d = 2). These vectors form a basis of L and the absolute values of their determinant is called the determinant d(L) of L. The parallelotope generated by the basis vectors is called a fundamental parallelotope of L. Its volume is equal to d(L). Note that there are countably many bases of L. In this book we shall not distinguish between vectors and points. Thus we may call the vectors of L (lattice) 14

MINKOWSKI'S FUNDAMENTAL THEOREM

15

points. An example of a lattice is the integer lattice Z' introduced in section 1.1.

r

°

° °

° ° °

Figure 3.1 Lattice in E2

By a convex body in E d we mean a compact convex subset of E d with non-empty interior. V( ) and S( denote Lebesgue measure and ordinary surface area measure in Ed. In case d = 2, we write A( ) (for area) and P( ) (for perimeter) instead.

3.2 The fundamental theorem (of Minkowski (1893)) or Minkowski's convex body theorem says the following. Let C be a convex body, symmetric about the origin o, and let L be a lattice in Ed. If V(C) ? 2dd(L), then C contains at least one pair of points ±p * o of L (see figure 3.2). If C is not a parallelohedron (see section 10.7 for a definition), there is a constant x < 2d depending on C such that the conclusion of the theorem still holds if V(C) % ad(L). For parallelo-

hedra, no such refinement exists. For some historical remarks see section 3.3 below.

0 °

0

0

0

0

0

0

0

0

C

O

O

0

0

°

0

0

Figure 3.2 The fundamental theorem. V(C)

4d(L)

LATTICE POINTS

16

In contrast with the precise information on the equality case in the fundamental theorem, in many inequalities, asymptotic bounds or, in

extremal problems of the geometry of numbers, the best possible constants or the extremal configurations, sets or lattices are not known.

We will sketch a proof of the fundamental theorem which clearly exhibits its geometric background. Let C be an o-symmetric convex body and L a lattice in Ed such that o is the only point of L in C. Then the convex bodies

ZC+p:peL are pairwise disjoint. The system of these convex bodies forms a so-called lattice packing of (2)C. The density of this packing is the proportion of space covered by the bodies or, more precisely, the quotient V((z)C)

d(L) (Packing is treated in more detail in chapter 8.) Since the bodies of the packing are disjoint the density of the packing is less than 1. This yields V(C) < 2dd(L), concluding the proof. For later purposes we shall give a different version of Minkowski's

convex body theorem. Let C be a convex body containing o in its interior or a star body, i.e. a closed subset of Ed with the origin as an interior point and such that each ray starting at o meets its boundary in at most one point. A lattice is admissible for C if it contains no interior point of C except o. Define the critical determinant or lattice constant O(C) of C by A(C) = inf {d(L): L is an admissible lattice for C}. Now the fundamental theorem can be stated in the following form. Let C be an o-symmetric convex body. Then V(C) < 2dO(C).

(3.1)

3.3

Among the numerous applications of Minkowski's fundamental theorem we shall present two classical ones due to Minkowski himself.

For other applications we refer to Cassels (1972) and Gruber and Lekkerkerker (1987).

Let q(x) = Iaikxixk be a real positive definite quadratic form in d variables and with discriminant b(q) = det(aik). For a > 0 the set {x: Za1kxIxk < CV)

MINKOWSKI'S FUNDAMENTAL THEOREM

17

is an ellipsoid with centre o and volume ad12Kd/S(q)1/2, where Kd is the volume of the solid euclidean unit ball in E d. Thus for a = (46(q)/xd)1/d

the ellipsoid contains a pair of points ±u * o of Zd by the fundamental theorem. This shows that the minimum of q(u) for integer values of the variables not all 0 is at most (46(q)) 1/d xdz

This result of Minkowski (1891) considerably improved a theorem of Hermite (1850). For Minkowski's original proof the fundamental

theorem was not yet available, but the idea behind his geometric proof led him to the discovery of the lattice point theorem soon afterwards. The theorem was formulated two years later in a letter of Minkowski addressed to Hermite (Minkowski (1893)). Minkowski's bound for the minimum of positive definite quadratic forms was essentially lowered by the work of Blichfeldt (1929) and in recent years by Kabatjanskii and Levenstein (1978). For a more detailed discussion of this problem in the

equivalent form of the problem of upper bounds for the density of lattice packings of balls we refer the reader to chapter 9. A second application of the fundamental theorem will give the linear form theorem of Minkowski (1896), §37: Let 11, ..., ld be d real linear

forms in d real variables. We require that the absolute value of the

..., ld, say 6, is positive. For reals

determinant of the coefficients of l1,

r1, ..., . rd > 0 the set

{x: 111(x)I < r1i ..., Ild(x)l < id}

is an o-symmetric parallelotope of volume 2 d tl .

.

id/S. If Z1

rd -- 6,

it contains at least one pair of points ±u =f o of Zd by Minkowski's fundamental theorem. Hence the system of inequalities Ill(u)I < 'G1,

.

.

., Ild(u)I < td

has at least one solution u = (u1, .,ud) with integers u1, . . .,ud not all 0 provided the real numbers i1, ..., id > 0 satisfy it Td 6. The linear form theorem has attracted a good deal of attention. The famous theorem of Hajds (1942) which will be discussed in geometric disguise in chapter 8 describes the systems of linear forms 11, ..., ld .

such that for suitable reals 7-1, of inequalities

.

..., rd > 0 with TI .

. Td = 6 the system

Il1(u)I < r,, ..., Ild(u)I < rd

has no solution u = (ul,

ui=.

=ud=0.

.

.

.,ud) with integers u1, ...,ud except for

Mordell (1936) considered a sort of converse problem for the linear

LATTICE POINTS

18

form theorem. Denote by K(d) the supremum of all numbers K > 0 with the following property: let 11, ... , Id be any system of d real linear forms in d variables and with absolute value of the determinant, say 6, positive. Then there are numbers Ti, ..., Td > 0 with T1 Td = KS such that the system of inequalities I11(u)I < T1,

.

.

., Ild(u)I < 'rd

has no solution u = (u I, . . ., U d) with integers u 1i ... , U d except for = ud = 0. It was shown by Szekeres (1936(a)) that U) = K(2) = i + 2U5 = 0.732606....

Alternative proofs and refinements are due to Sziisz (1956), Suranyi (1971) and Gruber (1971). Szekeres (1936(b)) and Ko (1936) proved that K(3) > 4. A local result of Ramharter (1980) supports the conjecture of Gruber that K(3) = ; cos

2, cos 7

7

= 0.578416.

...

Gruber and Ramharter (1982) proved that K(4) > 1116. Hlawka (1950(a)) and Gruber and Ramharter gave the following bounds for K(d): 2-0.5d2+o(d2) =

d (d!)22d(d+1)/2

< K(d) <

K(2)(d-1)/2 = 2-0466722...(d-1)

as d-> +oo. For further results see Gruber (1971), Ramharter (1980, 1981) and Gruber and Ramharter (1982).

In sections 10.8 and 12.3 we will return to this subject. An inhomogeneous counterpart of the converse problem was considered by Sawyer (1966) and Bambah et al. (1986).

3.4 In the following subsections we shall present generalizations and refinements of the fundamental theorem. This subsection contains results of a general character whereas in subsequent subsections the case d = 2 and the case of the integer lattice will be treated in more detail. Van der Corput (1935) proved the following result, announced earlier

by Blichfeldt (1921): let C be an o-symmetric convex body and L a lattice in Ed such that V(C) , k2dd(L) for a positive integer k. Then C contains at least k pairs of points ±p(1), ..., ±p(k) * o of L. For a slightly more precise result of this type we refer to White (1963) and Dumir and Hans-Gill (1977). If in van der Corput's theorem d(L) is replaced by d(C, L), where d(C, L) (-- d(L)) is the volume of the subset of a fundamental parallelotope of L covered by the sets 2 C + p:

MINKOWSKI'S FUNDAMENTAL THEOREM

19

p c- L, the conclusion still holds. This result follows from a theorem of Uhrin (1980).

Let C be an o-symmetric convex body and L an admissible lattice for C. Then C is contained in an o-symmetric convex polytope having at most 2d+1 - 2 facets (i.e. faces of dimension d - 1) and for which L is

still admissible. Hence one may replace the constant 2' in the fundamental theorem by 2da, where a (-- 1) is the supremum of the quotient V(C)/V(P) extended over all o-symmetric convex polytopes P containing C and having at most 2d+1 - 2 facets. Results of this type are due to van der Corput and Davenport (1946).

As we have pointed out in section 3.2 the fundamental theorem is related to a certain lattice packing. Since lattice packings are `periodic sets' it seems natural to use multidimensional Fourier series for their investigation. A classical refinement of Minkowski's convex body theorem due to Siegel (1935) is based on Parseval's theorem for Fourier series. Related results were given by Cassels (1947), Hlawka (1944,

1982) and - in some sense - by Uhrin (1981). Siegel's theorem is of interest for algebraic number theory - see Eichler (1963). If C is a convex body with o in its interior, the asymmetry coefficient a(C) of C is defined by a(C) = min {a , 1: -C C aC}.

Many other `measures of asymmetry' have been investigated in recent years. Earlier results in this area were carefully reviewed by Grunbaum (1963).

Let C be a convex body containing o as an interior point and let L be a lattice in Ed. If

V(C) > ((1 + a)d(1 - (1 -

CV-1)d )d(L)

where a = a(C), then C contains at least one point p * o of L according to Sawyer (1954).

Minkowski's fundamental theorem gives, in terms of volume and symmetry, a sufficient condition for a convex body to contain a point p * o of a lattice. The extensions of the fundamental theorem discussed so far are of a similar type. From the point of view of applications in integer programming there is interest in criteria which guarantee that a

convex body contains points of a lattice, and which make use of geometrical quantities different from volume and symmetry. Also from a more geometric and aesthetic viewpoint, working towards results of this

type is desirable. Unfortunately the known results of this type either concern the case d = 2 or apply in many cases to the integer lattice or some other lattices of very particular types. In the next two subsections we consider a series of theorems of this sort. The last subsection of this

LATTICE POINTS

20

chapter deals with results not directly related to Minkowski's theorem.

We conclude this subsection with a conjecture of Ehrhart (1955(b), 1964): a convex body C with centroid o contains at least one pair of points ±p * o of a lattice L, provided V(C) > (d d! l)d d(L).

Here the coefficient of d(L) may be replaced by a smaller one depending on C except in the case when C is a simplex. The conjecture

was proved by Ehrhart for d = 2 and remains open for d > 3. A common generalization of Ehrhart's theorem and the fundamental theorem for d = 2 is due to Scott (1976). In addition to Ehrhart's conjecture we ask whether the condition

V(C) ?

k(d + 1)d

d!

d(L).

for a positive integer k implies that C contains at least k distinct pairs of

points ±p(1), ..., ±p(k) of L\{o}.

3.5

Let d = 2. We first state criteria employing area and symmetry. Then results making use of the affine perimeter and the curvature are mentioned.

For the plane, Sawyer (1955a) gives the following refinement of his result cited in section 3.4. Let C be a convex body which contains o in

its interior and has symmetry coefficient a = a(C) and let L be a lattice. Then each of the following conditions implies that C contains a point * o of L:

1. A(C) ? (3a2 - 2a + 3 - 2(a - 1)(2ca2 + 2)'/2)d(L) and

1
2. A(C) ((9/2) - (3/2)(a - 2)2)d(L) and (8 + V15)/7 2; 3. A(C) >- ((a + 1)2/(2(a - 1)))d(L) and 2 < a < 3; 4. A(C) > (2a - (2a2 - 4a - 2)'/2)d(L) and 3 <- a. The fundamental theorem is a corollary of this result by putting a = 1. If o is the centroid of C then a 2 - see e.g. Bonnesen and Fenchel (1934), p. 34. Hence Ehrhart's theorem stated in the last subsection follows from the result of Sawyer.

Let L be a planar lattice. Choose a basis of L and consider the lines through o containing the basis vectors. The lines define four quadrants Qi, ..., Q4. Then any convex body C for which the conditions

A(C n Q;) = A(C)/4 for i e {1, ., 4} and A(C) ? 4d(L) hold contains a point p r o of L according to Scott (1978(a)). For .

.

a

MINKOWSKI'S FUNDAMENTAL THEOREM

21

related result see Scott (1974(b)).

A chord of a convex body is called a chord of symmetry if

it is

bisected by o. Arkinstall (1980) proved that a convex body C contains a

point p :f o of a lattice L provided one of the following conditions is satisfied:

1. C has either 2 or at least 4 chords of symmetry and A(C) ? 4d(L); 2. C has 3 chords of symmetry and A(C) > zd(L).

This clearly implies Minkowski's fundamental theorem. If o is the centroid of a convex body C, then C has at least 3 chords of symmetry, see e.g. Grunbaum (1963), p.254. Hence the theorem of Ehrhart cited in section 3.4 above is a consequence of Arkinstall's result.

Assume now that C is an o-symmetric convex body such that the boundary of C has continuous curvature K. Let the boundary be parametrized by the arc length s. Then the affine perimeter PA(C) of C is defined by

PA(C) =

J

where the integral is extended along the boundary. The affine perimeter and the corresponding concept of affine surface area of a convex body

in dimensions >3 are invariant with respect to equi-affinities. An equi-affinity is a volume-preserving affinity. The concepts of affine perimeter and affine surface area are due to Blaschke and Pick. For more information we refer the reader to L. Fejes Toth (1972). Making use of results of van der Corput and Davenport (1946), Fejes T6th (1972) proved the following. Let C be as in the last paragraph and let L be a lattice. If PA(C) , 288d(L), then C contains a pair of points p * o of L. The constant 288 is best possible. See also Groemer (1959). The results of this section presented so far have made use of affine or

equi-affine invariants. From the point of view of the geometry of numbers, results involving affine or equi-affine invariants are preferable,

but there are also several interesting results making use of isometric invariants. As an example of a result based on an isometric invariant we give a theorem of Groemer (1961(b)). Let C be an o-symmetric convex

body such that the curvature of the boundary of C exists and has positive infimum and supremum, say 1/R, 1/r. Then C contains a pair of points p 0 o of any lattice L of determinant 1 if C satisfies one of the following conditions:

1. A(C)?4-(21/3-rr)r2=4-0.322509...r2, 2.

A(C)?6R2arctanR2

=4-

27R4 +

= 4 - 0.592 592... R-4+ - ...

LATTICE POINTS

22

The first condition was obtained earlier by van der Corput and Davenport (1946) under slightly more restrictive assumptions for C.

All results of this subsection call for extension to d , 2 and for generalizations in the sense of van der Corput's generalization of the fundamental theorem considered in section 3.4.

3.6 In the following we will state for d ? 2 several results about the points of the integer lattice Zd contained in a convex body. These results form a small new branch of the geometry of numbers. Note that

we do not require that the convex bodies are symmetric or have a particular position in space.

A first theorem of this type is due to Bender (1962) (d = 2), Wills (1968, 1970) (d = 3, 4) and Hadwiger (1970) (general d). Let C be a convex body in Ed for which V(C) ? S(C)/2. Then C contains at least one point of Z d. This result is the best possible. If C is a large flat disk between two lattice planes of height almost 1, then V(C) is almost S(C)/2, but we still have that C n Zd = 0. We next consider the problem of specifying lower and upper estimates

for the number of points of Z d contained in a convex body. For the estimates we use quantities of the body which do not define it precisely.

Also the particular position in space is not taken into account. Hence one cannot expect the estimates to be very precise. Using the result of Hadwiger (1970) stated before, Hammer (1971) proved the following. Suppose that a convex body C satisfies V(C)

2kS(C) for some k E {-1, 0, 1, 2, ...}. Then C contains at

least d(2k+i - 1) + 1 points of Zd. A different generalization of Hadwiger's (1970) theorem can be stated as follows. Let C be a convex body

such that the smallest integer, say n, which is greater than V(C) 'S(C), is positive. Then C contains at least n points of Zd. The non-trivial proof of this estimate was achieved through the work of Nosarzewska (1948) (d = 2), Schmidt (1972) and Bokowski and Wills (1974) (d = 3) and - for general d - of Bokowski, Hadwiger and Wills

(1972). For Nosarzewska's result and a generalization of it due to Hadwiger and Wills (1976) see section 2.2.

The lower estimates for the number of points of Z d contained in a convex body stated above are rather satisfying. Considering upper estimates the situation is much worse in spite of many efforts. The best we can offer are conjectures. Let K be the cube {x: Ixil <_ z for i E {1, ..., d}}. The number of points of Z d contained in a convex body C is clearly bounded by V(C + K). A considerably smaller upper bound conjectured by Wills (1973) was confirmed for small dimensions. Unfortunately a counter-

MINKOWSKI'S FUNDAMENTAL THEOREM

23

example of Hadwiger (1979) shattered it in dimension 441. (For a modification of the original conjecture see Gritzmann and Wills (1986).)

In order to state another standing conjecture of Ehrhart (1977) we need to introduce the notion of quermassintegrals. According to a classical formula of Steiner for each convex body C there are quantities Wo(C) = V(C), W1(C) = S(C)/d, ..., Wd(C) = Kd, called quermassintegrals, such that we can express the volume of the parallel body of C at distance A > 0 in the form

dW;(C)Ai,

V(C + ABd) _ =0

l

see e.g. Hadwiger (1957), p. 214. Here, Bd is the solid euclidean unit ball in Ed and Kd = V(Bd). Steiner's formula is a special case of the theorem of Minkowski in section 2.3 and the quermassintegrals are special mixed volumes.

Ehrhart (1977) conjectured the following upper bound and proved it

for d = 2, 3: Let C be a convex body and let Q be the smallest parallelotope containing C and with edges parallel to the coordinate axes. Then the number of points of Zd contained in C is

V(C) + IS(C) + d ( d ) Wd-i(Q) =0

l

Kd-i

(K; is the i-dimensional measure of the solid euclidean unit ball in E'.)

3.7 For d = 2 there is a large body of criteria which make sure that a convex body contains one or more points of Z2. We shall give a list of pertinent results. Each result presents the challenge of generalizing it to higher dimensions.

Let C be a convex body and let A, P, D, R and W stand for area, perimeter, diameter, minimum circumradius and minimum width of C,

respectively. The minimum width of C is the minimum distance of parallel supporting lines of C. The conditions in the left column of table 3.1 imply that C contains k points of Zd. Sawyer (1953, 1955(b)) and Schaffer (1955) determined the planar convex body C of minimum area 3 such that any congruent copy of C contains at least one point of the integer lattice Z2. The corresponding a lattice with a basis {b(1), b(2)} such that IbG'I = b (2) I = I b 0> - b (2) was solved by Sawyer (1976), see also Scott

problem for

denotes the euclidean norm. For a covering set of (1978(b)). Here minimum perimeter see Reich (1973). I

Several papers are dedicated to particular convex bodies such as triangles and parallelograms - see e.g. Hsieh (1969), Maier (1969) and

LATTICE POINTS

24

Table 3.1 Sufficient conditions for a plane convex body to contain k points of Z2 Condition

Value of k

Author

A>D

k>A-D

A>-kAD,A= 1.142607...

k

A(D-1)?D2/2,D?2

1

Baiada and Tripiciano (1975) Scott (1974), Hammer (1979) Scott (1983) Nosarzewska (1948), Bender (1962) (k = 1), Hammer (1964)

A > kP/2

k

(W - 1)A ? W2/2

1

(W-1)D(2+\/3)/2

Scott (1980, 1983) Scott (1979)

1

= 1.866 025.. . (W - 1)(D - 1) >- 1 (W - 1)P > 3W (W - 1)R ? W/V3

Scott (1979) Scott (1980) Scott (1980) Sallee (1969)

C of constant width, W >_ constant (> 1.545)

W >_ (2 + \/3)/2 = 1/866025.

Scott (1973)

W>a

Elkington and Hammer (1976)

.a]2 a

2

C of constant width,

W?a C contains o in its interior,

(W-V2)(D-2)--2

1.5461

= [0.646 830. 2

.a]2

Elkington and Hammer (1976) Scott (1985)

Niven and Zuckerman (1967).

Finally, we mention Steinhaus' problem: does there exist a point set such that no matter how it is placed on the plane it covers exactly one point of Z2? Beck (1984) has the following partial answer: there is no bounded and Lebesgue measurable set satisfying Steinhaus' property.

4

Blichfeldt's theorem

4.1

A generalization of Minkowski's lattice point theorem, different from the results of section 3, is the theorem of Blichfeldt (1914), proved independently by Scherrer (1922). Let M be a measurable set and L a

lattice in Ed. If V(M > d(L) or V(M) = d(L) and M is compact, then M contains two distinct points p, q with p - q e L. If M is compact and Jordan measurable and does not give rise to a lattice tiling (see section

10.4), then there is a constant a < 1 such that V(M) , ad(L) still yields the conclusion of the theorem. Scherrer's proof makes use of Dirichlet's box principle, also called the

pigeonhole principle: if one has to put n + 1 objects into n boxes one must put at least two of the objects into the same box. This seemingly trivial proposition has applications in Diophantine approximation - see e.g. Minkowski (1907). We can formulate Blichfeldt's theorem in the following way. Suppose

M, L fulfil the assumptions of the theorem. Then there is a translate of

M, say M + t, which contains at least two distinct points of L. (See figure 4.1.) (Sawyer (1962) and Hammer (1968) obtain analogous results by applying rotation instead of translation.)

Like Minkowski's fundamental theorem, Blichfeldt's theorem also permits a simple geometric explanation. Consider the system of translates

{M+r:reL}. If V(M) > d(L), the `total volume' of these tranlates `exceeds' the `total volume' of Ed. Hence there must be overlappings. Assume for example that (M + r) r (M + s) * 0 for suitable r, s E L with r = s. Then

p + r = q + s for suitable p, q E M and thus p - q= s- r E L\{o}.

Using the concept of density the heuristic argument in this `proof' can be given a precise meaning. 25

LATTICE POINTS

26

O

O

O

Figure 4.1 Blichfeldt's theorem. V(M) , d(L)

4.2 The fundamental theorem is an easy consequence of Blichfeldt's theorem. Let C be an o-symmetric convex body and L a lattice such that V(C) % 2dd(L). Then V(ZC) % d(L). The body ZC is compact. Blichfeldt's theorem thus implies that there are distinct points p, q E Z C

with p - q E L. Since ZC is symmetric about o we have -q E 2C and thus p - q E 2 C + 2 C= C, concluding the proof. For other applications of the theorem of Blichfeldt we refer to Gruber and Lekkerkerker (1987).

4.3

Using essentially the idea of Scherrer's proof, van der Corput

(1936) generalized Blichfeldt's theorem in the same way as he general-

ized the fundamental theorem. Let M be a measurable set and L a lattice in Ed with V(M) > kd(L) or V(M) = kd(L) and M compact, where k is a positive integer. Then M contains k + 1 distinct points p(),

,p(k+ t) such thatp 0W-pWELfor i, jE{1,

.

.

.,k+1}.

Other generalizations of the theorem of Blichfeldt in E' fall into three categories. Firstly, there are elementary refinements due to Woods

(1958(a)) and Gruber (1967(a)) based on elementary set operations and simple properties of Lebesgue measure. These refinements apply to the so-called conjecture on the product of non-homogeneous linear forms see section 12.4. The second category contains generalizations employing functions, but still in an elementary way - see Rado (1946) and Uhrin (1981). Thirdly, there are generalizations and refinements using Fourier series due to Cassels (1947) and Bombieri (1962) which are similar in spirit to Siegel's refinement of the fundamental theorem; cf. section 3.4. Bombieri applied his generalization also to the conjecture on the product of non-homogeneous linear forms.

BLICHFELDT'S THEOREM

27

In essence the proof of Blichfeldt's theorem is based on the following modern version of Dirichlet's box principle: if on a measure space of total measure 1 the integral of a real integrable function is greater than

1, then the function assumes values greater than 1. Thus the idea suggests itself to extend Blichfeldt's theorem to spaces more general than Ed, such as topological groups or homogeneous spaces. Results of this type were obtained by Tsuji (1952, 1956), Macbeath (1959) and Santalo (1955), (1976), p. 175.

4.4 An interesting adjunct result of Blichfeldt's theorem in the plane is the following result of Beck (1988) which makes use of a theorem of W. Schmidt on Diophantine approximation: there is a function

f: R + - R+, f(x) - + - as x ---> + -, such that any planar convex

body C can be placed on the plane so

as to contain at least

A(C) + f(A(C)) points of Z2. Similarly C can be placed on the plane such that it contains at most A(C) - f(A(C)) points of Z2. This result confirms a conjecture of L. Moser. Beck conjectures that the true order of magnitude of f(x) is about x1/4, as x ---> + -.

5

Successive minima

5.1 In chapters 3 and 4 we have considered the lattice point theorem of Minkowski and several generalizations and results related to them. Yet another generalization is based on the notion of successive minima. In this section we will present this generalization due to Minkowski and also some related results. There are several important applications of these results in Diophantine approximation and to the reduction theory of positive quadratic forms. Among the numerous papers dealing with applications of successive minima we mention Mahler (1966), Schmidt (1969, 1970, 1980) and Jurkat and Kratz (1981). For further references the reader is referred to Gruber and Lekkerkerker (1987). Let S be a star body and L a lattice in Ed. The homogeneous or first successive minimum A = A1(S, L) of S with respect to L is defined by Al = inf{A > 0: AS contains a point

o of L}.

Clearly L is admissible for x.1S. If S is compact, then there are points of

L contained in the boundary of S. Define the successive minima Al = A1(S, L), ..., Ad = Ad(S, L) of S with respect to L by A, = inf{A > 0: AS contains i linearly independent points of L}. (5.1) Obviously,

0,Al ,A2 5.2

...,Ad<+co.

(5.2)

Let C be an o-symmetric convex body and L a lattice in Ed. A

version of the fundamental theorem of Minkowski says that AdV(C) s 2dd(L).

(5.3)

As mentioned in section 3.2, the extremal bodies for this inequality are the parallelohedra (see section 10.7).

Next we state a deep result, Minkowski's second theorem or the 28

SUCCESSIVE MINIMA

29

theorem on successive minima of Minkowski (1896), §50: Al

)dV(C) , 2dd(L).

(5.4)

Because of (5.2) this clearly refines Minkowski's fundamental theorem

(5.3). For the equality case in (5.4), we refer to Jarnik (1949). The extremal bodies are a sort of direct sum of lower dimensional parallelohedra.

There is no particularly simple proof of this result. Among the more modern proofs we mention the very satisfying proofs of Bambah, Woods

and Zassenhaus (1965) and the proof of Danicic (1969). Whereas the elegant proofs of Bambah et al. follow classical lines, the proof of Danicic is related to Scherrer's (1922) proof of the theorem of Blichfeldt.

5.3 This subsection contains refinements and companion results of Minkowski's second theorem.

Woods (1966) observed that Bambah's proof in the joint article of Bambah, Woods and Zassenhaus (1965) did not use all properties of the lattice L. Thus he could generalize the second theorem in the following way. Let L be a discrete subset of El containing d linearly independent

points and such that for p, qEL we have p -qEL or q - p E L or both. For each o-symmetric parallelotope P consider the limit inferior of

V(rP) as i - + number of points of rP n L and assume that the infimum of all these limits is finite and positive as P ranges over all such parallelotopes. Denote the infimum by d(L). If

L is a lattice then d(L) is simply its determinant. 1/d(L) is a sort of `density' of the point set L. The successive minima of an o-symmetric convex body C with respect to L are defined by (5.1), with S replaced by C. Then Woods proved that (5.4) still holds in this more general setting.

Let C be a convex body containing o in its interior. A conjecture of Davenport (1946) asserts that for any lattice L the inequality

Al .... d0(C) < d(L)

(5.5)

holds. For o-symmetric C we have V(C) <_ 2d0 (C) by the fundamental

theorem of Minkowski (see (3.1) in section 3.2). Thus Davenport's conjecture refines the second theorem of Minkowski (compare (5.4)). For two-dimensional o-symmetric convex bodies the validity of (5.5) can be deduced from a proof of Minkowski (1896), §50. The general case of

LATTICE POINTS

30

Davenport's conjecture for the plane was confirmed by Woods (1958(b)). For d = 3 Woods (1956) could give an affirmative answer for o-symmetric convex bodies. For ellipsoids or, what amounts to the same, for the unit ball Bd, the

conjecture follows from the work of Minkowski (1896), §51. For dimension d = 2, 3, 4 one can improve upon (5.5). The best possible results are easy consequences of results of Barnes (1978) on quadratic forms: )1/2

4-

31/2O(B2)

r

< d(L),

1

2 2

4 -)62

1/ 1

_A-k22

2/2

3)

2

-2_

; A2 2

O(B3)

1

< d(L),

2

A22

A1A2

Al

A2

j2

4A3A.4

2A4

3

4

A1A3 + 4A2A4

1/2

O(B4)

d(L). For more general sets an estimate similar to (5.5), but with the factor 2(d - 1)/2 on the right-hand side, was given by C. A. Rogers (1949) and Chabauty (1949) independently. Complementing his theorem on successive minima, Minkowski (1896), §50, proved that the inequality 2d

d(L) < Al

dd

.

.

. )'dV(C)

holds for any o-symmetric convex body C and any lattice L in Ed. Here, equality can hold only if C is a (linear image of the) cross polytope ({x: 11x;1 , 1}).

5.4 In this subsection we will consider polar convex bodies. For the sake of completeness a list of properties of polarity is given.

Let C be a convex body in E d which contains o in its interior. The polar (reciprocal) body C* of C is defined by

C* = {x: (x, y) <- 1 for all y E C). Polarity plays an important role in convexity and in other fields, see e.g. Leichtweiss (1979), §6, and Stoer and Witzgall (1970), §2.14. We state some general properties of polarity:

1. C* is a convex body and o is an interior point of C*.

2. C**=C.

SUCCESSIVE MINIMA

31

3. If D is a convex body with o in its interior, then C C D implies C* D D*.

4. If C is strictly convex (i.e. the boundary of C contains no line segment) then C* is smooth (i.e. the boundary is differentiable) and vice versa. If C is a polytope, C* is also a polytope. 5. If C is symmetric in o, then Kd

V(C)V(C*) ` x2d*

dd/2

Note that d

xd

7Tdd)

1 = 0.318309... (17.079468

...)dd-d - 1

as d - +- (xd = V(Bd)). The right-hand inequality is due to Blaschke and Santalo - see Santalo (1949). It is the best possible and equality is attained precisely for ellipsoids. The equality case was settled by Petty (1985); see also Lutwak (1985). The left-hand inequality was proved by Bambah (1954(a)). An essential improvement of Bambah's inequality is due to Bourgain and Milman (1987). For d = 2 Mahler (1939(b)) found the best inequalities:

8 , V(C)V(C*) for o-symmetric C, where equality is attained precisely for parallelograms (see Reisner (1986)) and

6.75 = 2a _- V(C)V(C*)

for general C, where equality

is

attained for triangles. An open

conjecture of Mahler asserts that for o-symmetric convex bodies C in Ed 4d

<_ V(C)V(C*),

dd

where equality holds if and only if C is a cube or a cross polytope.

Mahler (1948) proved the following result on critical determinants (see section 3.2). Let C be an o-symmetric planar convex body. Then

2 < A(C)A(C*) < 41 where equality is attained in the left-hand inequality precisely for parallelograms, see 5.5. In the right-hand inequality, equality holds for ellipses, but it seems to be an open problem whether it holds for ellipses only. Mahler stated a higher-dimensional result of this type but it seems to be difficult to specify the precise bounds. The covering constant IF(C) of a convex body C in Ed is defined

LATTICE POINTS

32

r(C) = sup{d(L): L a lattice such that U {C + p: p E L} = Ed). (Note that r(C) is related to the thinnest lattice covering of Ed with C see chapter 8.) A curious result of Bambah (1954(a)) says that for an o-symmetric convex body C in E2

2 < A(c)r(c*) . 4, where equality holds on the left-hand side for squares and on the right-hand side for regular hexagons. C also satisfies

r(C)r(C*) <_ 9.

a

These inequalities are also the best possible. Given a lattice L, its polar or reciprocal lattice L* is defined by

L* = {p: (p, q) integer for all q E L}. Consider a basis of L and form a d x d-matrix B the columns of which are precisely the vectors of the basis. Then L = BZd and

d(L) = detBI. (We do not distinguish between a matrix and the corresponding linear transformation.) From L = BZd we obtain L* = (B«)-1Zd and thus d(L*) = 1/d(L). (The superscript tr stands for transposition.)

With the notions introduced before we can enunciate the following theorem of Mahler (1939(a)): Let A1, ..., 1d and a,;, . . ., Ad be the successive minima of an o-symmetric convex body C with respect to a lattice L and of C* with respect to L*, respectively. Then 1 , A1Aa

1_i

, (d!)2 for i E {1, . . ., d}.

Note that (d!)2 - 6.283185 ... (0.135335 ...)dd2d + 1

as d - +oo.

Using Bambah's lower bound for V(C)V(C*), see (5.6), one easily shows that 4ddd/2

1 _- A Ad+1

Note that 4ddd/2 3 . 141592 K2d

...

(0 . 234199

... ) dd (3d/2) +1

as d - + °

.

A smaller upper bound is due to Bourgain and Milman (1987). Most probably it can still be lowered essentially as is the case when C is an ellipsoid with centre at o, see Lagarias, Lenstra and Schnorr (198?). (Dates of work given as ? are not yet published.)

SUCCESSIVE MINIMA

33

5.5 For successive minima for so-called compound convex bodies and algebraic lattices the reader may consult the papers of Mahler (1955(a), (b)) and Cook (1974) and the survey of Chalk (1983).

6

The Minkowski-Hlawka theorem

6.1 Minkowski's lattice point theorem shows that a lattice L in Ed which is admissible for an o-symmetric convex body C satsifies the inequality

2-dV(C) <_ d(L).

(6.1)

It thus seems natural to ask whether there are C-admissible lattices for which the determinant is not too large. Surprisingly, it turned out to be very difficult to give a satisfactory answer to this question, even in the case of very simple convex bodies such as euclidean balls. Minkowski (1905), §15, ingeniously showed the existence of a lattice in Ed which is admissible for the solid euclidean unit ball Bd and has determinant at most where denotes the Riemann zeta-function. More generally, Minkowski (1892) conjectured that this holds with Bd replaced by any o-symmetric star body. Minkowski's conjecture was proved by Hlawka (1944) for the class of bounded Jordan measurable sets. For each set J with V(J) > 0, there is a lattice L which is admissible for J satisfying

d(L)

V(J).

(6.2)

If J is o-symmetric one may insert the factor z on the right-hand side of

(6.2) and if J is a star set the factor

Here J is a star set if it contains o and if each ray starting at o intersects J in a point, a bounded interval or itself is wholly contained in J. This theorem of Minkowski-Hlawka remains true for (possibly unbounded) Lebesgue measurable sets.

A simple consequence of the Minkowski-Hlawka theorem is that for

any o-symmetric convex body C there is a lattice packing of C of 2-d+°(d) as d-+ +- - see chapter 8. For compact density a star bodies the Minkowski-Hlawka theorem can be expressed in terms of an inequality for the lattice constant. 34

THE MINKOWSKI-HLAWKA THEOREM

35

A result of Butler (1972) is the following: For any o-symmetric

is a lattice L such that simultaneously {C + p: p E L} is a packing and {21 + °(1)C + p: p E L} is a covering. Since the covering has density >-1, the packing has density ,2-d+o(d) see chapter 8. Thus we may consider Butler's result as an adjunct of the Minkowski-Hlawka theorem.

convex body C there

6.2

Being a counterpart of Minkowski's fundamental theorem, and because of its intrinsic interest, the Minkowski-Hlawka theorem drew much attention. Several alternative proofs were given. Among these are

proofs in the context of topological groups, integral geometry and reduction of positive definite quadratic forms. (For a simple classical proof see Hlawka et al. (1986).) There exists a multitude of refinements of the Minkowski-Hlawka theorem.

Making use of reduction of positive quadratic forms - and thus following the idea of Minkowski's proof for the ball - Siegel (1945) proceeded as follows. He considered a natural measure µ on the space of all lattices L of determinant 1 in Ed, normalized such that the whole space has measure 1. Using this measure he proved for each Riemann integrable function f on E d the formula

f

JE {f(p): p c- L\{o}}dµ(L) = ,,f(x) dx,

(6.3)

now called the Siegel mean value theorem. Call a point p of a lattice primitive if o and p are the only points of the lattice on the line segment with endpoints o and p. Complementing (6.3), Siegel proved that

f E(f(p): P E L\{o}, p primitive} dµ(L)

JEdf(x) dx.

(6.4)

This is in agreement with the well-known fact that for any lattice L the

`probability' that a point of L is primitive equals

(compare

section 13.3). Formulae (6.3) and (6.4) extend to Lebesgue integrable f as well.

The Minkowski-Hlawka theorem is an immediate consequence of Siegel's mean value theorem (6.3), resp. (6.4): Assume that J has measure 1 and let f be the characteristic function of J. (6.2) and its improvements for o-symmetric sets or star sets are not best possible. There is no conjecture for what the best possible inequality might be, even if M is a convex body. The sharpest known refinement of (6.2) is due to Schmidt (1963). It says that for a Lebesgue

measurable set M in Ed there is an admissible lattice L such that for

d=2

LATTICE POINTS

36

d(L) < 15V(M) = 0.9375 V(M)

(6.5)

and for general d d(L) <_ V(M)

where ad -* log \ = 0.346573.. .

as d ---> +c.

(6.6)

If M is symmetric in o one may replace the right-hand side of these inequalities by half of them.

For further references on refinements and alternative proofs of the Minkowski-Hlawka theorem the reader may wish to consult C. A. Rogers (1964) or Gruber and Lekkerkerker (1987).

6.3

We conclude this chapter with some remarks and open problems. Considering the large gap between the upper and lower bounds in (6.1) and (6.2), (6.5) or (6.6), respectively, it is a natural problem to find bounds which are closer to each other. For quite a period of time it was thought that it might be possible to find essentially smaller upper

bounds than were given in (6.2), (6.5) or (6.6), at least for convex bodies C instead of M. Now there are serious doubts whether such improvements are possible at all. One reason for this is that the classical (minor) refinements of the Minkowski-Hlawka theorem due to Rogers, Schmidt and others are extremely difficult to prove. A second reason is that recent progress in packing of balls indicates that lattices admissible

for B' might have a determinant at least of the order of magnitude of V(B"). In section 9.3 this is stated in terms of lattice packings of Bd. Siegel's mean value theorem shows that for any Lebesgue measurable set M with finite positive measure there are `many' admissible lattices of determinant , V(M). In principle - but only in principle - it is possible to find in a finite number of steps lattices admissible for a compact set

M having determinant , V(M). This can be seen from Hlawka's original proof or - more easily - from Rogers (1964), theorem 4.1. Since

the number of steps required makes this construction useless, the problem arises of designing `faster' algorithms for the construction of admissible lattices of `small' determinants for compact sets or convex bodies centred at the origin. Perhaps it is possible to find constructive proofs of the Minkowski-Hlawka theorem or of weaker versions of it in analogy to the work of Rush and Sloane (1987) and Rush (198?) on balls and convex bodies which are symmetric in the coordinate hyperplanes, which will be mentioned in sections 8.2 and 9.4.

Mahler's selection theorem

7.1 For many purposes it is of interest to know whether there exist lattices having certain extremal properties. We give an example: Consider

a convex body C. Does there

exist a

lattice L such that

{C + p: p E L} is a lattice packing of C of maximum density? While it is

intuitively clear that for many problems of this sort there exist

solutions, it is nonetheless very helpful to have a simple tool available which makes sure that extremal lattices exist. Such a tool with a wide range of applications is the compactness theorem of Mahler (1946). Before describing it we need to introduce some notions. On the space £ of all lattices in E' there is a `natural' topology. We describe two ways to introduce it: For a real d x d-matrix A = (a,k) define a norm by = (ak)

i/zIIAII

As we have remarked in section 5.4, any lattice can be written in the form BZd, where B is a d x d-matrix, the columns of which are the vectors of a basis of the lattice. On the other hand it is clear that for any non-singular d x d-matrix B the set BZd is a lattice in Ed. Now we may define the topology on £ by determining that for a lattice L = BZ d of £ a neighbourhood basis consists of the sets

{AZ d: A a d x d-matrix, I IA - B I I < e} where e> 0. A second way to introduce the topology of £' is through the following concept of convergence in £: a sequence L1, L 2, . . . of lattices of 1) is said to converge to a lattice L in £ if there are bases {b'(') b'(d)}, {b2(1),

.

.

., b2(d)},

.

.

.

and {b('), ..., b(d)} of L1, L2, ... and L,

respectively, such that in E d b M>, b 2(l),

.

.

.

b ('),

.

.

.,

b 1(d), b 2(d),

.

.

.

b (d).

Let £ be endowed with the topology thus defined. 37

LATTICE POINTS

38

7.2 The original form of Mahler's selection or compactness theorem is the following: a sequence of lattices with determinants bounded above

and which are all admissible for a neighbourhood of the origin o contains a convergent subsequence. This implies in particular that £ is locally compact.

A nice proof of Mahler's theorem was given by Groemer (1971). Since it exhibits a connection with another important geometric selection

theorem, the selection theorem of Blaschke (1916), we describe the main idea of the proof. For a lattice L define the Dirichlet-Voronoi cell D(o, L) of L by

D(o, L) = {x: xI < Ix - pl for all p E L}. (For more information see section 10.4.) Let L1, L2, ... be a sequence of lattices in El with determinants bounded above and which are all admissible for some neighbourhood of o. Consider the suitably topologized space of compact convex subsets of V. In this space the sequence D(o, L1), D(o, L2), . . has a convergent subsequence by Blaschke's selection theorem. The limit turns out to be the Dirichlet-Voronoi cell of a suitable lattice L and L is the limit of the corresponding .

subsequence of lattices.

7.3 In order to present an application of the selection theorem we outline a proof that for each convex body C in Ed there is a lattice L such that the system { C + p : p E L) is a lattice packing of C which has maximum density among all lattice packings of C, where the density is V(C)/d(L) (cf. sections 3.2 and 8.1). For a lattice L the system { C + p : p E L } is a packing if and only if

L is admissible for the convex body C - C = {x - y; x, y E C}. This

body is a neighbourhood of o. Let L1, L2, ... be a sequence of lattices such that (C + p: p c- L, } is a packing of C for each i c- { 1, 2, . . .} and

such that the densities V(C)/d(L,) tend to the supremum (>0) of the densities of the lattice packings of C. The lattices L1, L2, . . are all admissible for the neighbourhood C - C of o and their determinants are bounded above. Hence Mahler's selection theorem implies the existence of a subsequence of L1, L2, . . converging to a lattice L. It can be shown that L is still admissible for C - C and thus provides a .

.

lattice packing of C. The density of this lattice packing is maximal. For other applications of the selection theorem the reader is referred to the papers of Mahler (see the references in Cassels (1972) or Gruber and Lekkerkerker (1987)) and to Gruber and Lekkerkerker (1987).

7.4

Chabauty (1950) extended Mahler's selection theorem to a uni-

MAHLER'S SELECTION THEOREM

39

formly discrete sequence of subgroups of a locally compact topological group. Some other extensions and applications can be found in Macbeath and Swierczkowski (1960), Mumford (1971) and Harvey (1974). Hammer and Dwyer (1976) and Dwyer et al. (1983) consider compactness theorems for sequences of pairs each consisting of a convex body and a lattice. In particular they deal with the case when the lattices are packing respective covering lattices of the convex bodies (see section 8.1). Another example is provided by Dirichlet-Voronoi cells and the corresponding lattices, essentially considered by Groemer in his proof of

Mahler's selection theorem. These results provide direct proofs for existence theorems like the one in section 7.3.

8

Packing and covering

8.1 The theory of packing and - to a lesser degree - the theory of covering with convex bodies or star bodies, has attracted the interest of

several generations of mathematicians. There also exists a body of articles dealing with packing problems in physics, chemistry, biology and

technology. Early references are Reynolds (1885) (see also Coxeter (1969), §22.4) and Kelvin (1904). Whereas earlier results deal mainly with packing of balls and the problems of tiling (see chapters 9 and 10),

the general theory was developed starting with Minkowski's (1904) paper on lattice packing. The investigation of a more general type of packings was suggested by Hilbert who, in the context of his 18th problem, said the following: I draw your attention to the following question related to the preceding one, and important to number theory and perhaps sometimes useful to physics and chemistry: how can one arrange most densely in space an infinite number of equal solids of given form, e.g. spheres with given radii or regular tetrahedra with given edges (or in prescribed position), that is, how can one fit them together so that the ratio of the filled to the unfilled space may be as great as possible? (Hilbert (1900).)

Important later work is due to Fejes T6th, Bambah, Rogers, Zassenhaus and many others. The monograph of L. Fejes T6th (1972), first published in 1953, spurred a great deal of current research on packing and covering.

Basic references on packing and covering are Hlawka (1949), C. A. Rogers (1964), L. Fejes T6th (1964, 1972, 1984), Baranovskii (1969), G. Fejes T6th (1983) and Florian (1987).

A family of subsets of E' is a packing if any two of these sets have distinct interiors. We will consider mainly packings of congruent copies or of translates of a given convex body or a compact star body C. If a packing of C is of the form {C + p: p E L}, where L is a lattice, it will 40

PACKING AND COVERING

41

be called a lattice packing of C with packing lattice L. Call a family of subsets of Ed a covering if its union equals Ed. It is clear what is meant by covering with congruent copies or translates, by lattice covering and by covering lattice.

In order to avoid difficulties we define the concept of density only for packings and coverings with sets having uniformly bounded diameters. If {Ck: k E I} is such a packing, its density is defined as

Ck n {X: 14

lim supp

Q} * 0}

(2a) d

In case {Ck: k E I} is a covering, define its density by E{V(Ck): Ck C {X: Xij < Q}} lim inf

(20) d

a-_+.

The density of a packing can be interpreted as the `proportion of space' covered by the sets of the packing or as the probability that a randomly chosen point of Ed is contained in a set of the packing. The density of a covering is - cum grano salis - the `total volume' of all sets `divided' by the `total volume' of the whole space. Let C be a

convex body or a compact star body. If {C + p: p E L} is a lattice packing or a lattice covering, the density is V(C)/d(L) in each case. Let 5L(C), OT(C) and bc(C) denote the (attained!) suprema of the densities of the lattice packings of C, of the packings of translates of C and of

the packings of congruent copies of C, respectively. For coverings define similarly OL(C), OT(C), Oc(C) where the suprema are replaced by the infima. 5L(C) and OL(C) are called the densities of the densest lattice packing and of the thinnest lattice covering of C, respectively, and correspondingly for bT(C), bc(C) and OT(C), Oc(C). Obviously

OL(C) < 640 < MO

1 < OC(C)

OT(C) < 0J0-

(8.1)

For the first and last inequalities, it is not known whether there are convex bodies C for which the inequalities are strict. See the discussions in sections 8.3, 8.4 and 9.2. There are star bodies C for which the first and last inequality are strict, see section 8.4.

For a convex body C the problem of finding 6L(C) is equivalent to

the determination of the critical determinant 0(C - C) of the osymmetric convex body C - C. This can be seen from the formula 6L(C) =

V(C)

0(C - C)' A family of subsets of Ed is a k -fold packing or a packing of multiplicity k, k E {1, 2, ...}, if each point of Ed is an interior point of at most k of the sets. The multiplicity is precisely k if it is k and if some

LATTICE POINTS

42

point in E' is an interior point of k of the sets. Call a family of subsets of El a k-fold covering or a covering of multiplicity k if each point of Ed is contained in at least k of the sets. We say that the multiplicity is precisely k if it is k and some point of Ed is contained in precisely k of the sets. All concepts defined in the last two paragraphs can easily be transferred to this more general situation. Thus in particular we may , 9'(C). consider the quantities 8'(C), There is no exact duality between packing and covering but we will arrange the material in such a way that similar results are treated in the same subsections. This has the advantage of showing that progress is not uniform for packing and covering.

8.2

This subsection contains estimates for the quantities SA(C), ST(C) and OL(C), OT(C), where C is a given convex body in Ed. For the cases d = 2 and d = 3 we refer to 8.3. If C is symmetric in o, then the refinement of the Minkowski-Hlawka theorem due to Schmidt (1963) (see (6.6) in section 6.2) yields add < 5T(C) where ad -> logV2 = 0.346573 ... as d ---> +-. 2d-1 (8.2)

We should like to point out the large gap between the lower and upper bound for 8T(C) in (8.2) and (8.1). At present it seems to be difficult to tell whether (8.2) can be improved essentially. Our conjecture is that it probably cannot. Compare also the discussion for balls in section 9.3. Assume now that C is not necessarily symmetric. If { C + p ('), i E I) is a packing of translates of C, then an argument of Minkowski (1904), §1, shows that { 1(C - C) + p ('), i E I) is also a packing and vice versa. Hence SG(C) =

ST(C) =

V (C)

V(1z(C - C)) V(C) V(21

(C - C))

SL(z(C - C)), 8T(2(C - C)).

(8.3)

In a surprising recent work, Rush (198?) proved the following. Let C be a convex body which is symmetric through each of the coordinate hyperplanes. Then there is an explicit construction of a lattice packing of C of density 2-d + o(d)

as d ---> +

(for infinitely many d's).

This construction is based on a particular code and makes use of the

PACKING AND COVERING

43

contructions of Leech and Sloane (1971) for bal, packings, see section 9.4.

By inequalities of Rogers and Shephard (1957) (see also C. A. Rogers (1964), ch. 2) and Brunn and Minkowski (see, for example, Bonnesen and Fenchel (1934), §11) we have 2d V(C) < c 1. C)) V(((2 (C

-

))

Here equality holds in the first inequality if and only if C is a simplex and in the second inequality if and only if C is centrally symmetric. Together with (8.2) and (8.3) this shows that 2add where ad - log '\/2 = 0.346573 . . as d - +oo. (2d) C SL(C) Note that d3/2 2add -\/rrlog V2d3/Z = 0.614285... 4d as d (2d) 4d .

d 1

Thus for the non-symmetric case the lower estimates are even worse than the already very weak ones for the symmetric case.

For covering the situation is much better. While earlier attempts produced for OL(C) upper bounds of the form ad for suitable a > 1, C. A. Rogers (1957, 1959) succeeded in proving that for a convex body C

OT(C) <- d log d + d log log d + 5d OL(C) <

(d , 3),

dlog2logd+#

(8.4) (8.5)

where (3 is a suitable constant. Erdos and Rogers (1962) slightly

improved (8.4). A recent result of Gritzmann (1982) considerably refines (8.5) if the convex body C has k > 1092 log d + 4 perpendicular hyperplanes of symmetry. In this case OL(C) < yd(log d) i +1092e = yd(log d)2 442695

for a suitable constant y. (Compare the corresponding result for balls as exhibited in section 9.3.)

8.3

For d = 2, 3 much more is known than for general dimensions.

We consider the packing case first and give several general results and a list of densities of densest lattice packings of particular convex bodies. The covering case is treated afterwards.

Let C be a plane convex body and let H be a convex hexagon of minimum area circumscribed about C. A result of L. Fejes TOth (1971),

LATTICE POINTS

44

III, §10, says that

SA(C) < A(H)

A( H)

(8.6)

If C is centrally symmetric, then H may be chosen centrally symmetric according to a theorem of Dowker (1944). Considering a lattice tiling of

E2 with H (see section 10.4) we obtain at the same time a lattice packing of the body C which is contained in H. This lattice packing has density A(C)/A(H). Thus (8.1) together with (8.6) yields

6L(C) = 6T(C) = sc(C) = A(H)

(8.7)

for centrally symmetric plane convex bodies C. If C is not necessarily centrally symmetric then we still have

6L(C) = OT(C).

(8.8)

This is an immediate consequence of (8.7) and (8.3). In a different way (8.8) was first proved by C. A. Rogers (1951). An elegant concise proof of (8.8) was given recently by L. Fejes TOth (1983). There exists several refinements of (8.8). We cite only the paper of Graham, Witzenhausen and Zassenhaus (1972), the report of Zassenhaus (1961) and a recent

article of L. Fejes Toth (1985) in which he proves (8.8) for certain non-convex sets, too. One of the great problems of packing theory is to decide whether for

all convex bodies C in Ed, d >_ 3, we have SL(C) = 6T(C). Most probably the answer for d = 3 is yes but there are reasons to conjecture that for d as small as 10 or 11 it is no - see section 9.2. It is an open problem for what centrally symmetric plane convex bodies C does the value of SL(C) attain its minimum. Surprisingly the minimum is not attained for circular discs. Work of Reinhardt (1934) and Mahler (1947) supports the conjecture that the minimum is attained for regular octagons which are suitably smoothed at the vertices. Thus the conjecture says that for any centrally symmetric convex body C in the plane the following inequality holds:

6L(C)>

8-4\/2-log2 = 0.902414 2V2-1

A result of Tammela (1970) which improves upon an earlier theorem of Ennola (1961) shows that for all centrally symmetric convex bodies C in E2,

The bound in Tammela's result is surprisingly close to the conjectured optimal bound.

PACKING AND COVERING

45

For a convex body C in E2 that is not necessarily symmetric a result of Fary (1950) says that

6L(C) - 3', where equality holds for triangles only. (See figure 8.1.)

(a) SL(T) = ST(T) = 2/3

\Z\Z\/\/\/ VVV (b) bc(T) = 1

Figure 8.1 Most dense packing of translates and of congruent copies, respectively, of a

regular triangle T

The problem of determining the density of the densest lattice packing of a particular convex body or star body in general is a difficult task. There is no effective machinery available to tackle this problem. Table 8.1 contains an (incomplete) list of convex bodies for which the densities of the densest lattice packings have been determined. The densities are given only in the cases where they can be expressed in a simple manner. For further results consult Keller (1954) and Gruber and Lekkerkerker (1987). A recent result of Kuperberg (1987(a)) says that for a plane convex

body C we have

bc(C) ,

2s 32

Now we turn to coverings. Let C be a plane convex body and denote

by K (resp. T) a convex hexagon (resp. triangle) of maximum area inscribed into C. A general conjecture asserts that

Oc(C) , A(K). This has not yet been proved, but L. Fejes Toth (1972), p. 86, gave a

result which comes close to it. Fejes Toth's result implies that OT(C) -- A(K).

(8.9)

4

{x: ((x1)2 + (x2)2)1/2 + Ix31 < 1}

{x: IXI !5 1,x31 - Al

Ball

Cylinder with convex base C'

1x: IX1 + X2 + x31 Ixl<1

Tetrahedron Cubo-octahedron

d=3

... - 3)2 + 2A2 + 24A - 1))

fort - A _ 1

A2)/(4(-),3

=0 .

740 480

9

= 0.855 033

'(3 - A2)1/2

6 7rV6

3

7T

6L(C')

for t
9(A3 - 9A2 + 27X - 3)(8A(A2 - 9A + 27))

9A,(9 -

(9 - x,2)/9 for 0 < A << z

49 = 0.918367

198 = 0.367 346.. . 4

bL(C)

Table 8.1 Lattice packing densities of particular convex bodies

Whitworth (1951)

Chalk (1950)

Gauss (1831)

{Yeh (1948)

Chalk and Rogers (1948),

Whitworth (1948)

Groemer (1962(b)), Hoylman (1970)

Author

Parallel body of a rectangle Intersection of two congruent circles

pE(2.035,+-)

1}, {X: IX1I" + IX2IP p E (1.01, 1.99),

Circle

Regular 6k-gon

d=2

Table 8.1 (cont.)

2\

IT

6L(C)

=0.906899.

L. Fejes TOth (1972)

Kuharev (1966, 1968, 1971), Grisanovskaja et al. (1977), Malysev and Voroneckii (1977), Malysev (1977, 1979) L. Fejes T6th (1967)

Lagrange (1773)

Ohnari (1962/63)

Author

LATTICE POINTS

48

On the other hand a theorem of Bambah, Rogers and Zassenhaus (1964) shows that

A(C) (8.10) 2A(T)' Of the inequalities (8.9), (8.10) the first is stronger for certain convex bodies and the second for other convex bodies. A common generalizaOT(C) ,

tion of (8.9) and (8.10) was given by G. Fejes T6th (198?). If C is centrally symmetric the right-hand sides of (8.9) and (8.10) coincide and, moreover, OT(C) = BB(C).

(8.11)

As in the packing case, the problem arises whether (8.11) extends to higher dimensions.

From a result of Sas (1939) on polygons inscribed into centrally symmetric planar convex bodies L. Fejes T6th (1946) showed that for all centrally symmetric convex bodies C in the plane we have OL(C)

227

= 1.209199... ,

where equality is attained precisely for the ellipses. Within the class of all plane convex bodies C the inequality

OT(C)
Thus in the plane the ellipses and the triangles provide the least economical lattice coverings. See figure 8.2. Note that for centrally symmetric bodies the results for packings and coverings are not symmet-

rical to each other. For a plane convex body C a result of Kuperberg

(a) OL(C) = OT(C) = Oc(C) = 21T/N/27

(b) eL(T)= 3/2

Figure 8.2 Thinnest lattice covering with a circle C and a regular triangle T

PACKING AND COVERING

49

(1987(a)) says that Oc(C)

3

In the covering case the values of OL(C) or OT(C) have only been determined for a very small number of convex bodies or star bodies.

8.4 In the last subsection we started to investigate whether for some convex body or star body C there can be strict inequalities in the chain of the inequalities (8.1). This will be continued in the following. As remarked before, no example of a convex body C is known for which

SL(C) < OT(C),

(8.12)

but there are various examples of compact star bodies C for which (8.12) holds. The first such example is due to Stein (1972) in E5. A 3-dimensional example with interesting additional properties was given by Szab6 - see section 10.5. There are many convex bodies and compact star bodies C in any dimension for which OT(C) < Sc(C).

Any triangle provides an example in the plane. Schmidt (1961) proved that for any smooth convex body or smooth star body C we have Sc(C) < 1 < Oc(C).

(Smoothness means that the boundary is a differentiable surface.) As in the packing case there are many convex bodies or compact star bodies C with

Oc(C) < 6T(0A simple example in E2 is a triangle. While it is not yet known whether there are convex bodies C with OT(C) < OL(C),

(8.13)

the compact star bodies discovered by Stein (1972) and Szab6 (1987) satisfy (8.13). A 2-dimensional example of a compact star body for which (8.13) is fulfilled is due to Bambah et al. (1977). A second example results from the work of Kasimatis-Rooney (198?) and Loomis (1983).

8.5

Let C be a convex body and L a lattice in E'. When is L a

50

LATTICE POINTS

packing or a covering lattice for C or some other convex body related to C?

Concerning packings we note that L is a packing lattice for C if and only if it is admissible for the convex body C - C. Besides this very simple remark no non-trivial packing criterion seems to be known.

In the covering case much more can be said. We will state several covering criteria. The first is due to Chalk (1964, 1967) and Yates

(1970): If C contains d + 1 non-coplanar points of L, then L is a covering lattice for dC. Here d may be replaced by [(d + 1)/2] if C is centrally symmetric. (For real a we denote the integer part of a by [a].) The constant d or [(d + 1)/2] cannot be replaced by smaller numbers. From this criterion we obtain that a plane lattice which contains three non-collinear points of a centrally symmetric convex body is a covering lattice for that body.

If any translate of C contains at least one point of L, then L is a covering lattice for C. This simple remark enables us to interpret the results of Bender (1962), Wills (1968, 1970) and Hadwiger (1970) of

section 3.6, and the long list of results of section 3.7 in terms of covering. Thus, for example, the theorem of Bender, Wills and Hadwi-

ger yields the following criterion: if the convex body C satisfies the inequality V(C) ? S(C)/2, then Zd is a covering lattice for C.

8.6 Let C be a convex body and L a lattice in E d. If {C + p : p E L } is a packing, two distinct bodies of the packing are neighbours if they have non-empty intersection. Related to this is the concept of star number: if {C + p: p E L} is a covering, its star number is the number of bodies of the covering (including C) which intersect C. Obviously these two concepts can be defined for more general situations. From a classical result of Minkowski (1896), §31, one easily obtains the following: if (C + p: p E L) is a packing, then C has at most 3 d - 1 neighbours, where 3' - 1 can be replaced by 2 d + 1 - 2 if C is strictly convex. The bound 3 d - 1 is attained precisely in the case when C is parallelepiped and L has a basis consisting of vectors along the edges of C, see Groemer (1960(a)). There are extensions and refinements of these results by Woods (1958(b)), Groemer (1960(b), 1961(b), 1970) and Hadwiger (1969) related to successive minima, the non-lattice case and star bodies.

An elegant result of Swinnerton-Dyer (1953) which generalizes old 1873, 1877) and Voronoi (1908(a)) on balls (or rather: on positive quadratic forms, see section 11.3) is the following: if {C + p: p E L} is a lattice packing of C of maximum density, then C has at least d(d + 1) neighbours. A recent

results of Korkin and Zolotarev (1872,

PACKING AND COVERING

51

result of Gruber (1986) shows that in the sense of Baire categories `most' convex bodies have not more than 2d2 neighbours in any of their lattice packings of maximum density. It is an open problem whether 2d2

may be replaced by a smaller bound or - as is probable - even by d(d + 1). For more information on results about `most' convex bodies the reader is referred to a survey of Gruber (1985). An interesting result of Hadwiger (1969) which extends and generalizes a 2-dimensional theorem of Hlawka (1957) shows that if (C + p : p E L } is a lattice covering, then its star number is bounded above by

V(C - C + C) - V(C) d(L)

This equals (3d - 1) times the density of the covering in case C is centrally symmetric.

A result which goes in the reverse direction is due to Erdos and

if C is centrally symmetric and if {C + P: P E L} is a lattice covering, then the star number is at least 2d + 1 - 1. This also follows from a more general theorem of Groemer Rogers (1964). It says that

(1970). Presumably an analogous result holds for coverings with compact

sets of uniformly bounded diameters, but so far this has been proved only for dimension 2, by Boltyanskii (1950).

8.7 Several results on packings and coverings have been extended to multiple packings and coverings. In addition there are results dealing exclusively with multiple packing and covering.

Let C be a convex body in Ed. Then we have, for k E {1, 2, ...}, (8.14) bk(C) !!S bk(C) < 6k(C) <- k < O (C) .5 OT(C) < O (C). Generalizing a result of Schmidt (1961), Florian (1978) proved that for smooth C we have

6c(C) < k < Bc(C).

Given a packing or a covering of E d any system consisting of k translates of this packing or covering is a k-fold packing, respectively a k-fold covering. This remark yields the inequalities kSL(C)_ SL(C),

kST(C) <- ST(C),

kOc(C) < bc(C),

6c(C) < kOc(C),

BT(C) < kOT(C),

kOL(C).

A corollary of a very general but still elementary result of Groemer (1986) says that for all k we have

LATTICE POINTS

52

d-1

6i(C)

where a(d) _ E(d )d' K` , K.dd

k - a(d)k(d - 1)/d

1

and for k 6i (C)

Kdd5d/2

(8.15)

i

+1

k + l3(d)(k - 1)(d - 1)/d - 1, Kd

lid

dd +

where /3(d) = (_2

(8.16)

(K, is the i-dimensional volume of the solid euclidean unit ball in E'.) An earlier result of the type (8.16) is due to Cohn (1976). In the plane Groemer proved the following sharper results: for all k

6j(C) ? k -

6

(2k)1/2 = k - 2.700948... k1/2

(8.17)

7T

and for all k ? 33 6k (C) <- k + 4(k - 1) 1/2 - 1.

(8.18)

When C is a ball, more precise estimates can be given - see section 9.5. From (8.14)-(8.16) one immediately obtains that SL(C),

OT(C),

oC(C),

6C(C),

6T(C),

6L(C) - k

as k

If L is an arbitrary lattice then the system { C + p : p E L) can be considered as a packing of sufficiently high multiplicity (see the defini-

tion in section 8.1). We want to give an upper bound for its precise multiplicity. Similarly, {C + p: p E L} trivially is a covering of multiplicity 0 and our aim is to give a lower bound for the precise multiplicity.

It is easy to prove that {C + p: p E L} is a packing (covering) of multiplicity k if each translate of C contains at most (at least) k points of L. Applying this remark to the results of sections 3.6 and 3.7 one obtains a series of covering criteria and two packing results. Thus the theorem of Bokowski et al. (1972) yields the following result: the system {C + u: u E Zd} is a covering of precise multiplicity greater than V(C) - ZS(C). For d = 2 Dumir and Hans-Gill (1979a) (k = 2) and G. Fejes Toth (1984) (k = 3, 4) showed that for C centrally symmetric SL (C) = kSL(C)

for k = 2, 3, 4.

(8.19)

For k = 5 this no longer holds as can be seen from the example of circular discs (see Table 9.2). (8.19) does not extend to compact star bodies. This can be seen from an example of Dumir and Hans-Gill. For coverings it was proved by Dumir and Hans-Gill (1972(b)) that if C is centrally symmetric then 6'L(C) = kOL(C)

for k = 2.

PACKING AND COVERING

53

This does not extend to k > 2. An example is provided by the circular disc - see Table 9.2.

8.8 In this section we first consider different measures of density for packings and a measure of thinness of coverings. Then the so-called Minkowski-Hlawka packings or Minkowskian arrangements are treated. The closeness of a packing of congruent copies of a convex body C in

Ed is 1/p, where p is the supremum of the radii of the euclidean balls

which do not intersect the interior of the bodies of the packing. Similarly the looseness of a covering of E d with congruent copies of C is

1/a, where a is the supremum of the radii of the euclidean balls which are contained in the intersection of two or more of the bodies of the covering. This definition of L. Fejes TOth (1976, 1978) was slightly generalized by Linhart (1978) in the case of packings or coverings with translates. L. Fejes TOth (1978) proved that the closeness of an arbitrary packing of translates of a plane convex body C does not exceed the closeness of a suitable lattice packing of C, the so-called closest lattice

packing of C. We point out the similarity of this result with the corresponding result (8.8) employing the concept of density. For higherdimensional results concerning packings and coverings with euclidean balls see section 9.6. Let C be an o-symmetric convex body in Ed. A family of congruent copies of C is a Minkowski-Hlawka packing or a Minkowskian arrange-

ment if none of the bodies contains the centre of another body in its interior. Presumably it was Fejes T6th who first defined and investigated

this concept. A Minkowski-Hlawka packing of translates of C, say {C + p('): i E I) has density at most 2d. This follows from the fact that (' C + p (') : i E I) is an ordinary packing and this has density at most 1.

Answering a question of Fejes Toth, Bleicher and Osborn (1967) showed the existence of Minkowski-Hlawka packings of arbitrarily high density. For more information on Minkowski-Hlawka packings and the related concept of admissibility in the sense of Minkowski-Hlawka we refer to Gruber and Lekkerkerker (1987).

8.9 Given a packing of convex bodies which is very dense,

it is

intuitively clear that by comparatively small expansions of the bodies one obtains a thin covering. There are several results in the geometry of numbers which make this statement more precise. We will present two of these so-called transference theorems.

Let C be an o-symmetric convex body and L a lattice in Ed. The successive minima A 1, ... , Ad of C with respect to L were defined in

LATTICE POINTS

54

section 5.1. It is easy to see that {A1/2)C + p: p E L} is a packing. Define the inhomogeneous minimum µ = µ(C, L) of C with respect to L by

p = µ(C, L) = inf{v > 0: {vC + p: p E L} is a covering). Then clearly {µC + p: p E L} is a covering. A simple result of Jarnik (1941) says that Al+...+Ad.

Ad<2p

Other transference theorems are due to Hlawka (1952), Kneser (1955) and Birch (1956). Let 2dd(L) A1 V(C)

=m+s

where m(? 1) is an integer and 0

s < 1.

Then the results of Kneser and Birch say that A

1

p < 2 (m + Slid), µ

21(m+s)

form?d.

These inequalities cannot be improved essentially.

A related result of a different type is due to Butler (1972). Let C be an o-symmetric convex body. Then there exists a lattice L such that µ <_ (1 + 0(1))Al,

where o(1) -- 0 as d -3 +- uniformly for all C.

9

Packing and covering with balls

9.1 The problem of dense packing of balls and related problems have attracted the interest of mathematicians for many centuries. The interest of crystallographers, physicists, chemists, biologists, engineers and other scientists in these problems is more recent. Surprisingly enough, work on covering problems for balls started only a few decades ago. The earliest reference to packing of balls that we have been able to locate is Kepler (1611). Another early contribution is the discussion of Newton and Gregory on the problem of the thirteen balls: can a ball in E3 be touched by at most 12 non-overlapping balls of equal radii as asserted by Newton or by 13 as asserted by Gregory? The answer in favour of Newton was given as late as 1874 by Hoppe (1874), and in a more precise form by Schiitte and van der Waerden (1953); see Coxeter (1963) and Ehrhart (1985). Lagrange (1773), Seeber (1831) and Gauss (1831) essentially considered the problem of finding the densest lattice packing of balls in dimensions 2 and 3 which at present is solved up to dimension 8 only. Hales (1727) made experiments with random packings of peas by compressing them in a vessel. His experiment was repeated

in 1939 by Marvin and Matzke with small balls made of lead, see Matzke (1950). Later experiments of this sort are due to Bernal (1959) and G. D. Scott (1962). Professor Zassenhaus informs us that there are

hundreds of papers in natural sciences and technology dealing with packing of balls but it seems to be difficult to get hold of them. Melnyk et al. (1977) try to inform chemists on recent mathematical results on packing of `hard' and `soft' circles on the surface of the unit ball. A surprising connection between packing of balls and coding will

be discussed below. In section 11.3 we will give some background information on ball packing. The extremum problem for the zetafunction of section 15.2 can be interpreted as the problem of densest lattice packing of `soft' balls. Section 11.5 contains some more general remarks on coverings with balls. 55

LATTICE POINTS

56

Readers who want to obtain a more complete view of ball packing are referred to the comprehensive book of Conway and Sloane (1987) on sphere packing.

9.2

In this subsection we will discuss the classical low-dimensional results on packing and covering. By bL(B') and OL(B') we denote the densities of the densest lattice packing, respectively the thinnest lattice

covering, with the solid euclidean unit ball B1. Table 9.1 gives the known values of bL(Bd) and OL(Bd) respectively.

Professor Hlawka informs us that the complicated calculations of Blichfeldt were also confirmed by R. Brauer and himself. The lattices for which 6L(Bd), d E{2, . ., 8} are attained are unique .

up to rotations and reflections. This was proved by Lagrange (1773) (d = 2), Gauss (1831) (d = 3), Korkin and Zolotarev (1872, 1877) (d = 4, 5), Barnes (1957) (d = 6) and Vetcinkin (1980) (d = 7, 8). Similarly, the lattices for which OL(Bd), d E {2, . . ., 4} are attained are also unique up to rotations and reflections, as shown in the proofs of Kershner, etc. More generally Barnes (1956(b)), Baranovskii (1965(b), 1966), Delone and Ryskov (1963), Barnes and Dickson (1967(a)) and Dickson (1968) determined (up to rotations and reflections) the lattices in E2, E3, E4 which (among all lattices) locally give thinnest lattice coverings with Bd. There are 1, 1 and 3 such lattices, respectively. Non-lattice packings of balls have been considered since Thue (1892, 1910) indicated that 6L(B2) = 6 T(B2).

Thue's proof lacks certain compactness results. A first rigorous proof

was given by L. Fejes T6th (1940). The corresponding result for coverings is due to Kershner (1939): OL(B2) = OT(B2).

For more general results in this area we refer to L. Fejes T6th (1972) and to chapter 8 and the references cited there. For /d = 3 it is highly probable that bL(B3) = 6T//(B3), OL(B3) = 1T(B3),

but a proof seems to be exceedingly difficult. The following observation illustrates some of the difficulties. If one arranges 12 non-overlapping balls of unit radius in an icosahedral manner around a fixed unit ball,

then the density of this arrangement in its convex hull is greater than that of the corresponding lattice arrangement of 12 balls around a

PACKING AND COVERING WITH BALLS

57

central ball. It was proved by C. A. Rogers (1958) (see also Rogers (1964)) and later by Baranovskii (1964) that

0.740480... =

77

3-\/2

1

= 6L(B3) , 6T(B3) - V18 (arccos3

= 0.779635 .

.

IT

l

3)

..

Table 9.1 Packings and coverings with balls d 2

Author

oL(Bd) 7r

21/3

= 0.906899 ...

7T

3

4

3'\/2 = 7T

0.740480

...

Lagrange (1773) Gauss (1831)

2

- = 0.616 850 ... 16

Korkin and Zolotarev (1872, 1877) 2 7T

5

15 V2 7T3

6

7

48'\/3 IT3

105 =

= 0.465 257 . = 0.372 947 . 0.295 297

...

Blichfeldt (1925, 1926, 1934), Watson (1966), Vetcinkin (1980)

8

7T 4

384

= 0.253699 ...

d

OL(Bd)

2

3\/3 = 1209 199

3

5'\/57r = 24

4

5

Author

2rr

27r2

= 7/5 =

...

1.463503...

1.765528...

245-\/(35)7T2

3888-\/3

_ 2 124 285

Kershner (1939)

Bambah (1954(c)), Few (1956), Barnes (1956(b))

Delone and Ryskov (1963) R ys Vkov an d B aranovs kii ( 1 975)

LATTICE POINTS

58

A slightly smaller upper bound is due to Lindsey (1986), see also L. Fejes TOth (1972), VII, §2. A substantial first step towards the verification of the equality 6L(B3) = ST(B3) was obtained by Dauenhauer and Zassenhaus (1987). They showed that perturbations of a local type of

the densest lattice packing of B3 never increase the density of the packing. Professor Zassenhaus suggests that it might be possible to reduce this problem to a finite problem which, in principle, can be solved using a computer. Each lattice packing of Bd leaves `holes' which are not covered by the packing. It is conjectured that in dimensions 9, 10 or 11 the holes of the

densest lattice packing are so large that they can hold balls of unit radius. This would lead to a `double lattice packing' of BI having twice the density of the densest lattice packing. It is an open problem to verify this and to find the smallest dimension for which the densest packing of balls of unit radius has larger density than the densest lattice packing. Concerning coverings, a result of Coxeter et al. (1959) implies for d = 3 that 1.432 282

... = 3

3

(3 aresec 3 - iT) -- 0T(B3) -- OL(B3) =

5ir

5

2 503 ... = 1.463

2

9.3

Since one may not expect to be able to find the precise values of SL(Bd), ST(Bd), OT(Bd), OL(Bd) for all dimensions d (but see section

11.3 for SL(Bd)), it is of great interest to find asymptotic upper and lower bounds. A special case of the Minkowski-Hlawka theorem due to Minkowski himself (see the discussion in section 6.1) shows that S(d)21-d c 6L(Bd) < ST(Bd).

A slightly larger bound can be obtained from Schmidt's (1963) refinement of the Minkowski-Hlawka theorem. On the other hand the first significant upper bound for ST(Bd) was given by Blichfeldt (1929): SL(Bd)

bT(Bd) c d + 2

2-d12

2

Subsequent slight but difficult improvements of this result were given by

Rankin (1947), C. A. Rogers (1958) and Baranovskii (1964). Quite contrary to the expectations of many number theorists who believed that

2-d/2 should be the approximate order of magnitude of SL(Bd) and 6T(Bd), the following essential sharpenings were successively achieved by means of spherical harmonics ('linear programming method'):

PACKING AND COVERING WITH BALLS

59

ST(Bd)

2-05096d+o(d)

Sidel'nikov (1973, 1974)

bT(Bd)

2-0 5237d+o(d)

Levenstein (1975)

ST(Bd)

2-0599d+o(d)

Kabat'janskii and Levenstein (1978)

as d - -. There are some people who conjecture that the correct order of magnitude of bL(Bd), 6T(Bd) is 2-d+o(d) i.e. of the same order of magnitude as the lower bound. Part of the proof of Kabat'janskii and Levenstein was anticipated by Delsarte (1973) and by Delsarte et al. (1975).

Considering Siegel's mean value theorem in section 6.2 one obtains that for `many' lattices L the density of a packing of balls with packing lattice L is at least 2-d. Unfortunately it is rather difficult to explicitly

construct lattice packings or general packings of balls having this density, see section 9.4. There we describe several of the steps which finally led to the Minkowski-Hlawka bound.

For coverings we have the following lower and upper bounds for OT(Bd), OL(Bd) of Coxeter et al. (1959) and C. A. Rogers (1959):

0.223130... d + o(d) =

d

Or(Bd) < OL(Bd)

<_ O(d(log d)1/21o92(2rre)) = O(d(log d)2 047095. )

as d -. It is remarkable that in the covering case the gap between the lower and upper bound is much smaller than in the packing case. A second difference is worth mentioning. Although the upper bound of Rogers (1959) does not yield explicit constructions of thin lattice or non-lattice coverings, weaker earlier bounds of Davenport (1951, 1952) and Watson (1956) do. The idea for such a construction underlying the papers of Davenport (1951, 1952) was elaborated by Ryskov (1967).

9.4 Packing, tiling and covering have connections with several branches of mathematics which at first sight seem to be of completely different character. One such branch is coding. We will give some basic definitions of coding theory and then show how codes in principle may

be used for effective constructions of ball packings. For exhaustive information on material of this type see the monograph of Conway and Sloane (1987).

A (binary) code C of length d is a set of ordered d-tuples of Os and is, the so-called codewords. The Os and is are the letters of the codewords, their weight being the number of is. Given two codewords, the number of letters in which the codewords differ is their (Hamming) distance. The minimum of the distances of any two distinct codewords

LATTICE POINTS

60

of C is the minimum distance of C. A binary code of length d consisting of M codewords and of minimum distance m is called a (d, M, m)-code. A (d, M, m)-code C is linear of dimension k if it can be identified with

a k-dimensional subspace of the d-dimensional vector space over the Galois field GF(2), i.e. with coordinate-wise addition modulo 2. In this case we shall call C a [d, k, m]-code instead of a (d, M, m)-code. Given d, a `good' code is one for which for given m the number M, or for given M the number m, is `large'. Note that each code may be

repesented as a subset of the set of vertices of the unit cube {x: 0<xi<_1)of Ed. A reference on coding is McWilliams and Sloane (1977). Thompson (1983), Sloane (1977, 1982) and Conway and Sloane (1987) survey the relations between coding and packing of balls.

Codes are used for the transmission of data. First a message is translated into a finite sequence of codewords by an `encoder'. Then the codewords are transmitted by a `transmitter' and finally the codewords

are translated back by a `decoder' such as to obtain the original message. In the process of transmission errors can occur. Thus a codeword w may be transmitted into a d-tuple w' * w. If one knows in advance that the maximum number of errors per codeword is less than half the minimum distance of the code, then one can reconstruct the original codeword w from the d-tuple w' resulting from the transmission. Thus we may correct errors. (For more particulars see the book of Thompson (1983).) There are several constructions for packings of balls using codes, see Leech and Sloane (1971), Sloane (1977) and Conway and Sloane (1987). We will describe two of these which well reflect the basic idea. 1.

Let C be a linear [d, k, m]-code for which all codewords have even weights. Represent C as a set of vertices of the unit cube in E d and consider the lattice

L(C) = {2-1/2p: p = c + 2u for c e C, U E Z d) in Ed. It has determinant d(L(C)) = 2(d/2)-k and is a packing lattice for the ball pBd, where 2-3/2m 1/2

2.

ifm<--4,

P= -1/2 ifm?4. Let C be a linear [d, k, m]-code such that the weights of all codewords are multiples of 4. Again represent C as a set of vertices of the unit cube of Ed and consider the lattice

M(C) = (2-1/2p: p = c + 2u for c E C, U E Zd and such that E ui is a multiple of 4).

PACKING AND COVERING WITH BALLS

M(C) is a lattice of determinant d(M(C)) =

61

2(d/2)-k+1 and gives a

packing of Bd.

There are many codes which have been used for the construction of dense lattice and non-lattice packings of balls, e.g. Hadamard codes, Golay codes, Reed-Muller codes, Nordstrom-Robinson codes, BCH codes, Justesen codes, Preparata codes, codes based on algebraic curves,

codes over Galois fields, and others. In many cases they were used together with the constructions of Leech and Sloane (1971) described earlier. The results obtained typically give for infinitely many ds asymptotic values or lower bounds for the densities of lattice and non-lattice packings of balls which are explicitly constructed. The following is a short list of lattice results of this type, the last entry showing the sensational result of Rush (198?) in which he asymptotically attains the Minkowski-Hlawka bound of section 9.3. Density

Author

2 -d Iogz d+o(d 1og2 d)

Bos et al. (1982)

2-2.304+o(d)

Litsyn and Tsfasman (1985)

2-i 000000007719...d+o(d)

Rush and Sloane (1987)

2-d+o(d)

Rush (198?)

Here logy d is the smallest value of k for which the kth iterated 2-logarithm of d is less than 1, i.e. 19921092

.

.

.

loge d , 1,

19921092 .

k-1

.

.

loge d < 1.

k

The result of Rush (198?) is valid not only for balls but also for convex bodies which are symmetric in all coordinate hyperplanes. See also the discussion in sections 6.3 and 8.2.

9.5 Multiple packings and coverings with balls have been investigated intensively. For surveys we refer to Few (1967) and G. Fejes Tdth (1983). General results in this area are contained in section 8.7. Table 9.2 contains several explicit results.

An algorithm of Linhart (1983) allows the determination of 6i(B2) and 6i(B2) for each k with any prescribed accuracy. Temesvari (1984)

remarks that her method for k = 5 may be extended to determine O(B2) for any k, see also Temesvari et al. (1987). In particular she found the exact values of 67L(B2) (found earlier by Haas (1976)) and Bi(B2).

=

2

...

8

7

6

5

= 3 . 627 598

.

4V(220 - 21/193)1/(449 + 321/193) =

39697T

8827 = 6.489 245 ... 115

.

V7 = 4 749 641 ... V6= 5 611155 ...

V3

...

813 799

4

.

32 = 2.720 699 ...

1

3

v3

8L(BZ)

k

7 . 521 697

...

Table 9.2 Multiple packings and coverings of circular discs and balls

Yakovlev (1983) , Bolle (1976)

Blundon (1964), Krejcarek (1963), Bolle (1976)

Szirucsek (1960), Blundon (1963)

Heppes (1959)

Author

87T

...

..

=

4 . 363323

216

...

6 . 583 419

.

= 3 . 435627

= 2 784226 ..

...

V3\/(76U6 - 159)

87r

9L(B3)

=

k

27 V3

y7=5.428161...

18

7.672...

2

612

7rV(27138 + 2910\/97)

31/3 =

47r

.

2 . 418 399

=1

O (BZ)

3'\/3

7

6

5

4

3

2

k

2

k

Table 9.2 (cont.)

... (1957)

Few (1967)

Author

Haas (1976)

Subak (1960)

Subak (1960), Temesvari (1984)

l

B un don

Author

Few and Kana g asaba path y (1969)

Author

LATTICE POINTS

64

Bolle (1976) proved that for suitable positive constants a1, ..., a4 > 0 one has for k E N

k - a,015

SkL(B2)

k + a30/4

OL(B2) <_ k + a4k2"5.

<_ k - a2kl14,

The exponent a in these estimates is the best possible, as shown by Bolle (1984). Bolle (1982) extended the estimates to arbitrary dimension d. Compare also section 8.7 and the results of Groemer cited there.

9.6 This section deals with several more particular questions. Leech (1967) specified a lattice in E24 with many remarkable properties, the so-called Leech lattice. (In E8 the corresponding lattice is the Witt lattice.) It is related to the binary Golay code G24. Leech conjectured that it is the packing lattice of the densest lattice packing of balls in The Leech lattice has a very large group of rotations and reflections which map the lattice onto itself, and which turned out to be of great importance in the search for the sporadic simple groups - see Conway E24.

(1969, 1971) and Tits (1975).

The Newton or kissing number of a ball in E° is the maximum number of non-overlapping balls of the same radius as the given ball that can touch the given ball - see the surveys of Coxeter (1963) and Sloane (1982). The Newton number of a ball in Ed is known only for

d = 2, 3, 8, 24. For d = 2 it is 6. For d = 3 the problem arose in a conversation between Newton and Gregory - see section 9.1. The exact value is 12 and was found by Hoppe (1874). A more recent proof is due to Schiitte and van der Waerden (1953). For d = 8 the kissing number is 240. It is attained only by the arrangement of the balls touching a fixed

ball in a densest lattice packing of balls. For d = 24 the value of the Newton number of a ball is 196 560. It is attained only by an arrangement of balls in a lattice packing of balls where the packing lattice is (a congruent copy of) the Leech lattice that touch a fixed ball. The values 240 and 196560 have been determined by Odlyzko and Sloane (1979) and Levengtein (1979) independently. The uniqueness result is due to Bannai and Sloane (1981). For many more results and asymptotic upper and lower bounds see the book of Conway and Sloane (1987). The Leech lattice ball packing locally has maximum density. This was first pointed out and proved by Gruber, Hietler and Luptacik (1987),

but it soon turned out that simple proofs could be based on results known earlier, for example, on results of Barnes (1959) and Conway and Sloane (1982), as communicated by Neil Sloane. Hence, the authors

decided not to publish their computer proof, which is based on the theorem of Voronoi in section 11.3 and on the simplex algorithm.

PACKING AND COVERING WITH BALLS

65

The Leech lattice is an example of an (even) integer lattice, i.e. a lattice L such that jx12 is an (even) integer for each x E L.

Integer lattices and especially even integer lattices have attracted a good deal of attention for many decades. Important contributions towards a classification are due to Witt (1941), Kneser (1957), Niemeier (1973),

Vinberg (1985) and others; see Gruber and Lekkerkerker (1987) for references. It is plausible to suppose that lattices L which provide locally or

globally densest packings of balls are (multiples of) integer lattices, having large groups of rotations and reflections. (Compare Voronoi's theorem in section 11.3 and note that a large group of rotations and reflections of L implies that in the corresponding packing of balls of maximum radius any ball is touched by a large number of other balls; thus in the ball packing there are clusters of high density which yield that the packing itself has high density.)

Results of Heppes (1961), Horvath (1970), Hortobagyi (1972) and Ryskov and Horvath (1975) show that there exists a number Pd > 0 such

that for any lattice packing of Bd there is an unbounded `circular' cylinder with radius Pd which does not overlap any of the balls of the packing. For 3 -- d < 8 explicit values of Pd are given. We remind the reader of the notion of closeness of a packing of balls of equal radii in Ed. This is 1/p, where p is the supremum of the radii

of the balls in Ed which overlap none of the balls of the packing. Similarly, the looseness of a covering of Ed with balls of equal radii is

1/0, where a is the supremum of the radii of the balls which are contained in the intersection of two suitable balls of the covering. See section 8.8 for a more general case. Boroczky (1984) proved that among all packings of balls of unit radius

in E3 precisely those packings have maximal closeness which are translates of lattice packings of B3, where the packing lattice is a body

centred cubic lattice of side length 4/\/3. A lattice L is said to be a body centred cubic lattice of side length A if there are three orthogonal vectors b('), b(2), b3 ) E L with jb(') = Ib(2,1 = Ib(3) = A and

L = {uib(1) + u2b(2) + U30) + 2 (b(') + b(2) + b(3)): u; E Z}. (The body centred cubic lattices are precisely the lattices in E3 which provide the thinnest lattice coverings with balls. This was shown by Bambah (1954(a)), cf. Table 8.1.) Note that in the remarkable result of Boroczky there is no restriction to lattice packings in advance. From the packing result one easily deduces the corresponding covering result. It says that among all coverings of E3 with balls of unit radius

66

LATTICE POINTS

up to translations precisely those coverings have minimal looseness which are translates of lattice coverings of B3, where the covering lattice is a body centred cubic lattice of sidelength 4/V5. Surprisingly, in the packing maximum closeness is not attained for the densest lattice packing, whereas in the case of covering minimal looseness is attained by the thinnest lattice covering by B3. This remark again shows that there is no clear duality between packing and covering.

10

Crystallography, tiling and Hilbert's 18th problem

10.1 Crystallography is closely connected to other topics considered in this book. In particular it has strong relations to packing and tiling. In this section we first give an introduction to mathematical crystallography. Then we consider tiling problems and the relations of both subjects to Hilbert's 18th problem, part of which we have already encountered in the context of packing. Finally tiling problems in the context of discrete geometry but of a more general character are treated; we also include Penrose tilings and related completely unexpected recent results from crystallography. Kepler (1611) and Huygens (1690) seem to have been the first ones to

consider the possibility that lattice structures are the background for crystallography. Hauy (1784) though that crystals were constructed from small congruent parallelotopes filled with homogeneous matter. In the

opinion of Seeber (1824) crystals consist of parallelotopes in which spherical `atoms' are distributed. This concept of crystals is remarkably close to the modern view. In modern mathematical crystallography the central notion is that of a `symmetry group' of a crystal. A major problem is the classification of crystals by means of their symmetry groups. Among the workers in this branch of crystallography we mention Hessel, Bravais, Jordan, Sohncke, Fedorov and Schoenflies, Bieberbach, Delone, Hermann, Burckhardt, Heesch and Zassenhaus. For more material, interesting historical accounts, remarks and illus-

trations see Burzlaff and Zimmerman (1977), Brown et al. (1978), Schwarzenberger (1980), the monographs of Engel (1986) and Galiulin (1984) on geometric crystallography and the articles of Pedersen (1983), Grunbaum (1984), Hilton and Pedersen (1984) and Senechal (1986). Readers interested in ornamental arts are advised to read the discussion of Pedersen, Grunbaum and Hilton. Planar tilings have been considered in the arts and by craftsmen since 67

LATTICE POINTS

68

antiquity or even earlier. Not so well known is the fact that threedimensional tilings were also treated in antiquity. Aristotle (erroneously)

thought that it is possible to tile space by regular tetrahedra. This statement was repeated by Roger Bacon who, in addition, asserted that one can tile space with regular octahedra, which is false too. (See Struik (1928).) Important modern contributors to tiling theory are Dirichlet, Fedorov, Voronoi, Minkowski, Delone, Reinhardt, Venkov and many living mathematicians.

For material not treated in the following but close to our subject see Fejes Toth (1964), Grunbaum and Shephard (1980, 1986) and Gruber and Lekkerkerker (1987). We also refer to section 11.5. For a compre-

hensive exposition of planar tilings see the book of Grunbaum and Shephard (1986). Related material is presented in several articles contained in Hargittai (1986).

10.2 A crystal structure in E d is a subset of E d such that 1.

2.

among the rigid motions of Ed that map the set onto itself there are d translations of Ed in d linearly independent directions, and the translation vector of any translation different from the identity

mapping (id) and which maps the set onto itself has length ? b, where S > 0 is fixed.

A symmetry operation or a symmetry of a crystal structure is a rigid motion of Ed onto itself which maps the crystal structure onto itself. Clearly the set of all symmetry operations forms a group under composition, the so-called space group of the crystal structure. For a different way of introducting space groups see section 10.5. Represent a rigid motion m of E d onto itself, say

x -' rx+t

forxEEd,

by the pair (r, t), where r is the linear component of m (a rotation or a reflection) and t the translation vector, respectively. For rigid motions (q, s), (r, t), we have

(q, s) (r, t)

_ (qr, qt + s),

(q, s) -I

= (q-'

-q-1s)

Let G be a space group. The lattice L of G is the set of all translation vectors t with (id, t) E G. It can be shown that L (with ordinary vector addition as group operation) is isomorphic to the unique maximal normal subgroup of G. The lattice of G is also a lattice in the sense of section 3.1. The point group H of G is the set of all linear maps r of E d onto itself such that (r, t) E G for suitable t E L. Let (r, t) E G and

CRYSTALLOGRAPHY AND TILING

69

s E L. Then (id, s) E G and we obtain

(r, t) (id, s) (r,

t)-1

= (r, t) (id, s) (r

-r 1t)

_ (r, rs + t) (r-1, -r-1t)

=(id,-t+rs+t) _ (id, rs). Hence rs e L. From this follows

HL = L. As a consequence of this, one obtains that the point group of a space group is always finite.

10.3 Next we consider classifications of space groups and lattices and investigate some of their properties. The results stated are essential steps for the solution of the following part of Hilbert's 18th problem: Are there in n-dimensional euclidean space also only a finite number of essentially different types of groups of motion with a fundamental region? (Hilbert (1900))

(A group of motions with fundamental region is a space group - see section 10.5). Hilbert posed this problem after the 2- and 3-dimensional cases were already settled. These cases will be described in some detail. Two space groups G1, G2 are of the same space-group type if there is an invertible affinity a of E d such that G2 = a-1G1a.

(10.1)

That is, for suitable choices for the origin and bases of space, the symmetry operations of G1 and G2 are described by the same expressions. A result of Bieberbach (1912) shows that two space groups are of the same type if and only if they are isomorphic. Bieberbach (1910(a))

and Frobenius (1911) proved that in any dimension there are only finitely many space-group types, thus giving an affirmative answer to Hilbert's problem. In crystallography a slightly more refined classification of space-group

types is used: Two space groups G1, G2 are said to be of the same crystallographic or proper space-group type if there is an affinity, a, with

linear component having positive determinant and such that (10.1) holds. A space-group type either coincides with a crystallographic space-group type or else splits up into two crystallographic space-group types which are then said to form an enantiomorphic pair. For d = 2 the crystallographic space-group types were determined by

LATTICE POINTS

70

Fedorov (1891(b)). Later proofs of Fedorov's result are due to Fricke and Klein (1897), Niggli (1924) and P61ya (1924) - see Schwarzenberger (1980). There are 17 plane space-group types, none of which splits into

an enantiomorphic pair. The plane crystallographic space-group types are represented by the symmetry groups of the crystallographic structures in figure 10.1 (periodically continued). (pl, p2, . ., p6mm is a standard notation in crystallography for the space-group types.) .

o-

O--I

o-

-0- r O

.

o- o- o-

r0j r-o- r-O-

o- o-

-o- r O ro

o--I

p2

P1

0-1

o-+ O o- 0-,

0-4

0-. 0-4

O- 4

0-4 o o- 0_ 0

0 0_ OJ

pm

0--I 0 O-I

0-I

0-+ 0-

0-I 0-4 cm

FOiF0I

o-A

.

0

r-0- r-0-

-O '-o-, I--O pgg

r-0

ro

-o-- moo-, ro-

F-0-4 I-41 F-0-4

r-0- "o- r41

pmm

pmg

. L-0-, -0-,

r0- r-oI-- -, 0-, I-0

o+

F-o-, 1-b--I F-<>-1

pg

CI 0-1 0-I 0--I

O-I

F-0-4 F -

F 11 0 1

1

: 0-: :0. 0I

0

1

I-O-i : 0 1 F-o-1

cmm

p4

p4mm

p4gm

p3

p6

p3lm

p3ml

p6mm

Figure 10.1 Crystallographic structures representing the 17 plane space-group types

All 17 space-group types are represented by the space groups of ornaments. 13 of them were discovered in the Alhambra. Of the

remaining ones 3 appear in a Japanese book on pattern design and the

CRYSTALLOGRAPHY AND TILING

71

last one in a book on Chinese lattice. Of course, there are many more places where such ornaments have been found, see the discussion in Hilton and Pedersen (1984). According to Montesinos (1987) pp. 96, 97,

224-8, all 17 space-group types can be found in the Alhambra, if, in some ornaments, the colouring, floral ornaments and lettering are deleted. For d = 3 the 230 crystallographic space-group types were specified

by Fedorov (1891(a), 1892) and Schonflies (1891). Both authors gave incomplete lists first. The minor errors in their lists were eliminated by correspondence. Earlier contributions are due to Jordan (1867, 1868/69) and Sohncke (1879). The latter determined the 65 types of space groups where the linear components all have determinant 1. For an alternative proof of the result of Fedorov and Schoenflies see Rohn (1900). The 230 crystallographic space-group types may be obtained from the 219 spacegroup types, 11 of which split. For a listing of the (crystallographic) space-group types see, for example, the International Tables of Henry

and Lonsdale (1965) or the concise but carefully written book of Burzlaff and Zimmerman (1977).

While for d = 2 or 3 direct methods yield the complete lists of (crystallographic) space-group types, the determination of the crystallographic space-group types for arbitrary dimension d can be achieved by

an algorithm of Zassenhaus (1948). Unfortunately this algorithm requires the knowledge of a representative from each conjugacy class of finite subgroups of GL(d, Z) together with its normalizer in GL(d, Z). By the joint work of several authors the 4783 space-group types for d = 4, of which 112 split, were found by 1973, thus giving 4895 crystallographic space-group types altogether. For a listing and precise attributions we refer to Brown et al. (1978). Two space groups G1, G2 belong to the same geometric crystal class if

there is an invertible linear map r of Ed onto itself such that for the corresponding point groups H1, H2 we have

H2 = r-1Hlr. Unlike the situation for space-group types, space groups with isomorphic

point groups do not necessarily belong to the same geometric crystal class.

In analogy with the discussion before, one introduces the concepts of proper geometric crystal class and of geometrically enantiomorphic pairs

of these. Each (crystallographic) space-group type is contained in a (proper) geometric crystal class. For d = 2 there are 10 geometric crystal classes, none of which splits.

The 17 crystallographic space-group types in the plane are grouped together as follows to form the 10 geometric crystal classes:

LATTICE POINTS

72

pl; p2; pm, pg, cm; pmm, pmg, pgg, cmm; p4; p4mm, p4gm; p3; p6; p31; p3ml, p6mm.

The 32 geometric crystal classes in E3 were determined independently by Frankenheim (1826), Hessel (1830) and Bravais (1849). None of these classes splits up. In dimension 4 there are 227 geometric crystal

classes of which 44 split, thus giving 271 proper geometric crystal classes. This result is due to Bblow (1967). For lists by dimensions see Brown et al. (1978). As remarked before, the symmetry operations of a space group map the lattice of the space group onto itself. However, there can be more

symmetry operations that map the lattice onto itself. An example is provided by a space group of type p3, see figure 10.1. This group does not contain any reflection in a line but the lattice of the group has many symmetries which are reflections in suitable lines. We give a classification of lattices in Ed. Let L be a lattice in Ed (in the sense of the definition in section 3.1) and let H(L) denote the set of all rigid motions which keep o fixed and map L onto itself. (H(L) is the point group and L is the lattice of the space group G of the crystallographic structure L.) We say that two lattices L1, L2 in El are of the same Bravais type if there is an invertible linear transformation r of E d onto itself such that

L2 = r-1L1,

H(L2) = r-1H(Li)r.

As before it is obvious how to introduce the concepts of proper Bravais

type and of enantiomorphic pairs of proper Bravais types. From the

corresponding result on space groups one can deduce that in any dimension there are only finitely many Bravais types. In E2 there are five Bravais types, none of which splits up. They are represented by the lattices in figure 10.2.

Figure 10.2 Bravais types in E2

For d = 3 the 14 Bravais types of lattices were discovered by Bravais

(1850). Of the 14 Bravais types none splits up. In E4 there are 64 Bravais types of which 10 split up, giving 74 proper Bravais types. In sections 10.5 and 11.4 we will again return to related problems of mathematical crystallography.

CRYSTALLOGRAPHY AND TILING

10.4

73

In the following we first give the definition of tiling. Then the

so-called Dirichlet-Voronoi tilings are considered. A family of compact subsets of Ed with non-empty interiors is a tiling of Ed if the union of these sets equals Ed and any two distinct sets have disjoint interiors. The sets are called tiles. Let C be a compact set with non-empty interior. It is clear what is meant by a tiling of Ed with congruent copies or translates of (the prototile) C. If L is a lattice such that {C + p: p c- L} is a tiling, then this tiling is called lattice tiling of C with tiling lattice L.

By a (convex) polyhdedron we understand a subset of E d with the property that its intersection with any (convex) polytope is empty or a (convex) polytope. (Sometimes a more general definition is used. Then polyhedra in our sense are locally finite polyhedra.) If in a tiling all tiles are convex bodies we speak of a convex tiling. Call a tiling locally finite if for any point of Ed there is a neighbourhood which is intersected by at most finitely many tiles. It follows that any bounded set is also intersected by finitely many tiles only. It is easy to show that the tiles of a locally finite convex tiling are convex polyhedra (or polytopes if they are bounded). Let a locally finite tiling be given consisting of convex polytopes. The tiling is facet-to-facet if for any two distinct tiles with intersect in a (d - 1)-dimensional set, the intersection is a facet of both tiles. The tiling is called face-to-face if the intersection of any two tiles is a common face of both of them. Gruber and Ryskov (198?) showed that a facet-to-facet tiling is face-to-face. Thus a locally finite facet-to-facet tiling by polytopes gives rise to a (possibly infinite) cell complex, the cells of which are the polytopes of the tiling and their faces of all dimensions (cf. Alexandroff and Hopf (1935)). See figure 10.3 for some examples of tilings.

(a) not convex, not locally finite Figure 10.3 Tilings

(b) convex, locally finite, not facet-to-facet

LATTICE POINTS

74

A general method of constructing an important class of tilings goes back to Dirichlet (1850). It was later investigated by Voronoi (1908(a),

1908(b), 1909). Tilings of this class are of interest for packing and covering of balls (see C. A. Rogers (1964)), in metallurgy (Smith (1964)), crystallography (Koch (1973)) and geography (Rhynsburger (1973), Watson (1985)). For applications in combinatorics see the survey of Akimova (1984). Applications in numerical integration and quantiza-

tion of data are due to Babenko (1977) and Barnes and Sloane (1983(a)). Let L be a discrete (finite or infinite) point set in Ed (i.e. without accumulation points), for example a lattice. To it corresponds a Dirichlet-Voronoi tiling consisting of the Dirichlet-Voronoi cells

D(p, L) = {x: Ix - pj < Ix - qj for all q E L}, P E L. Instead of a Dircihlet-Voronoi cell one speaks also of Dirichlet or Voronoi cell or region, Wabenzelle, honeycomb, Wirkungsbereich, domain of action, Brillouin zone or Wigner-Seitz zone. Some DirichletVoronoi tilings are shown in figure 10.4.

(a) L finite Figure 10.4 Dirichlet-Voronoi tilings

A slightly more general concept is that of a power diagram - see Paschinger (1982) and Aurenhammer (1982). Related to DirichletVoronoi tilings is the notion of L-tiling or Delone triangulation due to Voronoi (1908(a), 1908(b), 1909) and Delone (1928, 1934(b)) - see e.g. Gruber and Lekkerkerker (1987) and section 11.5. Computational aspects are treated in the book of Preparata and Shamos (1985) on computational geometry. For a wealth of material, see Edelsbrunner's (1987) book on algorithms in combinatorial geometry.

Any Dirichlet-Voronoi tiling is convex, locally finite and facet-tofacet. Let {pltl, ..., pl"I} be a finite set in Ed. To each point p(i) we let correspond the half-space

CRYSTALLOGRAPHY AND TILING {(x1i

.

.

., xd+1): xd+1 ? 2(xlpl') + .

.

75

. + xdpe')) -

- ((p+ ... + p(di))2} The intersection of these half-spaces is an unbounded polyhedron in E d+1. Embed E d into E1+1 in the usual way using the first in

Ed+1.

d coordinates. Then the orthogonal projections of the facets of the polyhedron into Ed are precisely the Dirichlet-Voronoi cells correspond-

ing to L = {pM, ..., p(")}. Using this correspondence and known algorithms for finding the convex hull of a finite point set, it can be shown that the determination of the Dirichlet-Voronoi tiling corresponding

to

arbitrary

n

points

in

E'

can

be

achieved

in

O(nlogn + n«d+1)/2I) time and O(n«d+1)/2)) space (d >- 2). For the results in this paragraph, generalizations of it, related material and references we refer to Paschinger (1982), Aurenhammer (1982) and Edelsbrunner (1987). For some remarks on Dirichlet-Voronoi tilings for lattices see sections

10.1 and 11.2. (In section 11.2 we have chosen a version of DirichletVoronoi cells in terms of positive quadratic forms.)

10.5

Our next topic will be the relation of tiling and Hilbert's 18th

problem. First we will describe a different introduction of space groups. Then part of Hilbert's 18th problem is discussed. Let C be a compact subset of E d with non-empty connected interior such that for a suitable group G of rigid motions in E d the family

{m(C): m E G}

(10.2)

is a tiling. A result of Bieberbach (1910(a)) shows that there are d translations in G in linearly independent directions. Let S be a finite set

in the interior of C such that the identity is the only rigid motion mapping S onto itself and the diameter of S is less than half the radius of a ball contained in C. Then U{m(S): m E G} is a crystallographic structure with space group G. If, on the other hand, G is a space group, we can construct a tiling in the following way: since G consists of countably many rigid motions, the

set of points of Ed for which at least two of its images under the rigid motions of G coincide is contained in countably many hyperplanes. Hence we may choose p E E d such that its images under rigid motions of

G are all distinct. The set of its images consists of a finite set and all possible translates thereof by the vectors of the lattice of G. Thus the set of images is discrete. For the corresponding Dirichlet-Voronoi tiling we have

LATTICE POINTS

76

D(m(p)) = {x: x - m(p)l < Ix - l(p)l for all l E {m(m-'(x)): m-'(X) - m-'(m(p))l =

G}

<- 1m-'(x) - m-t(l(p))I for all I E G}

=m({y:ly-pl_ly-k(p)lforallkeG}) = m(D(p))

for m E G.

Hence

{m(D(p)): m E G} is a tiling. See figure 10.5.

Figure 10.5 Fundamental domain and corresponding tiling of a space group in E2 (p4)

Let G be a group of rigid motions and F a compact subset of Ed with connected non-empty interior. In accordance with common usage, F is called a fundamental domain of G if for any x E E d there is a rigid motion m E G with m(x) E F, where m is unique if m(x) is an interior point of F. Equivalently this means that {m(F): m c G} is a tiling of E d. See figure 10.5.

The last three paragraphs show that it is possible to introduce space groups as groups of rigid motions in E d with fundamental domain. In his 18th problem Hilbert also asked the question whether polyhedra also exist which do not appear as fundamental regions

of groups of motions, by means of which nevertheless by suitable juxtaposition of congruent copies a complete filling up of all space is possible (Hilbert (1900)).

An assistant of Hilbert, Reinhardt (1928), produced the first example of

CRYSTALLOGRAPHY AND TILING

77

a (non-convex) polytope P in El such that there

is a tiling with congruent copies of P but no tiling of the form {m(P): m E G}, where G is a group of rigid motions. The first planar (non-convex) examples are due to Heesch (1935) (see also Heesch (1968)) and the first convex example was given by Kershner (1968) (see figure 10.6). (It could be that the pentagon of Kershner is contained in earlier lists dealing with other properties, but it was Kershner who first recognized its peculiar

packing properties.)

(a) After Heesch (1935)

(b) a = b = e, c = d

a+/3+S=2mr, a=2y (After Kershner (1968))

Figure 10.6 Plane tiles

It is interesting to note that for Kershner's tile T there exists (up to rigid motion) only one tiling by congruent copies and all these copies are obtained by proper motions of T. In all these examples the congruent copies which form the tiling are not just translates. Hence a recent result of Szabo (1987) for d = 3 (and infinitely many higher dimensions) is of particular interest since it exhibits a (non-convex star) polytope P which tiles E3 with translates but admits no tiling of the form {m(P): m E G}, where G is a group of rigid motions. For a survey of related results we refer to Stein (1986). As will be seen below, each convex polytope P which provides a tiling with translates also admits a lattice tiling. Hence one may not expect convex examples of the type specified by Szab6.

In recent years a class of planar tilings specified by Penrose (1974, 1979) attracted much interest. The tilings consist of congruent copies of

two polygons, the `kite' and the `dart' with matching conditions described by colouring the vertices; adjacent vertices must have the same colour. (Penrose also described tilings with congruent copies of a `thick' rhombus with angles 27r/5 and 37x/5 and a `thin' rhombus with angles 7r/5

and 47x/5.) For a historical account which includes the work of Wang and Berger see the beautifully illustrated and highly readable article of Gardner (1977). For an algebraic treatment we refer to the papers of

LATTICE POINTS

78

de Bruijn (1981). The most complete discussion of Penrose tilings is contained in the book by Grunbaum and Shephard (1986).

The Penrose tilings have many unexpected properties. They are aperiodic in the sense that none of the (uncountably many) Penrose tilings is periodic. Any finite subset of such a tiling has infinitely many congruent images in each tiling. In any Penrose tiling there is a sort of hierarchy of the following type: by gluing together certain finite sets of tiles or parts of tiles one obtains a dilated copy of a Penrose tiling. This property easily yields the aperiodicity. The proportion of kites to darts is

(1 + V5)/2: 1, ((1 + x/5)/2 = 1.618034 ...). There are two Penrose tilings with pentagonal symmetry and Mackay (1981) suggested the construction of 3-dimensional aperiodic tilings where the tiles are congruent to two given parallelotopes with a suitable matching condition. Among these tilings are tilings with icosahedral symmetry. A more recent account of 3-dimensional tilings of Penrose type is due to Danzer (1988).

Surprisingly enough - and contrary to classical crystallography Shechtman et al. (1984) reported on `quasicrystals' of an alloy of aluminium, iron, manganese and chromium also showing icosahedral symmetry. It seems that these `quasicrystals' are related to Mackay's tiling but the matter is not decided at the moment. For more information on `quasicrystals' we refer to Nelson (1986), Cahn and Taylor (1986).

10.6 A convex polytope which is the fundamental domain of a group

of rigid motions in Ed is called a stereohedron or a stereon. The knowledge of stereohedra is rather limited.

Let S be a stereohedron in Ed and G a corresponding group. An aspect of the tiling {m(S): m E G} is a class of all tiles in the tiling which are translates of a fixed tile. Let a denote the number of aspects of the tiling {m(S): m c- G}. Then a result of Delone (1961) says that S is a polytope with at most

2d(a+1)-2 facets. Since a is bounded above by the order of the point group of G we obtain that for d = 2 the number of aspects is at most 12 and for d = 3 it is at most 48 (see, for example, Henry and Lonsdale (1965) or Burzlaff and Zimmerman (1977)). Thus in E2 a stereohedron has not

more than 50 edges and in E3 it can have at most 390 facets. (The smallest upper bound in the plane is 6.) The upper bound in E3 seems to be much too large. On the other hand an example of Engel exhibits a stereohedron in E3 with 38 facets, see Grunbaum and Shephard (1980).

CRYSTALLOGRAPHY AND TILING

79

For other results on stereohedra see Delone (1961), Grunbaum and Shephard (1977, 1980) and Delone, Dolbilin and Stogrin (1978).

10.7

Among the stereohedra the so-called parallelohedra have

attracted the greatest interest since the times of Dirichlet, Minkowski and, in particular, Voronoi. A parallelohedron is a convex polytope which admits a lattice tiling. A deep result of Venkov (1954), proved independently by McMullen (1980), says that a convex polytope for which there exists a tiling with translates is already a parallelohedron. Venkov and McMullen also give quite a precise description of parallelohedra. For a generalization and a refinement we refer to Alexandrov (1954) and Groemer (1962(a)). The determination of the different `types' of parallelohedra is still far

from a complete answer. Without going into details we mention that Fedorov (1885) specified the two `types' in E2 and the five `types' in E3. Representatives are given in figure 10.7.

D O (a) parallelogram

(c) parallelotope

(b) centrally symmetric hexagon

(d) hexagonal prism

(f) elongated octahedron

(e) rhombic dodecahedron

(g) cubo-octahedron or truncated octahedron

Figure 10.7 Parallelohedra in El and E3

LATTICE POINTS

80

For anaglyphes of these polytopes the reader may consult L. Fejes Toth (1964). In E4 there are 51 `types' according to Delone (1929) and Stogrin (1973).

A difficult problem in this context, which goes back to Voronoi, is the

any parallelohedron the affine image of a Dirichlet-

following: is

Voronoi cell of a suitable lattice? For d , 4 Delone (1929) proved that the answer is positive and for general dimensions important contributions towards an affirmative solution have been made by Voronoi (1908, 1909) and Zitomirskii (1929).

Readers interested in the topics treated in this subsection may consult Gruber and Lekkerkerker (1987) for additional results and many references.

10.8

In the last part of this section we deal with the theorem of

Haj6s on packing of cubes. Let K denote the unit cube {x: Ixil < Z} of Ed. Minkowski (1896), §37, conjectured that any tiling lattice of K has, after a suitable permutation of the coordinate axes, a basis of the form 1

P21

0

0 0)

1

0

I

I

0

,

.

.

.,

)

This means that in any lattice tiling of K there are cubes sharing whole facets. After many attempts this conjecture was finally proved by Haj6s (1942).

Keller (1930) proposed the following more general question: does there exist in any tiling with translates of K a cube which shares a whole facet with another cube? For d <_ 6 this was proved by Keller (1937) and Perron (1940/41) and remains open for the remaining cases.

Related problems concern multiple tilings with translates of K and tilings with translates of sets consisting of finitely many cubes. For problems and results of this types compare the survey of Stein (1986).

11

Geometry of positive quadratic forms: reduction, packing and covering with balls 11.1

Quadratic forms have attracted the interest of mathematicians

for more than two centuries in the context of various branches of number theory and geometry. From the vast literature on quadratic forms we have chosen a subject which is close to several other topics considered in this book. Our exposition will follow the program outlined by the fundamental papers of Minkowski (1905), Voronoi (1908(a), 1908(b), 1909), Delone (1934(a), 1937/38), Venkov (1940, 1952) and of several contemporary papers. In particular we will present some results and ideas of reduction and the theory of packing and covering with balls. As a by-product we will obtain some results on tiling. This chapter supplements chapters 9 and 10.

Let A = (aik) be a real symmetric d x d-matrix. We will not distinguish between the matrix A, the quadratic form q defined by d

q(x) = xtrAx = E, akxxk

for x E Ed,

i,k=1

and the point (all, a12,

.

.

., ald, a22,

.

.

., add)tr E Ed(d+1)/2.

(The superscript tr stands for transposition). The determinant of A is called discriminant 6(q) of q. The quadratic form q is positive (definite) if q(x) > 0 for x * o. The set of all points in Ed(d+1)/2 which correspond to positive quadratic forms on Ed represents an open convex cone

with apex at the origin, called the cone of coefficients of positive quadratic forms on Ed. As will be made more clear in the following, many properties, problems and results concerning positive quadratic forms can naturally be interpreted in terms of the cone (P.

Two quadratic forms q and r on Ed are equivalent if there is a d x d-matrix T with integer elements and determinant ±1, i.e. a 81

LATTICE POINTS

82

so-called integer unimodular matrix, such that

r(x) = q(Tx)

for x E Ed.

This clearly induces an equivalence relation on the space of all quadratic

forms and, in particular, on the space of positive quadratic forms, respectively on (P. Equivalent forms assume the same values on Zd. Between positive quadratic forms and lattices -there is a close connection. Let L be a lattice in Ed with basis {b(1), ..., b(d)} and let B be

the matrix with columns b(1), ..., b(d). Then L = {Bu: U E Zd}. To L or, more precisely, to the basis B of L we let correspond the positive quadratic form q defined by

q(x) = Bxl2 = (Bx, Bx) = xtrBtrBx = xtrAx for x E Ed, where A = (aik) _ ((b(i), b(k))) = BtrB is a symmetric matrix. q is

often called the metric form of L corresponding to the basis {b(1), ..., If {c(1), ..., c(d)} is a different basis of L and C the matrix with columns c(1), ..., c(d), then C = BT with a suitable integer unimodular matrix T. The metric form of L corresponding to the basis {c(1), ..., c(d)} is then the form q(Tx) equivalent to q. If, on the other hand, q is a positive quadratic form on Ed, then there is a lattice L with a basis such that the metric form corresponding to this basis is q. The lattice L and its basis are unique up to (proper or improper) rotations. Let T = (tik) be a non-singular d x d-matrix. The linear transformab(d)j.

tion

x-3 Tx

forx E Ed

induces the following transformation of (not necessarily positive quadratic) forms: xAx --> xtrTtrATx

or in terms of the coefficients d

aik -

tlitmkalm

for (a ii,

.

.

.)tr

c-

Ed(d+l)/2.

l,m=l

We have thus obtained a linear transformation .7 of Ed(d+1)/2 onto itself. Clearly = (-/).

11.2 When are two given (positive quadratic) forms (on E d) equivalent, or, geometrically, when do two lattices (in Ed) differ by a rotation only? This seemingly simple question constitutes one of the origins of reduction theory. It leads to the following more precise questions. As is customary we formulate them in terms of forms rather than for bases of lattices.

GEOMETRY OF POSITIVE QUADRATIC FORMS

83

1. Specify a set (Q of forms in (7) such that each form is equivalent to (a) one, or (b) a bounded number of forms of (Q. We call (Q a reduction domain and the forms of (Q reduced forms. In the cases considered in

the literature (Q is in general a convex polyhedral subcone of () with apex at the origin and with parts of the faces possibly deleted. (In many cases one does not care about the boundary of (Q.) Sometimes (Q is a finite union of such cones. Each form in () is equivalent to a form in (Q.

For an interior form, i.e. a form contained in the interior of (Q, we require either that it is not equivalent to any other reduced form or that it

is equivalent to a fixed number, say k, of reduced forms. In the

former case (Q is a simple reduction domain, whereas in the latter case (Q is called a multiple reduction domain of multiplicity k. (A reduced form on the boundary of (Q may be equivalent to more than one form in (Q even if (kis a simple reduction domain, and to a number (different from the multiplicity) of forms in (-if (Q is a multiple reduction domain.) By a tiling of (7 we understand, by analogy to the definition in section 10.4, a family of subsets of 7 whose union equals 7 and such

that any two distinct sets have disjoint interiors. If (Q is a simple reduction domain, then T an integer unimodular matrix}

is a tiling of (P. If (Q is a convex polyhedral cone or a finite union of such then a (not necessarily positive) form on an edge of (Q (which may be contained in the boundary of ()) is an edge form. 2. Specify an algorithm by means of which one can transform a given form into an equivalent reduced form.

3. Given a reduced form, what are the integer unimodular matrices by means of which one can transform it into all equivalent reduced forms?

There exist several different definitions of reduction due to Lagrange (1773), Seeber (1831), Gauss (1831), Dirichlet (1850); Hermite (1850); Korkin and Zolotarev (1872, 1873, 1877); Selling (1873/74), Charve (1882); Minkowski (1905); Hofreiter (1933(a)); Venkov (1940, 1952) and

others. For detailed information on reduction we refer to Delone (1937/38), Keller (1954), van der Waerden (1956), van der Waerden and Gross (1968), Ryskov (1980), Lenstra, Lenstra and Lovasz (1982) and Gruber and Lekkerkerker (1987). In the following we will study Minkowski reduction only. A form q(x) = I a;1x,xk is reduced (in the sense of Minkowski (1905)) if for each i E {1, ..., d} the inequality

q(u,,...,ud)aii =q(0,...,0,1,0,...,0) ith position

(11.1)

LATTICE POINTS

84

holds for any u = (u 1, ... , u d) cr E Z d for which the greatest common divisor of ui, ..., Ud is 1 and if for each i E {1, ..., d - 1} we have ai,r+i > 0.

(11.2)

(Note that it is not required that q is positive.) In more recent literature the conditions (11.2) are omitted in general. If this is the case then one speaks (somewhat unexpectedly) of classical

Minkowski reduction. In order to clarify the geometric meaning of Minkowski reduction for positive forms we state a geometric version of the definition: a basis {b('), ..., b(d)) of a lattice L in Ed is reduced (in the sense of Minkowski) if

b(') E {b E L\{o}: jbI is minimal),

b('+') E {b E L\{o}: 0),

...,

b''), b can be extended to a

basis of L, I b I is minimal), b(1)) > 0, (b('+l)

for i E {1, . ., d - 11. In the following we will use the first version of the definition only. .

If q is a positive reduced form, then

all = min{q(u): u E Z'\{o}). It was shown by Minkowski (1905) that for any positive form there is an equivalent reduced form. Minkowski (1905) proved two finiteness theorems. The first finiteness

theorem can be stated in the following way: There is a finite set Ul u u Ud C Zd such that a (not necessarily positive) form q satisfies all inequalities (11.1) if and only if it satisfies the finitely many inequalities

q(UI,...,ud)?aii

for(ul,...,ud)'E Ui, i E {1.... ,d}. (11.3)

Hence q is reduced if and only if it satisfies the inequalities (11.2) and (11.3). Minimal sets of such inequalities (11.3) were given by Lagrange (1773) (d = 2), Seeber (1831) and Gauss (1831) (d = 3) and Minkowski (1883) (d = 4). The set of inequalities specified by Minkowski (1887) for d = 5, 6 is sufficient as proved by Ryskov (1973) and Afflerbach (1982)

(d = 5) and Tammela (1973) (d = 6), but not minimal. A minimal subset of Minkowski's conditions for d = 5, 6 is due to Tammela (1973(a)). The case d = 7 was treated by Tammela (1977). For d = 8 minimal sets of such inequalities are not yet known. We state the results for d <- 7. A form q is reduced if it satisfies the conditions (11.2), the inequalities

GEOMETRY OF POSITIVE QUADRATIC FORMS

85

all -- a22 -- ' ' ' - add and all inequalities of the form g(u1, . ., Ud) > ail, where i E {1, ..., d} and (u1, ..., Ud) is an arbitrary permutation of one of the rows up to index d E {2, . . ., 7} in Table 11.1. (For d <_ 4 these conditions form a minimal set. For d >_ 5 some conditions have to be deleted.) .

Table 11.1 Reduction conditions

2: 3: 4:

5:

6:

7:

1

±1

0

0

0

0

0

1

±1

±1

0

0

0

0

1

±1

±1

±1

0

0

0

1

±1

±1

±1

±1

0

0

1

±1

±1

±1

±2

0

0

±1

0

1

±1

±1

±1

±1

1

±1

±1

±1

±1

±2

0

1

±1

±1

±1

±2

±2

0

±1

±1

±1

±1

±2

±3

0

1

±1

±1

±1

±1

±1

±1

1

±1

±1

±1

±1

±1

±2

1

±1

±1

±1

±1

±1

±3

1

±1

±1

±1

±1

±2

±2

1

±1

±1

±1

±1

±2

±3

1

±1

±1

±1

±2

±2

±2

1

±1

±1

±1

±2

±2

±3

1

±1

±1

±1

±2

±2

±4

1

±1

±1

±1

±2

±3

±3

1

±1

±1

±1

±2

±3

±4

1

±1

±1

±2

±2

±2

±3

1

±1

±1

±2

±2

±3

±4

LATTICE POINTS

86

Thus a binary positive quadratic form q(x) = a11x2 + 2a12xlx2 + a22x2 is reduced if and only if all -1 a22,

a12 % 0,

all ± 2a12 + a22 - a22.

This can be put in the following form: 0_12a12-1 all I- a22.

See figure 11.1. The finitely many conditions (11.2) and (11.3) actually are homogeneous linear inequalities for the coefficients alk of the forms q and thus define a polyhedral convex cone in E d(d+1)/2 with apex at the origin. A (not necessarily positive) form for which the inequalities (11.3)

and therefore also the inequalities (11.1) hold must be non-negative. Since the closure of is precisely the set of all non-negative quadratic forms, our polyhedral cone is contained in the closure of (P. This shows that the set (P+ of reduced (positive quadratic) forms, which is the intersection of with the polyhedral cone just considered, is a polyhedral cone with parts of its facets deleted.

Figure 11.1 Minkowski reduction domain for d = 2

From now on all forms are assumed to be positive quadratic forms unless stated otherwise. The second finiteness theorem of Minkowski (1905) says that there are only finitely many integer unimodular matrices T1i ..., T such that for

a reduced form q all equivalent reduced forms are among the forms q(Tlx), . . ., Tammela (1973(b), 1975) proved that for d 6

such matrices are among the integer unimodular matrices T = (tik) such

GEOMETRY OF POSITIVE QUADRATIC FORMS

87

that for each i E {1, ..., d} the row (til, ..., t;d) is a permutation of one of the rows up to index d E {2, ..., 61 of Table 11.2. Table 11.2 Equivalent reduced forms

±1

0

0

0

0

0

2:

±1

±1

0

0

0

0

3:

±1

±1

±1

0

0

0

±1

±1

±1

±1

0

0

±1

±1

±1

±2

0

0

±1

±1

±1

±1

±1

0

±1

±1

±1

±1

±2

0

±1

±1

±1

±2 ±2

0

±1

±1

±1

±1

±1

±1

±1

±1

±1

±1

±1

±2

±1

±1

±1

±1

±1

±3

±1

±1

±1

±1

±2

±2

±1

±1

±1

±1

±2

±3

±1

±1

±1

±2

±1

±1

±1

±1

±1

±1

±2 ±2 ±2 ±2 ±2 ±2

±1

±1

±2

±2

±3

4:

5:

6:

±2

±3

±4

Several algorithms have been proposed by means of which one can find for a given form an equivalent reduced form. These algorithms work well for d = 2 but even for moderately large d the number of necessary steps increases rapidly. A reduction algorithm for all d was indicated by Minkowski (1905) who also provided a more explicit version for d -- 4. Ryskov (1971) and Tammela (1975) considered the cases d = 5 and d = 6 respectively. Their algorithms are based on the result described before. We also refer to Afflerbach and Grothe (1983). Lenstra, Lenstra and Lovasz (1982) consider a concept of reduction which is related to Minkowski reduction but of a `less precise' form. For

LATTICE POINTS

88

this concept of reduction they devised the now famous so-called L3algorithm which is very effective and transforms a form (or rather a basis of a lattice) into a `reduced' one. The main difficulty arises of finding for q a point U E Zd such that q(v) is `close' to min{q(u): u E Zd\{o}}, see Dieter (1988), Knuth (1981) and Pohst (1981). The L 3-algorithm was successfully applied in several different contexts. Lenstra, Lenstra and Lovasz (1982) applied it for finding factorizations of polynomials with rational coefficients. Odlyzko and to Riele (1985) used it to disprove the Mertens conjecture, a stronger relative of the Riemann hypothesis. Applications to simultaneous Diophantine approximations are due to Babai (1986) and others. Several authors applied it in cryptography and in the ellipsoid algorithm of linear programming.

For more information see Babai (1986). Dieter (1988) applied his algorithm for calculating `short' vectors to test pseudo random number generators.

Let q and r be two forms. If an algorithm is available which permits us to find reduced forms q, and r, equivalent to q and r respectively, then the problem of deciding whether q and r are equivalent reduces to the simpler problem of deciding whether q, and r, are equivalent. Let q be a form. In analogy to the definition in section 10.4 we define the Dirichlet-Voronoi cell D(o, q) of q centred at o by

D(o, q) = {x: q(x) < q(x - u) for all u E Zd}. In general it is difficult to determine the Dirichlet-Voronoi cell of a given form. Tammela (1973(b), 1975) showed that for a reduced form q

in dimension d <- 6 the Dirichlet-Voronoi cell D(o, q) may be determined by using only the points u = (ul, ..., ud)" where ul, ..., Ud is a permutation of the rows in Table 11.2 or Table 11.3 up to index d.

Table 11.3 Dirichlet-Voronoi cells of reduced forms

5:

6:

±1

±1

±1

±2

±3

0

±1

±1

±1

±2

±3

±3

±1

±1

±1

±2

±3

±4

±1

±1

±2

±2

±3

±4

±1

±2 ±2

±2

±3

±3

GEOMETRY OF POSITIVE QUADRATIC FORMS

11.3

89

In this section we will consider the problem of locally densest

lattice packing of balls.

Let L be a lattice in Ed. Denote by p(L) the maximum p > 0 such that L is a packing lattice of pBd. We call p(L) the packing radius of L. Clearly

2p(L) = min{jpl: p E L\{o}}. The density of the lattice packing {p(L)Bd + p: p E L} is p(L)dV(Bd) d(L)

(see section 8.1). We say that L provides a locally densest lattice packing of balls if for all lattices M in a suitable neighbourhood of L (see section 7.1) we have p(M)dV(Bd) d(M)

p(L)dV(Bd)

d(L)

If q is a metric form of L (cf. section 11.1 above) then for the arithmetic minimum m(q) of q we have

m(q) = min{q(u): u E Zd\{o}} = (2p(L))2.

If L provides a locally densest lattice packing of balls, then for all metric forms r of the lattices in a suitable neighbourhood of L we have m(r)d

m(q)d

8(r)

S(q)

(11 4) .

A form q is extreme (sometimes also stable, critical or a limit form) if for all forms r in a suitable neighbourhood of q (in 7) the inequality (11.4) holds. It is easy to see that the problem of determining the locally densest lattice packings of balls is equivalent to the determination of the extreme forms (Minkowski (1905)).

An important tool for the determination of the extreme forms is the polyhedron Jt introduced by Ryskov (1970). For any u E Zd for which the greatest common divisor of ul, ..., ud is 1 consider in Ed(d+1)/2 the half-space d

{(v11, v12,

.

.

., Vld, V22,

.

.

., Vdd) tr E Ed(d+1)12: E Vikuiuk ? 1} i,k=1

(Vik = Vki)

with normal vector (u;, 2u1u2, ..., 2ud_lud, ud)tr. Let Jit denote the intersection of all such half-spaces. We list several properties of Jt, see Ryskov (1970):

LATTICE POINTS

90 1.

_//t is

a convex polyhedron in ' consisting of all forms with _//t is contained in a

arithmetic minimum >_ 1. Each facet of hyperplane of the form d

{(till,

V12,

.

.

., Vdd) tr

c-

Ed(d+1)/2: E UikUiUk = 1} i,k=1

(vik = Vki),

where u = (ul, ..., Ud) tr E Zd\{o} and U1, 2.

. ., Ud are relatively prime. The forms in the facet have arithmetic minimum 1 and attain it at u. Each ray in /-) starting at the origin intersects the boundary of J/1 in .

precisely one point. The discriminants of the forms in J/t are bounded below by a positive constant.

To any integer unimodular d x d-matrix T we assign a linear transformation 7 of E d(d+1)/2 according to section 11.1. Two sets in E d(d+1)12

are called equivalent if there is a transformation 7 mapping one of the sets onto the other one. 3.

J/t is invariant with respect to the group of all such transformation ,7. Any two facets of _//t are equivalent and there are only finitely many vertices of J/1 which are pairwise non-equivalent. Such a set of vertices can be found by the so-called algorithm of Voronoi (1908).

The (equi-)discriminant surface "/) = (/)(a) (a > 0) consists of all forms in O of discriminant a. It is a strictly convex smooth surface and each ray in (7) starting at the origin meets it exactly once. Using the concepts of J/2 and '/) Ryskov (1970) easily proved the following theorem: a form q on the boundary of J/1 is extreme if and only if it is a vertex and if the discriminant surface (/) through q contains

a suitable neighbourhood of q in J/t. The latter condition can be expressed by saying that the tangent hyperplane of (/) at q intersects J/1 at q only. (It is interesting to note that this deep theorem, and thus also Voronoi's theorem, which will be stated below, has an almost trivial

proof. This phenomenon is also encountered in Minkowski's fun-

damental theorem, in Blichfeldt's theorem and in some sense also in Mahler's compactness theorem. The latter is highly plausible; the difficulty in the proof is to find the right technical tool.) We draw some consequences of Ryskov's theorem: A vertex of J/1 is contained in at least d(d + 1)/2 facets of J/t. This implies that an extreme form q attains its arithmetic minimum in at least d(d + 1)/2 points of Zd\{o} (Korkin and Zolotarev (1872)). Since q is the intersection of the hyperplanes containing the facets in which q is contained, it is uniquely

GEOMETRY OF POSITIVE QUADRATIC FORMS

91

defined by its minimum points. In other words, q is perfect. The tangent hyperplane of (/) through a vertex (aik) of Al has the equation d

I bikVik = d

(Vik = Vki),

1,k=l

where (bik) _ (aik)-1. Using a well-known result from convexity we have that if the tangent hyperplane intersects J/l at (aik) only, its normal vector is a positive linear combination of the exterior normal vectors of the facets of Al containing (aik). Thus if 00, ..., u(^') E Zd\{o} are the points at which the form q(x) = Edk=1 aikxiXk attains its arithmetic minimum, then (b11, 2b12, .

.

., 2bd-ld, bdd)t

is a linear combination with positive coefficients of ((u(,'))', 2U1(1)ll2(1) ,

(1) (I) (1) ) 2 tr ., 2ud-lud , (ud ) ,

(N) (n) 2u1(N)u2(N) , . ., 2ud-lud , (ud(N)) 2)tr In other words, an extreme form is eutactic. Conversely it is easy to (((N)

ul

)

2

,

.

show that a perfect and eutactic form is extreme. We thus obtain the following fundamental theorem of Voronoi (1908(a)): a form is extreme

if and only if it is perfect and eutactic. Using this result, one can determine in principle whether a given positive quadratic form is extreme, or equivalently whether a given lattice provides a ball packing

of locally maximum density. For the Leech lattice this was done by Gruber et al. (1987) - see section 9.6.

In order to find all extreme forms it

is

sufficient to consider a

maximal set of inequivalent vertices of J/2 and to take those vertices q for which the tangent hyperplane of the discriminant surface through q intersects J/I at q only. (Compare property (3) of J/l.) In dimensions d = 2, 3, 4, 5, 6 there are (up to multiplication with a positive factor and up to equivalence) 1, 1, 2, 3 and 6 extreme forms respectively. For d s 5 the extreme forms were determined by Korkin and Zolotarev (1872, 1873, 1877). Their result was verified by Voronoi (1908(a)). Barnes (1957) determined the extreme forms for d = 6. For lists of certain extreme forms in higher dimensions and many results we refer to Ryskov and Baranovskii (1979) and Gruber and Lekkerkerker (1987). On a computer, the determination of all extreme forms even for

relatively modest dimensions at present seems to be out of reach, because of the huge amount of time and space required.

LATTICE POINTS

92

For a remark on the relation of locally densest lattice packings of balls and local minima of the zeta-function for lattices see the last paragraph in section 15.2.

We next consider the connection of the Minkowski reduction domain

iQ+ and J/l. For each facet of Jt consider the cone with apex at the origin generated by this facet. This gives a tiling of (P. A different tiling of (/) is obtained by considering the distinct 7/Q+ among the cones where T is an integer unimodular matrix. It is not difficult to show that the tiling derived from (Q+ is a refinement of the tiling connected with

A. From this one may deduce that each perfect form, i.e. a form contained on a ray from the origin through a vertex of J/1, is equivalent to a so-called edge form but not every edge form is perfect. An edge form (or Kantenform in German) is a form on a 1-dimensional edge of It follows that each extreme form is equivalent to an edge form (Q+.

(Minkowski (1905)).

11.4 Voronoi (1908(a)) did not use the polyhedron -/#. This was introduced by Ryskov in 1970. Instead he used the so-called Voronoi polyhedron II(d). It is defined as the convex hull of all points in E d(d+1)/2 of the form

/

z \u1,

ulu2,

, ulud,

z

uz,

., ud-lud, Ud) cr, z

(11.5)

where (u1, ..., ud)1` E Zd\{o} has the property that u1, ..., Ud have greatest common divisor 1. Note that the points of the form (11.5) are on the boundary of (P. One can prove that II(d) = {A E Ed(d+1)/2:

1 for all V E Jn),

where is the inner product in Ed(d+1)/2Thus II(d) is `dual' to Jn and each property of Al is reflected in a corresponding property of 1I(d).

We list some properties of II(d). 1.

rI(d) is a convex polyhedron (in a more general sense) in the

closure of (P. Each face of lI which is not contained in the 2. 3.

boundary of is a (compact) polytope (with finitely many vertices). The unbounded faces of 11 are of dimension -- d(d - 1)/2. Each ray in 7) starting at the origin intersects the boundary of II in exactly one point. II(d) is invariant with respect to the group of all transformations 7

GEOMETRY OF POSITIVE QUADRATIC FORMS

93

where T is an integer unimodular matrix. There are only finitely many faces of I7(d) which are pairwise non-equivalent.

The determination of the space-group types and of the Bravais types (see section 10.3) requires the knowledge of a representative from each class of integrally conjugate finite groups of integral unimodular d x dmatrices. Here two such groups are integrally conjugate or integrally equivalent if there is on isomorphism of the form

T-> S-'TS with a fixed integer unimodular matrix S mapping the first group onto the second.

A basic theorem of Jordan (1880) says that there are only finitely many classes of integrally conjugate finite groups of integral unimodular d x d-matrices. Such a class of group is also called an arithmetic crystal

class. For d = 2 there are 13 arithmetic crystal classes, for d = 3 there are 73 (see Delone (1934(a)), Neubuser and Wondratschek (1969) and Brown et al. (1978), and for d = 4 there' are 710 (see Dade (1965), Bulow (1967), Billow and Neubuser (1970) and Brown et al. (1978)). A

theorem of Maschke shows that for each finite group G of integer unimodular matrices there is a form q such that G is a subgroup of the

group of automorphisms of q, i.e. the group of integer unimodular matrices T such that

q(x) = q(Tx)

for all x E Ed.

Thus the problem of determining the arithmetic crystal classes reduces to the determination of the (integral conjugacy classes of) groups of

automorphisms of all forms. A result of Ryskov (1972(a), 1972(b), 1972(c)) says that these groups are among the groups of automorphisms of those forms which are the centroids of a maximal system of pairwise non-equivalent faces (of all dimensions) of 17(d) (or of Jit). Jordan's theorem is an immediate consequence of this. Thus using the algorithm of Voronoi (1908(a)) it is possible in principle to determine all arithmetic crystal classes in any dimension.

11.5 We next consider the problem of locally thinnest lattice coverings with balls. For reasons of simplicity we use lattices rather than forms. We will follow the papers of Delone, Dolbilin, Ryskov and Stogrin (1970), Delone and Ryskov (1971) and Ryskov and Baranovskii (1976). The main tools are L-tilings and L-types.

Let L be a lattice in Ed. Denote by a(L) the minimum a > 0 such that L is a covering lattice of aBd. Call a(L) the covering radius of L.

LATTICE POINTS

94

meant by the density of the lattice covering {a(L)Bd + p: p E L} and by a locally thinnest lattice covering with balls (see sections 8.1 and 11.3). The empty ball method of Delone (1928, 1934(a)), by means of which L-tilings are introduced, can be described as follows: a ball in Ed is an empty ball for the lattice L if it contains no point of L in its interior. If It is clear what is

an empty ball contains a system of non-coplanar points of L on its boundary the convex hull of these points is an L-polytope. The system of all L-polytopes forms a facet-to-facet tiling of Ed, the L-tiling or L-partition or Voronoi partition of L. It is also called Delone triangulation or partition. See figure 11.2. This concept is dual to the DirichletVoronoi tiling of L as introduced in section 10.4: An L-polytope is the convex hull of the centres of a system of all Dirichlet-Voronoi cells of L having a common vertex. It is easy to see that a(L) equals the maximum radius of a circumsphere of an L-polytope. L-tilings were first introduced and studied by Voronoi (1908(a), 1908(b), 1909). Later on they were intensively studied by Delone and his school.

Figure 11.2 L-tiling

We now describe the concept of L-type due to Voronoi. Two lattices

L and M are said to belong to the same L-type if there is a linear transformation of Ed which maps L onto M and at the same time the L-tiling of L onto the L-tiling of M. Positive quadratic forms q and r are of the same L-type if they are metric forms of lattices belonging to the same L-type. A lattice L (or a form q) belongs to a primitive L-type if the L-tiling of L (of a lattice of which q is the metric form) consists

GEOMETRY OF POSITIVE QUADRATIC FORMS

95

of simplices only. In dimension 2 there are one primitive and one non-primitive L-type (Fedorov (1885)); in dimension 3 there are one primitive and four non-primitive L-types (Voronoi (1908(a), 1909) and in dimension 4 there are three primitive and at least 49 non-primitive L-types (Delone (1929), Stogrin (1973)). The 221 primitive L-types for d = 5 were determined by Ryskov and Baranovskii (1976). In principle

- but only in principle - it is possible to determine all finitely many (primitive) L-types in any dimension - see Voronoi (1908(a), 1909) and Ryskov and Baranovskii (1976).

The primitive L-types of positive quadratic forms - considered as subsets of q) - are unions of disjoint open polyhedral convex cones with

apices at the origin. The non-primitive types are contained in the boundaries of these cones. The closures in (P of these cones form a tiling of (P.

A remarkable result of Barnes and Dickson (1967(a)) says the following: in any cone of this tiling of q) there is (up to multiplications with a positive factor) at most one form q such that a lattice with metric form q provides a locally thinnest lattice covering with balls. If T is an integer unimodular matrix, such that the cone which contains such a form q is invariant with respect to 9, then T is an automorphism of q. Delone, Dolbilin, Ryskov and Stogrin (1970) and Delone and Ryskov (1971) make the result of Barnes and Dickson even more transparent by giving a very clear geometric proof of it. In dimensions 2 and 3 the thinnest lattice coverings thus provide the only locally thinnest lattice coverings whith balls. In dimension 4 there are three locally thinnest lattice coverings with balls (Baranovskii (1965(a), 1966), Barnes and Dickson (1967(b)).

12

Selected problems of number theory

12.1 Geometry of numbers and lattice point problems of analytic number theory are branches of number theory in which the notion of lattice plays a central role. Several other parts of number theory either make use of results on lattices or may be interpreted in terms of lattice points. Examples are quadratic forms, Diophantine approximation and Diophantine equations. Continued fractions are related to lattice point results as well and the even more distant algebraic number theory still has connections with lattice point methods.

In this section we will first describe the so-called parallelogram algorithm and relate it to Mordell's converse problem for the linear form theorem of section 3.3, to a geometric theory of continued fractions which goes back to Smith, Klein and Minkowski and to the conjecture on the product of non-homogeneous linear forms. Then we touch on Diophantine equations and finally we will consider lattice point problems for `large' convex bodies.

12.2

In this subsection we will present a method, the so-called

parallelogram algorithm, which assigns to each planar lattice a sequence of rectangles. This algorithm and similar ones have been in use since the times of Klein (1895, 1895/96, 1897), Minkowski (1896) and Delone (1947). For historical remarks see Keller (1954). For applications see the subsequent sections. Unless stated otherwise we consider the planar case

only. A rectangle will always have centre at the origin o and edges parallel to the coordinate axes. Following Minkowski (1896), §45, we will describe the parallelogram

algorithm. Let L be a lattice in E2. A rectangle R is called extremal with respect to L if L is admissible for R and if L contains relative interior points of each edge of R. Extremal rectangles clearly exist.

Consider one of them. If no point of L on its horizontal edges is 96

SELECTED PROBLEMS OF NUMBER THEORY

97

situated on the x2-axis one may obtain a further extremal rectangle as follows: Contract the given extremal rectangle in the direction of the xl-axis until we obtain a rectangle such that the only points of L on its boundary are vertices. Then expand this rectangle in the direction of the x2-axis until the horizontal edges are `stopped' by a point of L. Thus we have obtained a second extremal rectangle. Clearly, one can make an

analogous construction with the roles of the horizontal and vertical edges, respectively the x2-axis and the xl-axis interchanged. Continuing this process in both directions we get a sequence of extremal rectangles

., R-1, Ro, R1, . . which may be finite or infinite in each of the two directions. .

.

.

(12.1)

Let (rn, sn), sn , 0, be the point of L in the relative interior of a vertical edge of R. (This is unique except for the case when Rn is the first or the last rectangle in the sequence (12.1). In these cases choose (rn, sn) respectively such that rnrn+l < 0 or rnrn_1 < 0). We obtain a sequence of points of L (12.2) ., (r-1, s-1), (ro, so), (r1, s1), ... lying alternately in the first and second quadrant. By renumbering (12.1), (12.2) if necessary, we may suppose that the points with even indices are all contained in the first quadrant. The construction of the sequences (12.1), (12.2) is called the parallelogram algorithm (figure .

.

12.1).

The following properties of the sequences (12.1), (12.2) are immediate: 1.

2.

Each extremal rectangle belongs to the sequence (12.1). We have (rn, SO = (rn-2, Sn-2) + an(rn-1, Sn-1) for n = 0, ±1, ±2, .

.

.

with suitable positive integers an. The point (rn, sn) is the last point of L on the ray starting at (rn_2i sn_2) and having direction (rn_1i sn_1) which is contained in the same quadrant as (rn-2, sn-2). 3. d(L) _ (-1)n (rnsn-1 - rn-1sn) for n = 0, ±1, .. . 4. A(Rn) = 4(-1)nrnsn+l for n = 0, ±1, ..., where A(Rn) denotes the area of R.

There exist several higher-dimensional versions of the parallelogram algorithm, see e.g. Minkowski (1907), §IV, and Keller (1954), §33.

12.3

This section contains several applications of the parallelogram algorithm. Before reading the first application the reader is advised to

I

RZ

(r2,s,)

R,

Figure 12.1 The parallelogram algorithm

(r3,s3)

Ro

(ro,so)

SELECTED PROBLEMS OF NUMBER THEORY

99

take a second look at section 3.3, where we considered Mordell's converse problem for the linear form theorem of Minkowski. In order to

be able to treat the converse problem in the planar case in a more geometric way we define for L a lattice in E2: TI T2

K(L) =sup

d (L)

: L admissible for {x: Ixi

Ti, i E {1, 2}}}

= sup{ A(R) : R extremal with respect to L}.

4d(L)

Using properties (1)-(4), Suranyi (1960/61, 1971) elegantly proved that for each plane lattice L

K(L) ,

2+

2

= 0.723 606... = K(2)

(12.3)

1

where K(2) is defined in section 3.3. Here equality holds precisely for lattices of the form DLO, where D is a diagonal transformation, and

_V5-1 1 1

2

_5+1 2

is a basis for Lo. (Lo is admissible for the star body {x: IXIX21 :!S; 1) and

has minimal determinant among all such lattices, i.e. is a critical lattice of this star body.) For lattices L with K(L) > K(2) the stronger inequality

K(L),2+21 =0.788675...

(12.4)

holds, equality being attained precisely for lattices of the form DL

1,

where D is a diagonal transformation and the lattice L 1 has basis

V3-1 (1)'

2

V3+1 2

Using the connection between the sequence (12.2), the numbers a in property (2) and continued fractions, Suranyi (1971) gave another proof

of the theorem of Szekeres (1936(b)) that, in the sense of Lebesgue measure in E4, for almost all bases {b(1), b(2)} (considered as points in E4) we have for the corresponding lattice L the equality K(L) = 1 - see

LATTICE POINTS

100

also Gruber (1971).

The phenomenon that the lattices for which (12.3) holds with strict inequality even fulfil (12.4) is an example of a so-called isolation theorem. Isolation occurs frequently in Diophantine approximation and geometry of numbers - see, for example, Cassels (1957). The close relation between the parallelogram algorithm and regular continued fractions was pointed out by Minkowski (1896), §45, who used this relation to prove some basic results on continued fractions. Instead of giving details we present a geometric interpretation of regular continued fractions due to Klein (1895, 1895/96, 1897).

Let a > 0 be irrational. We assume that the ray

S = {x: x2=ax1ix1>0) takes the role of the positive x2-axis, and instead of L we consider the integer lattice Z2 for the parallelogram algorithm. Then the points (0, 1), (1, [a]) appear in the sequence corresponding to (12.2). Introducing a new notation we consider the (unbounded) subsequences (12.5)

(q -i, p-i) _ (0, 1), (q1, p1), (q3, P3), ...

(12.6) (qo, Po) _ (1, [a]), (q2, P2), (q4, P4), ... on the left- and on the right-hand side of S. The (unbounded) polygonal lines with vertices in (12.5) or (12.6) are called Klein polygons (see

figure 12.2). The Klein polygons form part of the boundary of the convex hull of the points of Z2 in the (closed) first quadrant which are above (or below) the ray S, excluding o. Properties (2), (3) translate into

(qn, p,) = (qn-2, Pn-2) + an(gn-1, Pn-1) for n = 1, 2,

2'.

.

.

.

with

suitable positive integers al, a2, . . . The point (qn, pn) is the last point of Z2 on the ray starting at (qn-2i Pn-2) in direction (qn-1, Pn-1) before the ray intersects S.

gnPn-1 - q.-lpn = (-1)n for n = 1, 2, .. Using (2'), (3') and the geometric interpretation, one easily deduces 3'.

.

that

Po

P2

qo

q2

l

a-

< ... < a < ... <

Pn l < qn

<

1

gngn+l

1 qn

P3 q3

<

P1 qi

for n = 1, 2,

...,

1

ao +

1

al + a2 +

= [ao, al, a2, a3, .

.

.

SELECTED PROBLEMS OF NUMBER THEORY

101

Figure 12.2 The Klein polygons

where ao = [a]. The geometric interpretation of continued fractions led to new proofs of several well-known results. As an example we mention a proof of Fukasawa (1925) of the theorem of Hurwitz (1891) which says that 1 = 0.447213 ... < a_ for infinitely many n. 2 USRn 2 9n qn Any real a has, besides its unique expansion as a (regular) continued I

fraction 1

,a=ao+

ao E Z,

al +

a,, a2, .

.

.

E N,

1

a2 +

a unique expansion as a `semiregular continued fraction' of the special form 1

a = bo -

,

bl -

bo E Z,

b1, b2i

.

.

. E N\{1}

b2 (12.7)

and, conversely, each such expansion represents a real number - see Perron (1954). These special semi-regular continued fractions admit a nice geometric interpretation using the Klein polygons, described by H. Cohn (1973).

Let a > 0 be irrational and consider the Klein polygon corresponding

to the ray {x: x2 = axl, xl > 0} and starting at (0, 1). Consider all points of Z2 on this Klein polygon (not only the vertices), beginning

LATTICE POINTS

102

with (0, 1), say

(to, uo) = (0, 1), (t1, ul), .. Define integers bo E Z, b 1, b2, ... E N by

bo = -[-a] bn(tn, un) = (tn-1, un-1) + (tn+l, un+l)

for n E N.

If (4_1, un_1), (tn, un) and (tn+l, un+i) are on a straight line, then bn = 2, whereas bn > 2 if (tn, un) is a vertex of our Klein polygon. It is comparatively easy to see that

a=bo -

1 1

bl - b2-. By means of this geometric interpretation Cohn easily proves several well-known results on semi-regular continued fractions of the form (12.7).

12.4 Related to the parallelogram algorithm and the construction of the Klein polygons is Delone's divided cell algorithm (Delone (1947) see also Cassels (1972), X1.4.2). We will describe it and state one of its applications.

Let L be a plane lattice and p E E2 such that the grid p + L contains no point on either coordinate axis. Then it can be shown that there is a parallelogram Po with vertices belonging to the grid but containing no

further point of the grid and such that one of its vertices is in the interior of each of the four quadrants. Po is called a divided cell of the grid. Assuming that no vector :A o in L is parallel to a coordinate axis, the construction shown in figure 12.3 is possible.

Consider the lines containing the edges of P0 which intersect the xl-axis. One of these lines intersects the positive x2-axis, the other one the negative x2-axis. Choose on each of these lines a pair of neighbouring points of the grid which are separated by the x2-axis. This gives the vertices of a second divided cell. By repeating this step backwards and forwards we obtain a sequence .

.

., P-1, Po, P1, .

.

.

of divided cells of the grid p + L. It can be proved that each divided cell of p + L appears in this sequence. Unfortunately in dimensions ? 3 it may happen that a grid has no `divided cell'.

SELECTED PROBLEMS OF NUMBER THEORY

103

Figure 12.3 The divided cell algorithm

The existence of divided cells for planar grids easily yields the following result of Minkowski (1907), §11. Let 11, 12 be two real linear forms in two variables of determinant 1 and let al, a2 be arbitrary real

numbers. Then there are integer values of the variables u = (ul, u2) such that 1 (11(U) - al) (l2(u) - a2)I - 22.

A famous conjecture (curiously enough its origin is difficult to locate) generalizes this result: let 11, .. , ld be d real linear forms in d variables and with determinant 1. For any system of d arbitrary real numbers al, . ., ad there exist integer values of the variables u = (ul, ..., ud) for .

which

(ll(u) - al) . . (ld(u) - ad)I < .

2d

So far this conjecture has been proved up to d = 5 and there exist many

important contributions to the general case. Nonetheless it remains doubtful whether it holds at all. One reason for this is that a stronger conjecture was shown to be false by Gruber (1976) and Ahmedov (1977). For more information on known results related to this conjecture we refer to Woods (1965), Gruber (1967(a), 1970), Skubenko

LATTICE POINTS

104

(1973), Bambah and Woods (1980), Mukhsinov (1981) and the survey in Gruber and Lekkerkerker (1987). Other applications of the divided cell algorithm were given by Barnes and Swinnerton-Dyer (1954) and Barnes (1954, 1956(a)).

12.5 A Diophantine equation is an equation of the form (12.8) f(xl, . , xd) = 0, where f is a polynomial in the variables x1, ..., xd with integer coefficients. The aim is to find all solutions of (12.8) in integer or .

rational x1, ..., xd. Clearly (12.8) represents a curve or a surface in Ed and the problem consists of finding all points of Zd or all points with rational coordinates, i.e. rational points on it. For exhaustive information the reader may consult Lang (1962), Mordell (1969) and Mumford (1974). The following are some remarks on the case d = 2. If f is a homogeneous polynomial of degree 1 or 2, then the curve (12.8) is a line or a conic and much is known on the set of solutions see Mordell (1969). For f a homogeneous polynomial of degree >_ 3 a method of Baker (1967) gives a bound for the number of points of Z2 on the curve. For more applications of Baker's method and many open problems see Baker (1967/68), Ellison (1970/71), Stolarsky (1974). Besides the degree n of f, or equivalently of the corresponding curve, one can define the concept of genus g. If the curve has no singularities,

then g = (n - 1) (n - 2)/2. In other cases the definition is more complicated. If on a curve of genus 0 there is a rational point, then all

rational points may be obtained by means of a rational function. Mordell (1922) proved that on curves of genus 1 there can be infinitely

many rational points - see also Baker and Coates (1970). A famous conjecture of Mordell (1922) says that a curve of genus > 1 contains at

most finitely many rational points. Using deep tools from algebraic geometry, Mordell's conjecture was proved by Faltings (1983, 1984).

12.6 A classical number-theoretic problem which goes back to the circle problem of Gauss is to estimate the number of points of Z" contained in or on the boundary of large balls or ellipsoids, Important contributions to this difficult problem were given by Landau, Walfisz, van der Corput, Jarnik and others. In more recent times other sets besides ellipsoids have been considered. For surveys covering this more

general situation see Fricker (1982) and Gruber and Lekkerkerker (1987). In the following some results concerning convex bodies or more general sets are cited.

SELECTED PROBLEMS OF NUMBER THEORY

105

Let C be a convex body which contains o in its interior. A result of Hlawka (1950(b)) says that if C is sufficiently smooth and has positive gaussian curvature, then AdV(C) - L(AC) = O(),f)

(12.9)

as A ---> +

where /3 = d(d - 1)/(d + 1). (Recall that L(AC) is the number of points

of Zd in W. The number of points of Zd on the boundary of AC is denoted by L*(AC).) Hlawka also showed that /3 may not be replaced by

(d - 1)/2. There exist several alternative proofs and refinements of these results. An extension of Hlawka's estimate (12.9) to smooth compact o-symmetric star bodies is due to Berard (1978). Let C be a compact plane set bounded by a Jordan curve of class C?°° such that at any zero of the curvature the curvature vanishes at most of order k. A result of Colin de Verdiere (1977) says that A2A(C) - L(AC)= O(A2/3) A2A(C) - L(AC)= O(Ak+1)/(k+2))

as A - + as A - + -

for k = 0, 1, for k

2.

An example of Randol (1966) shows that if there is a zero of order k of the curvature at which the slope of the Jordan curve is rational, then

(k + 1)/(k + 2) may not be replaced by a smaller number, see also Nowak (1984). A classical result of van der Corput (1920) is the following: if p is an upper bound for the radius of curvature of a plane convex body C of class 0°°, then

A(C) - L(C) = 0(p2/3) (12.10) as p ---> + where O( ) may be chosen uniformly for all such convex bodies C. Jarnik (1925) proved that 0(p2/3) may not be replaced by o(p2/3). There are several generalizations of van der Corput's result and Chaix (1972) gave an elementary proof of (12.10). There exist several estimates for the number of points of Z I on the boundary of a strictly convex body C. Andrews (1963) and Chaix (1975) proved that

L'(C) <

yV(C)(d-1)/(d+1),

(12.11)

where the constant y depends on d only, supposing that the points of Z I

on the boundary of C are not coplanar. Applying the isoperimetric inequality S(C)d V(C)d-1

S(Bd)d V(Bd)d-1

to Andrews' result (12.11), one obtains that, for C satisfying the same conditions as before,

L(C)

SS(C)d/(d+l),

(12.12)

LATTICE POINTS

106

where S depends on d only.

For d = 2 and 3 some explicit values of the constants on the

right-hand sides of (12.11), (12.12) and other refinements have been given. Chaix (1977, 1978) proved for a strictly convex body C in E2 that L'(C) <_ max{2, 4(3/2)1/3A(C)1/3}

(4(3/2)1/3 = 4.578857 ...) and Jarnik (1925) showed that for a sufficiently smooth strictly convex plane body C we have

L'(C) :5

3

(2 )1/3

P(C)2i3 + O(P(C)1i3)

as P(C)

+-

(12.13)

(3/(2i-)1/3 = 1.625778 ...). Here O( ) can be chosen uniformly for all such bodies. An extension of (12.13) to d = 3 is due to Divig (1976).

13

Visibility

13.1 Among the most appealing results of combinatorial geometry are visibility theorems (which are often hidden under the disguise of illumination or transversality results) - see e.g. the booklets of Boltyanskii and Gohberg (1965) and Hadwiger et al. (1959) and the interesting survey of Danzer et al. (1963). To give the reader an idea of the flavour of such results we will cite one example. A point x of a subset S of E d is visible via S from a point p of S if the line segment with endpoints p, x is contained in S. Call S a star set if all points of S are visible from a suitable point of S. A classical result of Krasnosel'skii (1946) says that a compact subset S of E d is a star set if for any d + 1 points of S there is a point in S from which all d + 1 points are visible via S. It is not surprising that in the context of lattice points visibility problems have attracted interest too. In the following several types of

such results are presented ranging from results dealing with convex bodies to results of a purely combinatorial type.

13.2 This subsection deals with 'view-obstruction': a family of subsets of Ed obstructs view from a point p c- Ed (in direction r E Ed\{o}) if each ray starting at p (resp. starting at p in direction r) intersects at least one of the sets. Generalizing P61ya's (1918) `orchard problem' we state the following problem: let C be a convex body in Ed and p > 0. Determine min{). > 0: {)LC + u: u E Zd\{o}, Jul , p} obstructs view from o}.

The orchard problem is the special case where d = 2 and C is the euclidean unit disk. It was solved first by Speiser (cf. Polya (1918)). See also Honsberger (1973) and Allen's (1986) very general treatment of the problem.

Let e = (z, z, ..., z) and let C be a convex body. Cusick (1973) 107

LATTICE POINTS

108

stated the problem to find

min{A > 0: (AC + e + u: u E Zd} obstructs view from o in any

direction r = (r1,

For C = {x: IxJ

.

.

., rd) where r; > 0 for i E {1,

.

.

., d}}.

; for i c- {1, ..., d}} he found that in the cases

d = 2 and 3 this minimum respectively equals and Z. The latter result was found independently by Betke and Wills (1972) in the disguise of a theorem on simultaneous Diophantine approximation. For C the unit

disc Cusick showed this minimum to be equal to 1/U5 and he conjectured that the analogous result for d = 3 was 3/x/21.

Related to the notion of view obstruction is that of `blocking': a family of subsets of E d blocks a point p if p cannot be removed arbitrarily far from its original position along a continuous curve without hitting one of the subsets. L. Fejes T6th (1975(a)) posed the problem of

determining the convex bodies C of minimum volume such that the family of bodies { C + u : u E Z d } does not cover E d and blocks any

point in Ed not contained in the union of the bodies. He conjectured that for d > 2 the extremal bodies are parallelepipeds of volume z and the extremal configurations form a sort of `d-dimensional chess boards'.

For d = 2 L. Fejes T6th (1973, 1975(b)) and Groemer (1966) proved that the extremal bodies are parallelograms and triangles of area z . The solution of the `chess board conjecture' in general dimensions is due to Barany et al. (1986).

A different visibility problem connected with packings of balls was discussed in section 9.6.

13.3 We say that a point v E Zd is visible from u E Zd if V * u and on the open line segment with endpoints u, v there is no point of Zd. A point u E Zd is visible (or primitive) if it is visible from the origin o. A classical folklore theorem says that the proportion of visible points among all points of Zd is where is the Riemann zeta-function.

There exist many counting results of a similar character. We will mention one of them.

For n = 1, 2.... let f(n) denote the minimum number of points of a subset of the set of points u = (u 1, u 2) E Z d with 1 -- U1, U2,---n, such that every point of the set is visible from at least one of the points of the subset. Then a result of Abbott (1974) says that log n 2log log n

f(n) < log n.

Abbott's proof is an existence proof and gives no indication how to

VISIBILITY

109

construct small subsets from which any point of the set is visible. It

would even be of interest to construct such subsets of cardinality O(log n).

Given a finite subset of Z d, can it be translated such that all its points become visible or invisible from o? A satisfying answer to this result is due to Herzog and Stewart (1971): by a pattern P in Zd we mean the following: there exists a positive integer n such that P consists of the points u = (ut, ..., ud) E Zd, where 1 -- ut, ..., ud n and to any

such point there is assigned one of the symbols , +, 0. This gives a representative of P as disjoint union of three sets, P°, P+, P°, say. A pattern P is realizable in Zd if there is a vector v E Zd such that for each point u E P° (resp. U E P+) the point u + v is visible (resp. invisible). Then the following criterion holds: a pattern P in Z d is realizable if and

only if for each prime p we have Zd * P° + pZd (_ {u + pw: U E P°, W E Zd}) (see figure 13.1). 0

0

0

0

0

+

+

+

0

+

0

0

0

+

+

+

0

+

0

0

0

+

+

+

0

+

+

+

0

0

0

0

0

0

0

0

0

0

0

0

(a) non-realizable

(b) realizable

(c) realizable

(v = (1307, 1273))

(v = (53, 19))

Figure 13.1 Patterns

As a corollary we obtain that for d = 2 any pattern consisting only of crosses or of one, two or three squares and any number of crosses can be realized.

13.4 A series of unsolved problems is connected with the `graph of visible

points'. Two points of Z2 are neighbours if one of their

coordinates coincide and the other ones differ by 1. The graph whose nodes are the visible points of Z d and whose edges are the line segments connecting neighbouring visible points is the graph of visible points of Z2 (figure 13.2). By a remark of Herzog and Stewart (1971) this graph is not connected. A close investigation of this graph would be of interest, and there are many natural problems: for example, the number of its components in a

bounded set, paths with particular properties contained in the graph,

LATTICE POINTS

110

11

+

0++ +

+

+

Figure 13.2 The graph of visible points of Z2

density questions, etc. - see, for example, Erdos (1981) or Winfee (1965). See also chapter 16 on lattice graphs.

For more problems of the sort described earlier we refer to Abbott (1974), Erdos (1958), Rearick (1966) and Rumsey (1966).

14

Lattice point problems of integral geometry

14.1 Buffon's needle experiment of 1733 is generally considered as the first landmark of integral geometry. Buffon's result was published only in 1777 (see Buffon (1777)). Despite interesting sporadic results in

the 19th and the beginning 20th centuries, due to Crofton, Czuber, Poincare and others, systematic research in this area started only with the work of Blaschke and his school. To acquaint the reader with the sort of problem considered in integral geometry we state a typical question: what is the `measure' of the set of lines in E d which meet a given convex body? For exhaustive information on integral geometry we refer to Hadwiger (1957) and SantalO (1976). The underlying concepts for integral geometry are topological groups

and Haar measure, but most authors prefer to use the flexible tool of differential forms. In this section we will exhibit several planar lattice point results from

integral geometry by Hadwiger and add some remarks on the higherdimensional case.

In order to formulate two central results of integral geometry due to

Poincare and Blaschke several notions are needed. The kinematic density dK in E2 is the Haar measure on the group of proper rigid motions m = (r(4), t) in E2, suitably normalized. (Here r(t) is a rotation about the origin with angle 0 and t is a translation - see section

10.2.) Equivalently, dK can be defined as the product of Lebesgue measure on [0, 27r] and Lebesgue measure in E2, that is dK = do dt. Let C be a compact set in E2 bounded by finitely many simply closed piecewise smooth curves which can be oriented such that C is on the left-hand side of each of these curves. The Euler characteristic X(C) of C can then be defined as the number of positively minus the number of negatively oriented curves. If C is a polygon, this is equivalent to the definition in section 2.1. If C consists of k disjoint simply connected pieces, then x(C) = k. We shall write A(C) and P(C) for the area and 111

LATTICE POINTS

112

the length of the boundary (perimeter) of C.

Let A, B be two piecewise smooth curves in E2, of finite lengths P(A) and P(B) respectively, and denote the number of points of A n B by n(A n B). Then the formula of Poincare (Poincare (1912), p. 143; Barbier (1860)) says that

f n(A n m(B)) dK(m) = 4P(A)P(B)

(14.1)

where the integral is extended over all motions m of E2 for which A n m(B) * 0. (See Santal6 (1976), p. 111.) Let C, D be two compact subsets of E2, which are equal to the closure of their interiors and with boundaries consisting of finitely many disjoint simply closed piecewise smooth curves. The fundamental kinematic formula of Blaschke (see e.g. Blaschke (1955)) says that

f X(C n m(D)) dK(m) = 27rA(C)X(D) + P(C)P(D) + 2ITX(C)A(D).

(14.2)

Here again the integral is extended over all rigid motions m for which C n m(D) 0 0. (See Santal6 (1976), p. 114.) The results in (14.1) and (14.2) have many applications ranging from the isoperimetric inequality to the following results of Hadwiger (1941).

14.2 Let C, D be compact subsets of E2, each bounded by a simply closed piecewise smooth curve. Let L be a plane lattice and assume that {C - p: p E L} is a packing. (We have written -p only for convenience.) Then, up to boundary points, C is contained in a fundamental domain F of L. The Euler characteristics of C and D are equal to 1. Thus (14.2) yields

27r(A(C) + A(D)) + P(C)P(D) =

f X(C n m(D)) dK(m)

= I L(r,p+t) X(C n m(D)) dK(m) tEF

pEL

fm=(r,t)X((C

- p) n m(D)) dK(m),

rEF

(where X((C - p) n m(D)) = 0 if (C - p) n m(D) = 0). Since C - p and m(D) both are simply connected, their intersection consists of simply connected disjoint pieces. Let k(m) denote the total number of all such pieces of m(D) as p ranges over L. Then

LATTICE POINT PROBLEMS OF INTEGRAL GEOMETRY

22r(A(C) + A(D)) + P(C)P(D) =

JtEF(r, t)

k(m) dK(m).

113

(14.3)

T

Noting that

ftEF

m =(r, t)

dK(m) = 27Td(L),

this formula can be interpreted as follows: the average a of the number

of connected pieces into which a random congruent copy of D is dissected by the packing {C - p: p E L} equals

27r(A(C) + A(D)) + P(C)P(D) 27rd(L)

See figure 14.1. As a consequence of this we obtain that if {C - p: p E L} is a tiling, then there is a congruent copy of D which can be covered by at most [a] tiles. (Note that then A(C) = d(L)). (See figure 14.1.)

Figure 14.1 Integral geometric lattice point problems I

Particular cases of the latter result are the following: A suitable congruent copy of D can be covered by at most 2P(D) A(D) P(D) + A(D) = I1 + 0.636619... [1 + + a2 a2 Ira

1

L

or

squares of edge-lengths a belonging to the standard lattice tiling with squares of edge-lengths a. Similarly, a suitable congruent copy of D can be covered by at most 2P(D) + 2A(D) I1 + V37ra 3\/3a2 11

A(D)] =I1+0.367552... P(D) a +0.384900... a2 L

LATTICE POINTS

114

regular hexagons of edge-length a which belong to the standard lattice tiling with regular hexagons of edge-length a. By covering the hexagons with circular discs, one obtains a similar covering result with discs. There exist higher-dimensional analogues of these results of Hadwi-

ger, due to Santald (1944), but the proofs seem to hold for convex bodies only - see Groemer (1986). In order to give the reader a feeling for these extensions we cite the following result: any convex body C in E d can be covered by at most 1 +

V(C) ad

1

d-1

+ - E Kd

1

x; i

Wd-i(C)1 J

a'

cubes of edge-lengths a, all of which belong to a suitable lattice tiling. Here Ki denotes the volume of the solid eculidean unit ball in E' and Wi(C) is the ith quermassintegral of C - see section 3.6. It remains an open problem to extend this to topological balls.

The results of this subsection are of importance for results of Groemer (1986) on multiple packings - see section 8.7.

14.3 Let A, B be two piecewise smooth curves in E2 of finite lengths. Assume that L is a lattice and that A is contained in a fundamental domain F of L, where by a fundamental domain of L we understand a subset F of E2 such that for each x E E2 there is exactly one p E L such

that x + p E F; note that this is slightly more precise than the corresponding concepts of fundamental parallelotope, fundamental domain and simple reduction domain defined respectively in sections 3.1, 10.4 and 11.2. For a rigid motion m let 1(m) denote the number of points of the intersection of B with the (disjoint) curves {A + p: p E L}. Then reasoning similar to that which led to (14.3) but using (14.1) instead of (14.2) shows that

4P(A)P(B) =

J m-(r,t) teF

l(A n m(B)) dK(m).

This permits the following interpretation: if a random congruent copy of B is put on the plane, the mean value of the number of points which this curve has in common with the system of curves {A + p: p E L} is equal to 2P(A)P(B) = P(A)P(B) 0.636619. 7rd(L) d(L) See figure 14.2. If, for example L has an orthogonal basis {b(1), b(2) } with I b (1) J = a, l b (2) = i and A is the polygonal line with vertices b (1) , o, b (2), then the mean value of the number of points which an arbitrarily

LATTICE POINT PROBLEMS OF INTEGRAL GEOMETRY

115

placed congruent copy of B has in common with the double grid of lines corresponding to the basis {b(1), b(2)} of L is

2(a + r)P(B)

T

= 0.636619.. a + P(B). (14.4) Iroi ai By taking for B a line segment of length A i and letting a - +oo we see that the probability that a randomly chosen line segment of length A meets any of a system of parallel lines which are at distance A apart is 2A

=0.636619.....

This is the classical result of Buffon.

The mean value in (14.4) can be used to determine the approximate length of a curve B by placing at random a double grid of orthogonal lines on B and counting the number of intersection points. See figure 14.2.

Figure 14.2 Integral geometric lattice point problems II

15

Applications to numerical analysis

15.1 The notion of lattice and several lattice point results are important for purposes of numerical analysis and applied mathematics. In this section we first introduce zeta-functions on lattices and discuss their relevance for numerical integration. We then present the basic idea of the multi-grid method for boundary-value problems of partial differential equation. We have inserted the multi-grid method in this book for two reasons. The first is its recent importance in numerical analysis.

The second is that the multi-grid method again exhibits the fact that even very simple ideas about lattice points may have far-reaching consequences in other areas. (A different example of this is Minkowski's

fundamental theorem which applied to many problems of number theory.) We are optimistic that there exist many more applications of lattice point results in other branches of mathematics.

15.2 A direct generalization of the Riemann zeta-function is the following concept of zeta-function for lattices, first introduced in a different context by Epstein at the beginning of this century and more recently defined independently by Sobolev: IEL\{o)

1

ill's

for lattices in L in E' and s > d/2. Here I I denotes the euclidean norm on Ed.

The lattices of given determinant for which, for a fixed value of s > d/2, the zeta-function attains its minimum have been determined so far for d = 2, 3 only. For d = 2 this was done by Rankin (1953), Cassels (1959), Ennola (1964(a)) and Diananda (1964). The extremum lattice is unique up to rotations and has a basis (b('), b(2)) such that b (1) I = I b,2)1 = I b (') - b (2) 1, i.e. it is an `equitriangular' lattice. In case 116

APPLICATIONS TO NUMERICAL ANALYSIS

117

d = 3 Ennola (1964(b)) proved that the extremum lattice is unique up to rotations and gives a lattice packing of balls of maximum density. For several further properties of the zeta-function the reader may consult the article of Delone and Ryskov (1967). For large values of s only the points of L\{o} which are closest to o contribute significantly to (L, s). Hence if L provides a local minimum

of (L, s) among all lattices of given determinant and arbitrarily large values of s, the points of L\{o} closest to o must be as far away from o as possible. This shows that L provides a locally densest lattice packing of euclidean balls.

15.3

The importance of the zeta-function for numerical integration seems to have been first recognized by Sobolev, see Sobolev (1974): Consider the space of all real functions f on E' with compact support contained in a fixed bounded domain D and with bounded continuous partial derivatives up to order s. How should one choose nodes x(l), . . ., x(k) E D such that the error which one makes when replacing the integral fdxl...dxd JD

by an expression of the form V(D) y, Ax (0) k ,_,

is - in a well-defined sense - as small as possible? For d > 2 it seems to be hopeless to give an answer, unless some additional assumption about

the nodes is made. If the nodes are the points of a lattice L of given determinant, say S, which are contained in D, then, neglecting the contribution of the points near the boundary of D, the best choice for L is when the following is obtained: for the given value of s the c-function attains its minimum on the set of all lattices of determinant 1/6 for the polar lattice L* of L. The results described before thus show that for

d = 2, 3 the `best' lattices are the `equitriangular' lattices and the lattices which provide the thinnest covering of E3 with balls, i.e. the `body centred' cubic lattices.

15.4 For different classes of continuous functions the problem of numerical integration was considered by Babenko (1976, 1977) using Dirichlet-Voronoi cells as a tool. We give a description of a special case

of one of Babenko's (1976) results: Let D be a bounded Jordan

LATTICE POINTS

118

measurable set in Ed. For 0 < p < 1 consider the class 'P of real functions f on D which are Holder continuous in the following sense:

for x,yED.

If(x) - f(Y)I , (I Ix - YIIm)P

Here I . denotes the maximum norm: I Ix I x E Ed. Then I

I

I

1

s

f dxl ... dxd - V(D)

UPp

k

If(x( =1

I

= max{ Ix 1 1, dP (D)

... ,

1+pld

2 (d + p)

Ixd I } for P1

k

Id

for any choice of x(l), ..., x(k) E D. If the nodes x(1), ..., x(k) are (essentially) the points of

(vD))lIdzd nD then the lower bound on the right-hand side is asymptotically best

possible as k -. A simple refinement and generalization of Babenko's result is due to Gruber (199?).

15.5 There exist many discretization methods in numerical analysis by means of which various types of (partial) differential equations and corresponding boundary-value problems are transformed into systems of linear equations for the (approximate) values which the solution of the differential equation assumes at certain points of some fixed lattice (see e.g. Collatz (1966)) and Gerald and Wheatly (1985). Now the problem arises of finding the solution of the system of linear equations. Consider an approximate solution. The error, that is the difference between the exact and the approximate solution of the system of linear equations, is

decomposed into a 'high-frequency' and a `smooth' component. To reduce the error of the high-frequency component the given grid is used whereas for the smooth component a coarser subgrid is sufficient. This reduces the number of steps required to find an approximate solution of the linear system of equations having a smaller error (two-grid method).

Iterating this idea leads to the multi-grid method. (See Hackbusch (1985) and Jespersen (1985).) The multi-grid method was first introduced by Fedorenko (1964) in the case of Poisson's equation. Since for

more than one variable the presentation of the multi-grid method becomes technically involved we will consider a one-dimensional model

case first and then indicate the necessary modifications in the twodimensional case. Our exposition follows Hackbusch (1985). Consider the 1-dimensional Dirichlet.boundary-value problem

-u"(x) = f(x)

on (0, 1) and u(0) = u(1) = 0,

(15.1)

APPLICATIONS TO NUMERICAL ANALYSIS

119

where f is a given real continuous function on (0, 1). For the level 1 = 0, 1/21+1. The grid of level I consists of the points 1, 2, ..., let h, = xi = ih,, i E (1, 2, ., 21+1 - 1). .

.

Next, at every grid point xi replace the derivative in (15.1) by a suitable difference expression, for example by

h (-u(xi - h1) + 2u(xi) - u(xi + h,)) = -u"(xi) + 0(h 2). Setting

ur = (u(xl), ..., u(x2,.'_l))", fl = (f(x1), ..., and omitting the O(hl)-terms, this gives the following linear system for u1:

Lrur = fl,

(15.2)

where

2 -1 2 -1

-1 I1 =

2

-1

h1 l

2 -1I

-1

2

The matrix L1 of the system (15.2) is tridiagonal and thus (15.2) may be solved easily in a direct way. Since our aim is to show the basic idea of the multi-grid method we proceed instead as follows. For the solution of (15.2) one can use for example the Jacobi iteration starting with some vector us°): 2

u(k+1) = 11

hr (L1ulk) 2

- fl),

or the damped Jacobi iteration with damping factor 2, say, ur

k+1) = usk)

- hr 4

'k) -

(Lrur

4 Lr) ur hr

(k)

h

+ 4 fr

(15.3)

Here I, denotes the (21+1 - 1) x (21+1 - 1) unit matrix. The eigenvalues and the eigenvectors of the iteration matrix I, - (h2/4)L1 are A11 = 1 - sin2(jTrh1/2),

e(,i) = Ugh, (sin(jnhr),

..., sin((21+1 - 1)Jn.hr))", jE{1,...,21+1-1).

Since the maximum of the absolute values of the eigenvalues, the so-called spectral radius, 1 - 7r2h1/4 + 0(h°), is almost equal to 1, the overall convergence of the damped Jacobi iteration is slow.

LATTICE POINTS

120

If one represents the initial error u}") - u, of the damped Jacobi iteration in the form

u,°1 - u, _

aieli)

then k steps of the damped Jacobi iteration yield the error l ail for j small, u, _ /3iey), where lf3i l = l ai l IA,i I k {l <« l ai l for j large.

This shows that the iteration efficiently reduces the 'high-frequency' components of the error, i.e. the components corresponding to large values of j. This is often expressed by saying that the damped Jacobi iteration `smoothes' the error. In order to reduce the whole error one should combine the damped Jacobi iteration with a second method which reduces the 'low-frequency' components, i.e. those corresponding to small values of j.

A number of steps of the form (15.3) reduce the high-frequency components of the initial error u °) - u, and at the same time produce a smoother error, uSm) - u, = vi, say. The smooth error v1 satisfies the linear system

L,v, = (L,u,(') - L,u, = L,uim) - f, _) gr,

(15.4)

say. This equation has the same form as the original equation (15.2) but since its solution v, is comparatively `smooth', one can approximate it

without making a substantial error on a `coarser' grid, say the grid corresponding to h1_1 = 2h,. Define g,-1 as the vector whose components are the second, fourth, sixth etc. components of g, and consider the linear system (15.5)

L1_1V1_1 = g1-1.

Find an exact solution v1_1 of (15.5) and construct from it an approximate solution v1 of (15.4) in the following way: The first and the last components of v, are half of the first and last components of v1_1. The

second, fourth, sixth, etc. components of 51 equal the first, second, third, etc. components of v1_1 and the remaining components of v, are the arithmetic means of the already defined adjacent components of v,. The vector W1 = of m) - 5, is called the coarse-grid correction of u,. We may write the damped Jacobi iteration which led from u(,°) to u(,m) together with the coarse-grid correction which led from u(lm) to u , in the form (15.6) M,,,,ui°) = u with a suitable matrix M,,,,. It can be shown that the spectral radius of

ui°

1

APPLICATIONS TO NUMERICAL ANALYSIS

121

M1, tends to 0 at least as fast as 1/em = 0.367879 .../m as m - -. This implies that the two-grid method just exhibited is essentially more efficient than the simple damped Jacobi iteration. Some remarks are appropriate.

1. The two-grid method proved to be a very fast iteration. v1_1 is needed to find an approximate solution 51 of (15.4). Since 51 is only an approximate solution of (15.4), it is sufficient to find an approximate solution 51_1 of (15.5). This approximate solution 151_1 may be obtained

by applying the two-grid method to (15.5). This gives a three-grid method. Continuing this until grids of level one are reached for which the corresponding linear equations can be solved trivially, we obtain the multi-grid method.

2. As an example of a linear partial differential equation of second order in d variables we consider the 2-dimensional Poisson equation on a compact plane domain D with smooth boundary,

-(uXX + u,) = f

in the interior of D

(15.7)

with Dirichlet boundary condition

u=g on the boundary of D. (15.8) Here f, g respectively are continuous real functions on D and the boundary of D.

For l = 0, 1, 2, ... let h1 = 1/21+1 and consider the lattice h1Z2. The grid of level 1 consists of the points of h1Z2 on D enumerated in a suitable way, for example in lexicographic order:

(xt, yt), (x2, y2),

.

.

., (xi, yi),

.

.

.

.

If (xi, yi) is a grid point such that the `cross' with endpoints (x, - h1i

yi), (xi + h1, yi) and (xi, yi + hi), (xi, yi - hi) is contained in the interior of D we replace -(ux(xi, yi) + uyy(xi, yi)) by the difference expression hI2

(-u(xi - hl, yi)

- u(xi + h1, yi) - u(xi, yi + hi)

1

- u(xi, yi - hi) + 4u(xi, yi)). (15.9) Besides this simple difference expression others have been used, for example the Mehrstellenverfahren of Collatz (1966). Next we have to take the boundary values g into consideration. Consider the `cross' (xi ± hi, yi), (xi, yi ± h) and suppose that (x1 - sh1, yi) and (xi, yi + th1) are boundary points of D for suitable s, t E (0, 1) while the open line segments with endpoints (xi - shli yi), (x, + h, yi) and (xi, yi + th,), (xi, yi - h1) are in the interior of D.

LATTICE POINTS

122

Figure 15.1 Difference formula for partial derivatives

Then instead of (15.9) we take 2

h2

s(s+1) u(x,-shi,yi)

+su(x`+hi,yj)

(= g(x, - shi, y1)) 2

t(t + 1)

u(x,, y, + thl)

2

-

t+1

u(x,, y,)

(= g(x,, y, + th1)) + (s + t)

u(xi,

Yi))

(15.10)

as an approximation for -(ux,(x,, y,) + uyy(x,, y,)). (See figure 15.1.) Note here other difference expressions have been used. Setting u1 = (u(x1, yl), u(x2, Y2), .

.

.)tr

.)tr fl = (f(x1, y'), f(x2, y2), the boundary-value problem (15.7), (15.8) leads via (15.9), (15.10) to a .

.

linear system

L1ul = fl to which the two-grid or the multi-grid method may be applied.

16

Lattice graphs

16.1 In this section we exploit the fact that the lattice Zd (and any other lattice as well) gives rise to so-called lattice graphs. We hope the material chosen will give a first idea of the flavour of the various types

of the problems on lattice graphs ranging from chess to binomial coefficients and crystal physics.

For the investigation of lattice graphs we need some graph-theoretic terminology, which for the reader's convenience will be outlined below. For more information on graphs the reader may consult, for example, Wilson (1985).

A graph G is an ordered pair (V, E) consisting of two finite or infinite sets, the set V = V(G) of vertices and the set E = E(G) of edges of G. There is a relation called incidence between the edges and the vertices, such that any edge is incident with exactly two distinct

vertices, and we say that the edge joins these vertices and that the vertices are adjacent. If two distinct vertices u, v are joined by an edge e, the latter is unique. (We do not allow multiple edges or loops.) Thus we may use the symbol uv for e. In an abstract sense a graph consists of a set of vertices and a set of (non-ordered) pairs of distinct vertices. More intuitively one may think of a graph as a set of vertices contained in some space where certain pairs of vertices are joined by curves. In the following a graph G will be called a lattice graph if V(G) C Zd

for suitable d. We note that there are several more restricted concepts of lattice graphs in use.

Let G be a graph. The valency of a vertex v of G is the (cardinal) number of the set of vertices u adjacent to v. A graph H is called a subgraph of G if

V(H) C V(G)

and

E(H) C E(G). 123

LATTICE POINTS

124

A spanning subgraph of G is one which contains all vertices of G. Paths and circuits are particular subgraphs. A path consists of a finite

. . ., V(k), say, such that v(0)00, o(k-1)v(k) are all edges of G. If additionally, v(k)v(0) also is

sequence of distinct vertices 00), V(I)V(2),

...,

an edge of G, then the sequence represents a circuit. It is clear what is

meant when we say that a path joins two (distinct) vertices of G. If every pair of vertices of G is joined by a path then G is connected. A circuit on G is a hamiltonian circuit if it includes every vertex of G. A tree is a connected graph without circuits.

16.2 An important problem in graph theory is to characterize the graphs for which there exist hamiltonian circuits. While a general solution of this problem seems to be out of reach, there are some results of this type for particular classes of graphs. Let d integers n 1, ... , n d > 1 be given and let G be the lattice graph the vertices of which are precisely the points v 1 E Z d with 1 < v 1: n 1 for i E {1, . . ., d} and where two vertices are joined by an edge if their

(euclidean) distance equals 1. (One may think of an edge as a line segment of length 1.) A result of Kotzig (1964) states that G has a hamiltonian circuit if and only if n1 nd is an even number. Several more properties and applications have been found on hamilto-

nian circuits for lattice graphs by Kurschak (1927), Nash-Williams (1960), Kotzig (1964), Thompson (1977), Binz (1978), Myers (1981),

Cannon and Dolan (1986) and many others are scattered in the literature on recreational mathematics, disguised as puzzles, board games, etc. One such problem is the `knight tour problem' on a chessboard, using the legal moves only, visiting every square exactly once. How many different tours can be designed?

16.3 Many problems of chess can be translated into lattice graph problems and can sometimes be solved by graph-theoretic machinery. An example is the problem how to place the maximum number of queens on the chessboard so that no two attack each other. (It is interesting to note that this problem was known already to Gauss.) Consider any chess piece P (not a pawn) and a lattice graph G whose

vertices correspond in an obvious way to the 64 (or in general, n) squares of a (possibly rectangular or even more general) chessboard. Two vertices u, v are joined by an edge if and only if P placed on the square corresponding to u can attack the square corresponding to v. The problem now is to find a set S of vertices of G with the properties that 1.

no two vertices in S are adjacent and

LATTICE GRAPHS

125

the number of vertices in S is maximal.

2.

For solutions to this problem and relevant references see Foulds and Johnston (1984).

Originally most of these problems were designed as puzzles and games. Later people found interesting applications in many parts of combinatorics. Perhaps the best known of these applications is the rook polynomial.

For a given chessboard with n squares let rk, k = 0, 1, 2, ..., denote the number of ways of placing k rooks on the board in such a way that no two attack each other. Define its rook polynomial r by n

r(x) _ I rkxk

for x E R.

k=0

The rook polynomial and some of its extensions are related to several parts of combinatorics, for example, the principle of inclusion-exclusion, permutations with forbidden positions, latin rectangles and matching problems. For particulars see Kaplansky (1946), Riordan (1958) and Roberts (1986).

16.4 Let S be a finite set of points in the plane. A tree in the plane whose vertices include all points of S and whose edges are non-crossing

line segments is a Steiner tree (figure 16.1) on S if the sum of the (euclidean) lengths of all edges is minimal (among all such trees). Let L(S) denote this sum. Steiner trees have a long and venerable history, dating back at least to Maxwell. They have a wide range of applications in optimization. For information see Gilbert and Pollak (1968) and Maculan (1987).

Figure 16.1 Steiner tree

Let

L(n) = max L(S) taken over all sets S of n points in the unit square.

L. Fejes TOth (1940) showed, by considering points arranged in a

LATTICE POINTS

126

regular hexagonal lattice, that

L(n) ? (3/4)1/4 \/n + 0(1). On the other hand Chung and Graham (1981) obtained for sufficiently large n

L(n) < 0.995Vn. They conjectured that

L(n) < 0.99\/n. They also considered the corresponding problem when the edges of the tree are parallel to the sides of the square and obtained

Vn + O(1) < L*(n) < 1/n + 1 + o(1) and conjectured

L*(n) !S Vn + 1

for all n. (L*(n) is defined similarly to L(n) but using the linear rectilinear distance Ix-y111=Ix1-x21+Iy1-Y21.)

Even the latter conjecture seems so complicated that it has not been verified even in the case when the points are the points of tZ2 contained in the unit square. Gardner (1986) exhibits trees on checkerboards of several different sizes but is unable to prove that they are Steiner.

16.5 Several important quantities in combinatorics can be interpreted as the number of certain lattice paths. These interpretations lead in many cases to very simple proofs of combinatorial identities. The following interpretation of binomial coefficients was apparently

first introduced by P61ya (1962). By a (staircaselike) path in Z2 we understand a finite sequence of vertices u(°) = o, v(1), U(2), ... such that

fork=0,1,2,... V(k+l)

= (Ulk) + 1, v?)), or (v

,

Uzk) + 1),

where U(k) = lUlk), V?));

then the number of paths from o to (x, y) e Z2\{o}, x, y ? 0 is xxy).

(If this number is denoted b(x, y), then b and the binomial coefficients satisfy

LATTICE GRAPHS

1y)+(x+X

0/-1'11y(XXI=(xX

b(0,y)=1, b(1,y)=y,

127

b (x, y) = b (x - 1, y) + b (x, y - 1)

for (x, y) E Z2, x, y ? 1 and hence coincide.) Since the number of path

from o to (x, y) equals the number of paths from o to (y, x) the identity (XY+ Y

xxy) is obtained. With similar ease one can derive many other identities involving binomial coefficients using this method. Next we consider the more general concept of a (non-decreasing) path

in Z2, viz. a finite sequence of points 0) = o, U(1), v(2), ... such that for k = 0, 1, 2, .. U(k+1) = (U(k) + 1, V2(k) + 1) or (V (k), v + 1) or (v + 1, v.k) + 1) .

for v(k) = (Ujk), V M.

Kasinsky and Bryant (1983) denote the number of these paths from o

to (x, y)EZ2\{o}, x, y?0by

\xxy) and derive a number of identities for these numbers, for example

x- 1

x

x- 1

x

for x, y ? 1, (x - 1, y - 1) E Z2\{o}. Thus one can arrange the values of

x in

X

yl

a triangular array similar to Pascal's triangle for the binomial

coefficients but with a different rule for the passage from two consecutive rows to the next one: 1

3

1

/

1

\\

5

5.

1

I:

1

UV

\

1

7

13

7

1

LATTICE POINTS

128

The numbers and

\x X y) are related. By considering the paths with 0, 1, 2, ... `diagonal edges' one arrives at the relation

E (x zy k2k)(x+y-k). k=O

Similarly one can show that ((1x+y-k).

( xxy)-E1k) Related but different arrays of `triangular number' were introduced by Shapiro (1976) and D. G. Rogers (1978).

Numbers based on paths the edges of which join mutually visible lattice points were introduced by Mohanty and Handa (1968). They are related to multinomial coefficients. More material of this sort including extensions to higher dimensions and lattice-path interpretations of other

problems can be found in Sved (1984), Breach (1985), Garsia and Remmel (1985) and in the monographs of Mohanty (1979), Narayana (1979) and Goulden and Jackson (1983).

16.6 It is well known how much the so-called `spreadsheet' is applied in computing. A spreadsheet is a two-dimensional square lattice which appears on the screen of a cathode-ray tube of a computer. The squares are usually called cells. The `value' of each cell can be made to depend on any other cell or a group of cells. In general the cells can be used to describe mathematical expressions or some logical relations. The spreadsheet provides a suitable medium for handling diverse problems from accounting to mathematical games. In this section we discuss only the relationship between a continuous region and its computer image. As an example we define a digital convex set and state a digital version of Helly's theorem. For other results we refer to the following sources: Minsky and Papert (1968), Sklansky (1970), Doignon (1973), Rosenfeld and Kak (1976), (1982), Sklansky and Kibler (1976), Kim (1981), Kim and Sklansky (1982), Hammer (1984), Arganbright (1985) and Horn (1986).

A digital region is a finite set S of integer lattice points. A finite set of lattice squares in E2, i.e. of squares of edge length 1 whose vertices

belong to Z2, is called a cellular region. It is clear that there is a one-to-one correspondence between digital and cellular regions by

LATTICE GRAPHS

129

assigning to each point u E Z2 the lattice square with u at its lower left vertex.

Given a `continuous' object C in E2, such as a compact set or a convex body, its digital (resp. cellular) image is the digital region

CnZ' (resp. the set of lattice squares which have non-empty intersection with C). Instead of C a computer `sees' the digital or cellular image of C. It thus seems to be important to define geometric and topological properties of digital or cellular regions in terms of their `continuous preimages'.

If a, b are two points of a lattice, then the digital line segment with endpoints a, b is the intersection of the lattice with the closed continuous line segment [a, b]. A digital region S is digitally convex if there exists at least one convex

set of which the digital region is the digital image. In other words, S is digitally convex if it is the intersection of the lattice with a convex set. This definition implies that for a digitally convex set S the digital line segment with endpoints a, b is contained in S whenever a, b E S. (The converse of this statement does not hold, as may be seen by considering the digital region S = {(0, 0), (1, 2), (2, 1)).) Doignon (1973) proved the following remarkable digital version of

Helly's theorem in dimension d: A finite collection of at least 2d digitally convex sets has a non-empty intersection if each subcollection of 2d members has a non-empty intersection. (The number 2d is best possible.)

References on digital processing are Horn (1986) and Rosenfeld and Kak (1976, 1982).

16.7 In this section we present problems which have interpretations in physics and other sciences. For many more such problems see the surveys and monographs cited below. Some of these problems can be interpreted in terms of covering, packing or tiling. We use these interpretations, for they prevail in the literature.

1. The cell growth problem. A one-celled animal is simply a lattice square. A one-celled animal can `grow' by successively adding further one-celled animals to its free sides. In this way one obtains n-celled animals for n = 1, 2, . ., also called polyominoes. How many simply (or multiply) connected n-celled animals (for given n) are there up to isometries of the plane? For problems of this sort and generalizations to animals consisting of regular triangles or hexagons we .

LATTICE POINTS

130

refer to Golomb (1965, 1966), Harary (1967) and Temperley (1981).

An appealing tiling result on four-celled animals is due to Coppersmith (1985). It says that one can tile the plane with congruent copies of any four-celled animal. The corresponding problem for five-celled animals remains open, while there are six-celled animals which admit no tiling of the plane. 2. The dimer problem. If a biatomic gas molecule of the right size is adsorbed on the surface of a crystal it occupies two neighbouring sites.

Thermodynamical problems of systems of such molecules led to the following mathematical problem: Consider a lattice graph G in Z2 having n vertices and such that two vertices are adjacent precisely if they have distance 1. A dimer on G is a subgraph consisting of two adjacent vertices and the edge joining them. A dimer covering of G is a spanning subgraph of G each vertex of which has valency 1 (this means that it consists of disjoint dimers which contain all vertices of G). The problem is to determine the number of different dimer coverings of G or to give asymptotic estimates for increasing sequences of Gs. One can show that if m, n are non-negative integers then (using some difficult analysis) for the rectangle {u E Z2: 1 <_ ul 2m, 1 < u2 2n} there are m 2m 22mn

1111 i=1 j=1

{cos2(2m

i+

7) + cos2( 1

j

2n + 1

)

1 i/2

different dimer coverings. This asymptotically approaches e4mny/n

as m, n -* + oo, where y = 0.915 965

...

(Catalan's constant). For various methods of solution for this dimer covering problem see Fisher and Stephenson (1963), Kasteleyn (1967), Percus (1971) and Temperley (1981) and the references in these sources. It is clear that the dimer covering problem can be formulated in any dimension and for arbitrary finite graphs. (The case Z3 is related to the problem of mixtures of molecules of different sizes.) In most cases no satisfactory solution has been obtained so far.

3. The Ising problem. Consider a lattice graph G in Zd (a crystal). Let each vertex v of G be occupied by an atom which can be in two states s(v) = 1, s(v) = -1. (For example, consider a crystal consisting of atomic magnetic dipoles and let s(v) = 1 if the dipole of v is oriented parallel to some fixed direction, and let s(v) = -1 if it is parallel to the opposite direction; or consider a binary alloy consisting of elements A, B, and let s(v) = 1 if element A is at site v and let s(v) = -1 if element

B is at site v.) Assume that only atoms of adjacent sites can interact and that the interaction energy of atoms at adjacent sites u, V is Js(u)s(v) for some constant J. Moreover, there may be a contribution

LATTICE GRAPHS

131

-Is(v) of the atom at site v to the energy, where I is a constant. Then the total energy of the configuration s on G is E{s}

J > {s(u)s(v): u, v E V(G) adjacent}

- I E {s(u): u E V(G)}. The weight of configuration s at absolute temperature T is e-E{s}/kT where k is the Boltzmann constant. The thermodynamical properties of the crystal G can all be derived from the function Z = I {e-E{s}/kT: s configuration on G}. The calculation of this function and in particular the determination of its

asymptotic behaviour as the number of atoms of G tends to +00 constitutes the Ising or Ising-Lenz problem. One can show that if G has n vertices and I = 0, then

Z = (cos-j%)"h(m) (tanh-)m, M=0

where h(m) is the number of spanning subgraphs of G containing m edges such that each vertex has even valency. Explicit solutions of the Ising problem are known only in a small number of cases, for example

by Onsager (1944) for the planar case with I = 0. A result due to Kasteleyn (1967) shows that the Ising problem can be reduced to a modified form of the dimer problem. For this as well as for other results we refer to Montroll (1964), Kasteleyn (1967), Percus (1971), Harary (1971) and Temperley (1981). 4. The random walk problem. Let a point perform a move randomly on Z d according to the following rules: At time t = 0 it is at the origin and if at time t - 1 it is at site v, say, then at time t it will be one of the

2d lattice points nearest to

v;

the probability of its being at any

specified one of those points is 1/2d. Polya (1921) discovered the remarkable fact that the randomly moving point will with probability 1 return to the origin if d <- 2, while if d > 2 then the probability of returning to o is less than 1. More precisely, this probability is n

1-

1

P(1)'

where P(1) =

1

(27r) d

J

n .

.

.

or asymptotically

(2d-2-(3/2d)+ )-'

as d-> -.

dcpl

.

.

. dcpd

1 - (1/d) Ed COST,

132

LATTICE POINTS

One generalization of this problem is to consider different probabili-

ties for different directions; another is to consider random walks on other lattice graphs where not only nearest vertices are adjacent. These problems are connected with enumeration problems for paths of given length and starting at the origin: among the voluminous body of papers

and books on random walks we mention Montroll (1964), Spitzer (1964), Kasteleyn (1967), Percus (1971), Mohanty (1979) and Snell and Doyle (1984).

17

Extremal combinatorial problems

17.1 In this section we discuss extremal problems on sets of points in which a lattice structure provides or might provide a solution. Some of the problems will be posed specifically on sets of lattice points of Z". Usually we will restrict ourselves to the plane, though many interesting questions can be posed in higher dimensions.

1. Let there be given n points, x(l), ..., x(n), in the plane. Assume their mutual minimum distance is 1. Put

6(n) = min (E Jx(') - x(')J),

(17.1)

1a,
the minimum of the sum of all possible distances between the points. Fejes TOth poses the problem of determining 6(n). Let {x(1), ..., x(n)} be a set which implements (17.1). Fejes Toth conjectures that in this case the xWs are the vertices of an equilateral triangular lattice of side 1. Horvath (1969) gave an affirmative answer to the conjecture in case

3-- n-- 7. 2. For points x(l),

...,

x(n)

in the plane with minimum mutual

distance 1 denote by D(x('), ..., x(n)) the number of distinct distances.

(Note that if the number of points is not finite then the number of different distances D is not finite either, as can be shown easily.) Put

f(n) = min D(x('), ...,

X( n)),

where the minimum is to be taken for all possible choices of xM, ..., x(n). Denote by g(n) the largest number of pairs in {x(1), ..., x(")} for which

Jx(') - xWJ = v,v a constant. As far as we know problems

concerning f(n) and g(n) were first asked (and some of them answered)

by Erdos (1946). It is easy to see that f(2) = f(3) = 1, f(4) = f(5) = 2,

f(6) = f(7) = 3. To determine f(n) for all n seems hopeless at the moment. The strongest conjectures for upper and lower bounds for these functions are 133

LATTICE POINTS

134

f(n) < an exp(c 1 log n/log log n)

(17.2)

and

g(n) >

(17.3)

fan/(logn)1/2.

(In this section all a, /3, y, 6, a, i are positive constants.) It was shown that if (17.2) and (17.3) are true, they are best possible. Also it was proved that

(n - 1)1/2 - 1 < f(n) < an/(logn)]n.

(17.4)

The upper bound in (17.4) is obtained by considering the number of lattice points {(x, y): 0 < x, y < Vn} (see Erdos (1946)). L. Moser (1952) improved the lower bound in (17.4) to 6n2/3 and Chung (1984) improved this to bn5/7. For g(n) it was proved that n1+r/loglogn < g(n) < 2n2/3.

Again the latter is obtained by considering the lattice points {(x, y): 0:5 x, y "Vn). See Erdos (1946). Beck and Spencer (1984) improved the first inequality to

3. Let {x('),

...,

n312-,, for a certain E > 0.

x(n)} be a set which implements f(n), i.e. D(x(1), x(n)) = f(n). Is it true that {x(1), ..., x(')) has lattice structure?

The first step to answer this question would be to decide if there is always a line which contains an 1/2 of the x Ws (and in fact n' instead of

n1/2 would be interesting), where a > 0 is an absolute constant. A stronger result would be that there are an 1/2 (or n1-E) lines which contain all the x0s. The only result in this direction, due to Szemerddi,

states that if D(x(1), ..., x(n)) is o(n) (more precisely we should consider a sequence of sets {x(n'1) x(n'n)}, n = 1, 2, ..., but we adopt the usual sloppy notation) and k is a positive integer, then there

is always a line which contains at least k of the points supposing n is sufficiently large (see Erdos (1975)). In fact Szemerddi's result gives that

such a line can be chosen as the perpendicular bisector of two of our points, and also that there are o(n) lines which contain all our points. 1

4. Erdos and Guy (1970) ask: how many lattice points (u;'), up), k with coordinates 0 < u;'), u(j) n may be chosen with all

mutual distances distinct? They conjecture k < an 2/3 (log n) 1/6

and showed that n2/3-E < k < an /(log n) 1/4 for any E > 0 and sufficiently large n.

EXTREMAL COMBINATORIAL PROBLEMS

135

Further related problems can be found in Erdos (1975, 1986) and in Moser and Pach (1986).

17.2 A well-known theorem of van der Waerden (1927, 1971) states that for all positive integers 1, h there is a number w(l, h) such that if the positive integers not exceeding w(l, h) are partitioned into l classes, at least one class contains an arithmetic progression containing h terms.

The geometric interpretation of the theorem makes it plausible to extend this to higher dimensions: For any finite subset H of the lattice points of Ed and any partition of the lattice points of Ed into 1 classes,

at least one class contains a subset which is homothetic to H. This theorem was established by Grunwald (see Rado (1938)). There are several other generalizations and extensions of van der Waerden's theorem - see Erdos and Graham (1980), Guy (1981) and Sos (1983) and the references there. For the purpose of obtaining estimates for the number w(l, h), Erdos and Turan (1936) introduced the quantity rk(n), defined to be the least

integer r so that if

l <_ a 1 <

< a. < n, then the sequence of a;s

must contain a k-term arithmetic progression. After several estimations of the bounds of rk(n) by various people Szemeredi (1975) proved that

for fixed k we have that rk(n) = o(n) as n

-. This result has also

been proved by Furstenberg (1977, 1981) using ergodic theoxy; Furstenberg and Katznelson (1978) established the higher-dimensional version of Szemeredi's theorem.

17.3

We restate the theorem of van der Waerden in a slightly

different form: for any positive integer h there is a positive integer w(h) so that if we partition the integers from 1 to w(h) into two classes, at least one of them contains an arithmetic progression of h terms. In this section we have collected problems of similar character: For h > 0 let f(h) be the smallest integer f so that every set of cardinality f contains a subset consisting of h elements with a given structure.

1. Zarankiewicz (1951) posed a problem, which can be stated as follows: Let 2 < a < p, 2 < b < q be integers: determine the least number z = z(a, b; p, q) so that a p x q (0, 1)-matrix, containing z ones and pq - z zeros, no matter how distributed, contains an a x b submatrix consisting entirely of ones. Or in colourful terminology: determine the least z such that if in the above matrix we paint any z elements red, and the remaining ones blue, then the matrix will contain

LATTICE POINTS

136

an all red `spot' of size a x b. Sierpinski (1951) showed that z(3, 3; 4, 4) = 14, z(3, 3; 5, 5) = 21 and z(3, 3; 6, 6) = 27. Kovari et al. (1954) (see also Reiman (1958)) showed that

limn-3/2 z(2, 2; n, n) = 1.

There are some more exact values for z but it seems hopeless to determine the general z. The interested reader can find information about the `many faceted' nature of z in the review paper of Guy (1969) and in Bollobas (1978).

2. Determine the smallest integer f(r, d) such that any set of f(r, d) lattice points in Ed contains a subset of r points the centroid of which is

itself a lattice point. Clearly f(2, d) = 2d + 1, as an arbitrary set of 2d + 1 lattice points in Ed, contains at least one pair of lattice points such that the centre of their connecting line is again a lattice point, and this number is minimal as shown by the unit cube. Harborth (1973) has shown that

(r - 1)2d + 1 <- f(r, d) < (r - 1)rd + 1. He also showed that

2) = 4.3 and f(2', d) = (2' - 1)2d + 1.

See also the papers of Kemnitz (1983) and Erdos, Ginsburg and Ziv (1961).

3. Many interesting results and problems exist which are of similar type to the above, but since they are not directly related to lattice points we indicate just a few of them: Ramsey's theorem (see Parsons (1978)

and Graham, Rothchild and Spencer (1980)), Esther Klein's convex n-gon problem (see Erdos and Szekeres (1935, 1960/61), Korte and Lovasz (1984) and Roberts (1986)), Turan's problem in graph theory (see Bollobas (1978) and Simonovits (1983)).

17.4 The van der Waerden type theorems are the forerunners of a new branch of combinatorics which goes under the name euclidean Ramsey theory. This theory deals with questions of the following type. Given any finite set S in some euclidean space, S is called Ramsey if the following situation holds: for any k there is an nk (depending only on k and on the finite set S) so that if you partition the points of E"k into k classes (or if you wish, you colour by k colours), then at least one class contains a set which is isometric to S (i.e. it contains a monochromatic copy of S). Note that this set is not homothetic to S as in Grunwald's theorem, but isometric to S. The simplest thing which is Ramsey is (the

EXTREMAL COMBINATORIAL PROBLEMS

137

set of vertices of) the unit square. For example in E5 one can give fifteen points so that if they are coloured by two colours at least one colour contains a unit square. More generally, for every k there is an nk

so that one can give a finite set in E"k so that if it is coloured by k colours one of them contains a unit square. This clearly implies that (the

set of vertices of) the unit square is Ramsey. Finally we mention a theorem of Graham (1980): for any k there is a number g(k) so that for

any k-colouring of the lattice points of the plane there is always a monochromatic right-angled triangle with area g(k). For results of a similar character, see the papers of Erdos et al. (1973) and Erdos and Graham (1980).

17.5 A theorem of Roth (1964) states that if we colour the integers from 1 to n by two colours, say red and blue, in any way, there always exists an arithmetic progression such that the difference of the numbers of red and blue terms in this progression has absolute value greater than an 1/4 for an absolute constant x > 0. This is a discrepancy-type problem in the theory of uniform distribution (see Sos (1983)). A basic problem in the theory of uniform distribution is to colour the elements of a set with two colours as uniformly as possible with respect to a given family of subsets such that each colour meets each subset in approximately the same number of elements. Now, let X be a finite set and X be a family of subsets of X. Given a two-colouring f: X -> {+1, -1 } of X, and a set SEX , call

E = E(S, f) = Zf(y)l yES

the error off relative to S. Set

D(X), f) = max(E(S, f)); S E X} and define the discrepancy Disc(X, f) of f by

Disc(X, f) = min(D(X), f); f a two colouring of X}. The basic problem is to estimate and, if possible, to determine Disc(X)). Beck (1982) conjectures that

Disc(i) = O((a log a)1/2) where

a = max IS E ti: y E SI. YEX

Beck (1982) has the following generalizations of Roth's theorem: If we

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{(ul, u2): the lattice points of the n x n-square 0 < u; < n} C Z' with red and blue in any way (n = 1, 2, 3, ...), there exists a line segment with E at least /3n'/'-f, where (3 > 0 is a suitable absolute constant. On the other hand there exist two-colourings of the n x n-square (n = 1, 2, 3, ...) such that for any line segment colour

E = O(n'13(log n)'I'). A more general theorem of Beck (1985) is the following: There is an absolute constant y > 0 such that for convex bodies C in E' for which the radius of the largest inscribed eucidean ball is at least 1 and for any colouring f: Zd -b {1, -1} there is a proper orthogonal transformation A, a real number A E [0, 1] and a vector t E E d such that

E flu): u c- (AA(C) + t) n Zdl ? yS(C)'/'. Here S(C) denotes the surface area of C. Uniform distribution and, in particular, results dealing with discrepancy have applications in geometry, number theory, numerical analysis and several parts of combinatorics. See Sos (1983), Beck (1985) and Lovasz et al. (1986). For general information on uniform distribution we refer to the books of Kuipers and Niederreiter (1974) and Hlawka (1984).

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Nouvelles applications des parametres continus a la theorie des formes quadratiques. Deuxieme memoire. Recherches sur les para1leloedres primitifs. J reine angew Math 134: 198-287 Nouvelles applications des parametres continus a la theories des formes quadratiques. Domaines de formes quadratiques correspondant aux different types des paralleloedres primitifs J refine angew Math 13: 67-181 Collected works (Russ) (Vols 1-3), Izdat Akad Nauk Ukrain SSR, Kiev.

Voskresenskii V E and Klyachko A A 1985 Toroidal Fano varieties and root systems Izv Akad Nauk SSSR 48:, Math USSR Izv 24: 221-44 Waerden B L van der 1927

Beweis einer Baudet'schen Vermutung Niew Arch voor Wisk 15: 212-16

1956

Die Reduktionstheorie der positiven quadratischen Formen Acta

Math 96: 265-309 How the proof of Baudet's conjecture was found. In: Studies in pure mathematics. 251-60 Academic Press. Waerden B L van der and Gross H (eds) 1968 Studien zur Theorie der quadratischen Formen. Birkhauser, Basel. Wagon S 1985 The Banach-Tarski paradox. Cambridge University Press. Watson D F 1985 Natural neighbour sorting Austral Computer J 17: 189-93 Watson G L 1971

1956

The covering of space by spheres Rend Circ Mat Palermo (2) 5: 93-100

1966

On the minimum of a positive quadratic form in n (< 8) variables (verification of Blichfeldt's calculations) Proc Cambridge Philos Soc 62: 719

White G K 1963

A refinement of van der Corput's theorem on convex bodies Amer J Math 85: 320-6 Whitworth J V 1948 On the densest packing of sections of a cube Annali Mat Pura Appl (4) 27: 29-37 1951 The critical lattices of the double cone Proc London Math Soc (2) 53: 422-43

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Gitterpolytope mit inneren Gitterpunkten. In: 2 Kolloquium fiber diskrete Geometrie (Salzburg 1980) 231-5. Inst Math University of Salzburg.

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On an analogue to Minkowski's lattice point theorem. In: The

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Kugellagerungen and Konvexgeometrie, Jber. Deutsch. Math. Verein., In print.

Wilson R S 1985

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Winfee A 1965

Advanced problem 5263 Amer Math Monthly 72: 192-3

Witt E 1941

Spiegelungsgruppen and Aufzahlung halbeinfacher Liescher Ringe

Abh Math Sem Univ Hamburg 14: 289-322 Woods A C 1956

The anomaly of convex bodies Proc Cambridge Philos Soc 52: 406-23

1958(a) 1958(b) 1958(c)

On a theorem of Tschebotareff Duke Math J 25: 631-8 On two-dimensional convex bodies Pacific J Math 8: 635-40 On a theorem of Minkowski Proc Amer Math Soc 9: 354-5 1965 Lattice coverings of five space by spheres Mathematika 12: 143-50 1966 A generalisation of Minkowski's second inequality in the geometry of numbers J Austral Math Soc 6: 148-52 Yakovlev N N 1983 On the densest lattice 8-packing in the plane Vestn Mosk Univ Ser 1: 8-16; Moscow Univ Math Bull 38: 7-16 Yates D L 1970

Suprema for a class of finite sums with applications to lattice coverings Proc London Math Soc (3): 21 731-56

Yeh Y 1948

Lattice points in a cylinder over a convex domain J London Math

Soc 23: 188-95 Zarankiewicz K 1951 Problem 101 Colloqu Math 2: 310 Zassenhaus H 1948

Uber einen Algorithmus zur Bestimmung der Raumgruppen Com-

1961

ment Math Helvet 21: 117-41 Modern developments in the geometry of numbers Bull Amer Math Soc 67: 427-39

2itomirskii 0 K 1929 Verscharfung eines Satzes von Woronoi Z Leningrad Fiz-Mat Obsc 2: 131-51 Zolotarev E I (= G Zolotareff) 1931

Complete collected works (Russ). Izdat Akad Nauk SSSR, Moscow.

Subject index

adjacent vertices 123 admissible in the sense of MinkowskiHlawka 53 admissible lattice 16 affine perimeter 21 algorithm of Voronoi 90 Alhambra 70, 71 animals 129 aperiodic tiling 78 arithmetic minimum 89 arithmetical crystal class 93 aspect of a tiling 78 asymmetry coefficient 19 Baire categories 51 basis 14 binary code 59 binomial coefficient 126, 127 blocking a point 108 body centred cubic lattice 65 Bravais type 72, 93 Brillouin zone 74 Buffon's needle experiment 111 cell growth problem 129 cellular region 128 chess 124, 125 chord of symmetry 21 circle problem of Gauss 104 circuit 124 classical Minkowski reduction 84, 86 coarse grid correction 120 code 59 cone of coefficients of positive quadratic forms 81 conjecture on non-homogeneous linear forms 103 conjecture of Davenport 29

conjecture of Ehrhart 20, 23 conjecture of Gruber 18

176

conjecture of L. Moser 27 conjecture of Mahler 31

conjecture of Mordell 104 connected graph 124 continued fractions 100, 101, 102 convergence in the space of lattices 37 converse problem of Mordell 17, 99 convex body 10 convex tiling 73 covering 41 covering constant 32 covering criterion 50, 52 covering lattice 41 covering of multiplicity k 42 covering radius 93 covering with congruent copies/translates 41 critical determinant 16 crystal structure 68 crystallographic or proper space group type 69

damped Jacobi iteration 119 Delone triangulation 74, 94 density 16, 41 density of the densest lattice packing 41 density of the thinnest lattice covering 41

determinant of a lattice 14 digital (cellular) image 129 digital line segment 129 digital region 128 digitally convex 129 dimer problem 130 Diophantine equation 104 Dirichlet boundary value problem 118 Dirichlet-Voronoi cell 38, 74, 88, 117 Dirichlet-Voronoi tiling 74, 75 Dirichlet's box principle 25 discrepancy 137

SUBJECT INDEX discrepancy-type problem 137 discretization method 118 discriminant 16, 81 discriminant surface 90 dissection 1 divided cell algorithm 102, 103, 104 domain of action 74 dual cone 11

edge form 83, 92 edge of a graph 123 edge of a polytope 6 ellipsoid algorithm 88 embedding into Zd 12 empty ball 94 empty ball method of Delone 94 enantiomorphic pair 69, 71, 72 equi-affinity 21 equidissectable 1

equivalent quadratic forms 81 error 117, 118 Euler characteristic 6, 111 eutactic form 91 extremal rectangle 96 extreme form 89 face-to-face tiling 73 facet-to-facet tiling 73 fan 11 finiteness theorems 84, 86 first successive minimum 28 formula of Hofreiter 3 formula of Poincare 112 formula of Steiner 23 fundamental domain of a lattice 114 fundamental kinematic formula of Blaschke 112 fundamental lattice 2 fundamental parallelotope 14 fundamental theorem of Minkowski 15, 26

G-equicomplementable 1 G-equidissectable 1

genus 104 geometric crystal class 71

Hilbert's 3rd problem 1, 2 homogeneous minimum 28 honeycomb 74

inequality of Blaschke and Santal8 31 inhomogeneous minimum 54 integer lattice 65

integer unimodular matrix 82 integrally conjugate groups 93 interior form 83 Ising (-Lenz) problem 130, 131 isolation theroem 100 k-fold covering 42 k-fold packing 41 kinematic density 111 kissing number 64 Klein polygons 100, 101, 102

L-polytope 94 L-tiling 74, 93, 94 L-type 94 L3-algorithm 88 L3-reduction 87, 88 lattice 14 lattice constant 16 lattice covering 41 lattice graph 123 lattice packing 16 lattice point 14 lattice point enumerator 6 lattice polytope 4, 6 lattice square 128 lattice tiling 73 Laurent polynomial 11 Leech lattice 64, 91 length of a code 59 letters of a codeword 59 linear code 60 linear form theorem of Minkowski 17 linear programming method 58 locally densest lattice packing 89, 91 locally finite tiling 73 locally thinnest lattice covering 94, 95 losseness of a covering 53, 65

geometry of numbers 14 graph 123 graph of visible points 109

grid 102 grid of level 1, 119

Hadwiger's criterion for Zd equidissectability 3 Hamilton circuit 124 Hamming distance 59 Hilbert's 18th problem 40, 67, 69

Mahler's selection (or compactness) theorem 38 measure of density 53 measure of thinness 53 Mehrstellenverfahren 121 Mertens conjecture 88 metric form of a lattice 82 minimum distance of a code 60 minimum width 23 Minkowski reduction 83, 84, 85, 86

177

LATTICE POINTS

178

Minkowski's convex body theorem 15, 26

Minkowski's second theorem 28, 29 Minkowski-Hlawka packing 53 Minkowski-Hlawka theorem 34, 35, 36, 42, 59, 61 Minkowskian arrangement 53 mixed volume 10, 23 multi grid method 121 multiple reduction domain 83 neighbour in a packing 50 neighbourhood basis for the space of lattices 37 neighbouring lattice points 109 Newton number 64 norm of a matrix 37 normalized internal angle 2 numerical integration 116, 117, 118 orchard problem 107

packing 40 packing lattice 41 packing of multiplicity k 41 packing radius 89 packing results 52 parallel body 10 parallelogram algorithm 96 prallelohedron 15, 28, 29, 79, 80 path 124 pattern 109 Penrose tiling 77 perfect form 90 pigeonhole principle 25 Poisson equation 121 polear (reciprocal) body 30 polar (reciprocal) lattice 32 polygon 6 polyhedron 73 polyominoes 129 polytope 6 positive (definite) quadratic form 81 power-diagramm 74 primitive L-type 94 primitive point 11, 35, 108

problem of the thirteen balls 55 problemm of Voronoi 80 proper Bravais type 72 proper convex lattice polytope 4 proper geometric crystal class 71 proper polytope 1, 6 prototile 73

pseudo random numbers 88 quermassintgrals 23

Ramsey theory 136 random walk problem 131 rational point 104 realizable pattern 109 reduced form 83 reduction domain 83 reduction theory 82 Riemann hypothesis 88 rook polynomial 125

selection theorem of Blaschke 38 selection theorem of Mahler 38 short vectors 88 Siegel's mean value theorem 35, 59 simple reduction domain 83 simplex algorithm 64 simplicial complex 6 simplicial decomposition 6 smooth body 31 space group 68, 69, 93 space group type 69, 70, 71, 93 spanning subgraph 124 sporadic simple group 64 star body 16 star number 50, 51 star set 34, 107 Steiner tree 125 Steinhaus' problem 24 stereohedron 78 stereon 78 strictly convex body 31 subgrah 123 successive minimum 28 symmetry (operation) 68 theorem of Bieberbach 69, 75 theorem of Blichfeldt 25 theorem of Faltings 104 theorem of Hajos 17, 80 theorem of Mahler 32 theorem of van der Waerden 135 theorem of Venkov and McMullen on parallelohedra 79 theorem of Voronoi 64, 65, 91 tile 3, 73 tiling 73, 83 tiling lattice 73 tiling of congruent copies or translates 73

topology on the space of lattices 37 toric variety 11 transference theroems 53, 54 transmission of data 60 tree 124 two grid method 21 types of parallelohera 79, 80

SUBJECT INDEX uniform distribution 137 valency 123

vertex of a graph 123 vertex of a polytope 6 view obstruction 107 visible point 107, 108 Voronoi polyhedron 92

Wabenzelle 74 weight of a codeword 59 weighted lattice point enumerator 3 weighted number of points 3 Wigner-Seitz zone 74 Wirkungsbereich74 Witt lattice 64

zeta-function for lattices 116

179

Author index

Abbott 108, 110 Afflerback 84, 87 Ahmedov 103 Akimova 74 Alexandroff 73 Alexandrov 12, 79 Allen 107 Andrews 105 Arganbright 128 Aristotle 68 Arkinstall 21 Aurenhammer 74, 75 Babai 88 Babenko 74, 117 Bacon 68 Baiada 24 Baker 104 Bambah 18, 29, 31, 32, 40, 48, 49, 57, 65, 104 Banach 2 Bannai 64 Baranovskii 40, 56, 57, 58, 91, 93, 95 Barany 108 Barbier 112 Barnes 30, 56, 57, 64, 74, 91, 95, 104 Batyrev 12 Beck 24, 27, 134, 137, 138 Bender 22, 24, 50 Berger 77 Bern§tein 10

Betke 4, 5, 9, 10, 11, 108 B6rard 105 Bieberbach 67, 69, 75 Binz 124 Birch 54 Blaschke 21, 31, 38, 111, 112 Bleicher 53 Blichfeldt 17, 18, 25, 26, 27, 29, 56, 57, 58 180

Blundon 62, 63 Bokowski 22, 52 Bolle 62, 64 Bollobas 136 Boltjanski 2, 51, 207 Bolyai 2 Bombieri 26 Bonnesen 10, 20, 43 Bos 61

Bourgain 31, 32 Boroczky 65 Brauer 56 Bravais 67, 72 Breach 128 Brown 67, 71, 72, 93 Bruijn, de 78 Brunn 43 Bryant 127 Buffon 111, 115 Burckhardt 67 Burzlaff 67, 71, 78 Butler 35, 54 Biilow 72, 93 Cahn 78 Cannon 124

Cassels 16, 19, 26, 38, 100, 102, 116 Chabauty 30, 38 Chaix 105, 106 Chalk 33, 46, 50 Charve 83 Chung 126, 134 Coates 104 Cohn H. 101, 102 Cohn M. 52 Colin de Verdiere 105 Collatz 118, 121 Conway 56, 59, 60, 64 Cook 33 Coppersmith, 130

AUTHOR INDEX Corput, van der 18, 19, 21, 26, 104, 105 Coxeter 8, 40, 55, 58, 59, 64

Crofton 111 Cusick 107, 108 Czuber 111

Dade 93 Danicic 29 Danilov 12 Danzer 78, 107 Dauenhauer 58 Davenport 14, 19, 21, 29, 59 Dehn 2 Delone 14, 56, 57, 67, 68, 74, 79, 80, 81, 83, 93, 94, 95, 96, 102, 117

Delsarte 59 Demazure 11 Diananda 116 Dickson 56, 95 Dieter 88 Dirichlet 14, 25, 38, 39, 68, 73, 74, 75, 79, 83

Divil 106 Doignon 128, 129 Dolan 124 Dolbilin 79, 93, 95 Dowker 44 Doyle 132 Dumir 18, 52 Dwyer 39

Edelsbrunner 74, 75 Ehrhart 9, 11, 20, 21, 55 Eichler 19

Fricker 104 Frobenius 69 Fukasawa 101 Furstenberg 135 Furtwangler 14 Galiulin 67 Gardner 77, 126 Garsia 128 Gauss 1, 14, 46, 55, 56, 57, 83, 84, 104, 124 Gerald 118

Gerwien 2 Gilbert 125 Ginsburg 136

Gohberg 107

Golomb 130 Goulden 128 Graham 44, 126, 135, 136, 137 Gregory 55, 64 Griganovskaja 47 Gritzmann 23, 43 Groemer 21, 38, 39, 46, 50, 51, 52, 64, 79, 108, 114 Gross 83

Grothe 87 Gruber 18, 26, 51, 64, 73, 100, 103, 118

Gruber, Lekkerkerker 9, 16, 26, 28, 36, 38, 45, 53, 65, 68, 74, 80, 83, 91, 104

Grunbaum 21, 67, 68, 78, 79 Grunwald 135 Guy 134, 135, 136

Elkington 24

Haas 61, 63

Ellison 104 Engel 67, 78 Ennola 44, 116, 117 Erdos 43, 51, 110, 133, 134, 135, 136,

Hackbusch 118

137

Euler 6, 7 Ewald 12 Faltings 104 Fary 45, 48 Fedorenko 118

Fedorov 14, 67, 68, 70, 71, 79, 95 Fejes Toth G. 40, 48, 52, 61

Hadwiger 2, 3, 4, 7, 8, 22, 23, 50, 51, 107, 111, 112, 114

Haj6s 14, 17, 80 Hales 55 Hammer 22, 24, 25, 39, 128 Handa 128 Hans-Gill 18, 52 Harary 130, 131 Harborth 136 Hargittai 68 Harvey 39

Fenchel 10, 12, 20, 43 Few 57, 61, 63 Fisher 130 Florian 40, 51 Foulds 125

Hairy 67 Heesch 67, 77 Helly 128 Henry 71, 78 Heppes 62, 65 Hermann 67 Hermite 14, 17, 83 Herzog 109

Frankenheim 72

Hessel 67, 72

Fricke 70

Hietler 64

Fejes Toth L. 13, 21, 40, 43, 44, 45, 47, 48, 53, 56, 58, 68, 80, 108, 125, 133

181

182

LATTICE POINTS

Hilbert 1, 2, 40, 67, 69, 75, 76 Hilton 67, 71 Hlawka 18, 19, 34, 35, 36, 40, 42, 51, 53, 54, 56, 58, 59, 61, 105, 138 Hofreiter 3, 83 Honsberger 107

Hopf 73 Hoppe 55, 64 Horn 128, 129 Hortobagyi 65 Horvath 65, 133 Hoylman 46 Hsieh 23 Hurwitz 101 Huygens 67 Ising 130, 131

Kuperberg 45, 48 Kiirschak 124 Lagarias 33 Lagrange 14, 47, 55, 56, 57, 83, 84 Landau 104 Lang 104 Leech 43, 60, 61, 64 Leichtweiss 10, 30 Lekkerkerker see Gruber Lenstra A. K. 83, 87, 88 Lenstra H. W. 33, 83, 87, 88 Lenz 131 Levenstein 17, 59, 64 Lindsey 58 Linhart 53, 61 Litsyn 61 Liu 8

Jackson 128

Jarnfk 29, 54, 104, 105, 106 Jespersen 118 Johnston 125 Jordan 67, 71, 93 Jurkat 28

Kabat'janskii 17, 59 Kak 128, 129 Kanagasabapathy 63 Kaplanski 125 Kasimatis-Rooney 49 Kasinski 127 Kasteleyn 130, 131, 132 Katznelson 135 Keller 45, 80, 83, 96, 97 Kelvin 40 Kemnitz 136 Kempf 11 Kepler 14, 55, 67 Kershner 56, 57, 77 Kibler 128 Kim 128 Klein E. 136

Klein F. 14, 70, 96, 100 Klyachko 12 Kneser 4, 54, 65 Knuth 88 Ko 18 Koch 74 Korkin 14, 50, 56, 57, 83, 90, 91 Korte 136 Kotzig 124 KOv3ri 136

Krasnoselskii 107 Kratz 28 Krejcarek 62 Kuharev 47 Kuipers 138

Lonsdale 71, 78 Loomis 49 Lovasz 83, 87, 88, 136, 138 Lucas 12 Luptacik 64 Lutwak 31

Macbeath 27, 39 Macdonald 8, 11 Mackay 78 Maculan 125 Mahler 28, 31, 32, 33, 37, 38, 39, 44 Maier 23 Malysev 47 Marvin 55 Maschke 93 Matzke 55 Maxwell 125

McMullen 2, 9, 10, 11, 12, 79 McWilliams 60 Melnyk 55 Milman 31, 32 Minkowski 10, 14, 15, 16, 17, 25, 28, 29, 30, 34, 35, 36, 40, 42, 43, 50, 53, 58, 59, 61, 68, 79, 80, 81, 83, 84, 86, 87, 89, 90, 92, 96, 97, 100, 103

Minsky 128 Mohanty 128, 132 Montesinos 71 Montroll 131, 132 Mordell 17, 99, 104 Moser L. 27, 134 Moser W. 135 Mukhsinov 104 Mumford 39, 104 Myers 124

Narayana 128

AUTHOR INDEX Nash-Williams 124 Nelson 78 Neubiiser 93 Newton 55, 64 Niederreiter 138 Niemeier 65 Niggli 70 Niven 24 Nowak 105 Nozarzewska 8, 22, 24

Oda 12 Odlyzko 64, 88 Ohnari 47 Onsager 131 Osborn 53

Rosenfeld 128, 129

Roth 137 Rothchild 136 Rumsey 110 Rush 36, 42, 61 Ry9kov 56, 57, 59, 65, 73, 83, 84, 87, 89, 90, 91, 92, 93, 95, 117

Sah 2 Sallee 24 Santalo 27, 31, 111, 112, 114 Sas 48 Sawyer 18, 19, 20, 23, 25 Schaffer 23 Scherrer 12, 25, 26, 29 Schmidt 22, 27, 28, 35, 36, 42, 49, 51, 58

Pach 135 Papert 128 Parsons 136 Paschinger 74, 75 Patruno 12 Pedersen 67, 71 Penrose 77 Percus 130, 131, 132 Perron 80, 101 Petty 31 Pick 7, 8, 21 Pohst 88 Poincare 111, 112 Pollak 125 P61ya 70, 107, 126, 131 Preparata 74

Rabinowitz 9, 13 Rado 26, 135 Ramharter 18 Ramsey 136 Randol 105 Rankin 58, 116 Rearick 110 Reeve 7, 8 Reich 23 Reiman 136 Reinhardt 44, 68, 76 Reisner 31 Remmel 128 Reynolds 40 Reznick 8 Rhynsburger 74 Riele, to 88 Riordan 125 Roberts 125, 136 Rogers C. A. 30, 36, 40, 43, 44, 46, 48, 51, 56, 58, 59, 74 Rogers D. G. 128 Rohn 71

Schneider 2, 9 Schnorr 33 Schoenberg 12 Schoenflies 67, 71 Schiitte 55, 64 Schwarzenberger 67, 70 Scott G. D. 55 Scott P. 20, 21, 23, 24 Seeber 55, 67, 83, 84 Selling 83

Senechal 67 Shamos 74 Shapiro 128 Shechtman 78 Shephard 43, 68, 78, 79 Sidel'nikov 59 Siegel 14, 19, 26, 35, 36, 59 Sierpinski 136 Simonovits 136 Sklansky 128

Skubenko 103

Sloane 36, 43, 56, 59, 60, 61, 64

Smith 96 Smith F. 74 Snell 132 Sobolev 116, 117 Sohncke 67, 71 S6s 135, 137, 138 Speiser 107 Spencer 134, 136 Spitzer 132 Stanley 11, 12

Stein 49, 77, 80 Steiner 23 Steinhaus 24 Stephenson 130 Stewart 109 Stoer 30 Stogrin 79, 80, 93, 95 Stolarski 104

183

184

LATTICE POINTS

Struik 68 Subak 62 Suranyi 18, 99 Sved 128 Swierczkowski 39 Swinnerton-Dyer 50, 104 Szab6 49, 77 Szekeres 18, 99, 136 Szemer6di 134, 135 Sziruczek 62 Sziisz 18

Tammela 44, 84, 86, 87, 88 Tarski 2 Taylor 78 Teissier 12 Temesvari 61, 63 Temperley 130, 131 Thompson 60, 124 Thue 56 Tits 64 Tripiciano 24 Tsfasman 61 Tsuji 27 Turan 135, 136 Uhrin 19, 26

75, 79, 80, 90, 91, 92, 93, 94, 95 Voskresenskii 12 Waerden, van der 55, 64, 83, 135, 136 Wagon 2 Walfisz 104 Wang 77

Watson D. F. 74 Watson G. L. 57, 59 Wheathly 118 White 18 Whitworth 46 Wills 7, 8, 9, 11, 22, 23, 50, 108 Wilson 123 Winfee 110 Witt 64, 65 Witzenhausen 44 Witzgall 30 Wondratschek 93 Woods 26, 29, 30, 50, 103, 104

Yakovlev 62 Yates 50 Yeh 46 Zarankiewic 135 Zassenhaus 29, 40, 44, 48, 55, 58, 67, 71

Venkov 14, 68, 79, 81, 83 Veteinkin 56, 57 Vinberg 65 Voroneckii 47 Voronoi 14, 38, 39, 50, 64, 68, 73, 74,

Zimmermann 67, 71, 78 Zitomirskii 80 Ziv 136

Zolotarev 14, 50, 56, 57, 83, 90, 91 Zuckermann 24