Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Z0rich Series: Scuola Normale Superiore, Pisa Adviser: E. Vesentini
202 John Benedetto Scuola Normale Superiore, Pisa/Itatia University of Maryland, College Park, MD/USA
Harmonic Analysis on Totally Disconnected Sets
Springer-Verlag Berlin.Heidelberg. New York 1971
AMS
Subject Classifications (1970):
43 A 45, 4 6 F xx, 42 A
48, 42 A 72, 42 A 36
I S B N 3-540"05488-X Springer Vertag Berlin • H e i d e l b e r g - N e w Y o r k I S B N 0-387-05488-X Springer Verlag N e w Y o r k • H e i d e l b e r g • Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg I971. Library of Congress Catalog Card Number 77-163741. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
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PREFACE
These and
Scuola
totally gin
to
notes
Normale
properties
the
of
is
the
background
a spectral
and
problems
[43]
, the
[31,
Vol,
2]
a deeper
many
of the
cussion
can be
study
• I have have
change.
pact
abelian
groups
ings.
thoughts ject
or
matter.
Brownian
important
masterful
new
are
developed
as
the
include
some
basic
with
standard
setting
For
notes,
for
of
intermediate
of K a h a n e
are
and
treatise necessary
other
along
with
and v a r i o u s
field seven
Appendix
that
extensions
included
two
it b e g i n s
Many
on
are
which
hand a dis-
with that
a self-contained with
one
the
latter
topic
attractive Katznelson
although
admittedly
refreshing
or
become
a
are
more
or
locally
certain
one
for
problems
another
groups
approxi-
is n a t u r a l l y
since
com-
the
in-
Galois
disconnected.
self-explanatory are
really
not
of
meant
extraneous
elementary
definition
and m e a s -
and
of the
available
are
[22]
problems
totally
of text
be
whereas
is
appendices
techniques A is
might
on [ 0 , 2 w )
structure
sections
which,
on p r o f i n i t e
than
this
Edwards
of the
different
analysis
and p a r t i c u l a r l y
I thought
example,
simpler
as u s e f u l
motion;
work
on the
here,
analysis;
is on [ 0 , 2 w ) .
conceptually
further
also
motivation
general
[54]
functional
Fourier
these
effect,
generally;
are
book
Helson
problems.
of d e t a i l
are
synthesis
of m a n y
questions;
deal
and
and to be-
pioneering
and Ross'
various
if e v e r y
is the
monograph
arithmetic-synthesis techniques
known
classical
of H e w i t t
discuss
spectral
source
a good
infinite
I have
the
the
included
group.
There
With
of R.E.
in the
of
here.
and
problem
books
theorems
terested
and this
of M a r y l a n d
analysis,
is not
in the
more
important
it
found
Further,
less
arithmetic
sections
a stultifying
meaningful
mation
of
major
are
The
is to
in h a r m o n i c
is p r e s e n t l y
Kahane's
concurrently
presentations
could
relevant
as w e l l
read
[57]
area
prerequisites
theory,
set,
at U n i v e r s i t y
purpose
in p a r t i c u l a r ,
research.
, and
of the
between
given
The
arise
presented
fundamental
The ure
this
I have
Pisa.
which
synthesis
much
Salem
for
sets;
material
Salem
and
sets
relations
such
set
lectures
Superiore,
disconnected study
Kahane
are
sub-headas
after-
to the
treatment
a probability
sub-
of space,
VI
develops This
the W i e n e r
inequality
portant
in the
In A p p e n d i x existence basic
process,
and the t e c h n i q u e s of h a r m o n i c
analysis
B I give
a leisurely
account
of non
spectral
and p r e r e q u i s i t e s , only
harmonic
synthesis
analysis
(outside
subset
(in c l a s s i c a l
The r e m a i n i n g (hopefully)
problems
since
of the terms)
sections
motivation
for
the
further
is u s u a l l y
has
disconnected
sets.
t h e o r e m makes
the
up n o t a t i o n
standard,
it re-
of e l e m e n t a r y
a closed totally
§ 2 gives
a general
arithmetic-synthesis
fundamental
im-
t h e o r e m on the
a knowledge
group E / 2 ~ Z ) .
inequality.
have b e c o m e
§ 1 sums
B, E is always
to deal with
present
Malliavin's
meaningful.
of A p p e n d i x
motion
of M a l l i a v l n ' s
our n o t a t i o n
circle
Salem-Zygmund
on t o t a l l y
sets.
a s c a n n i n g by the r e a d e r who
disconnected proach
and,
the
from Brownian
study
arithmetic-synthesis
quires
and p r e s e n t s
results,
ap-
questions.
techniques,
and
study.
J.B. , 1970
This
w o r k was
che"
of Italy.
supported
in part by the
"Consiglio
Nazionale
delle
Ricer-
TABLE
from
CONTENTS
i
Preliminaries
i i
General
1 2
Synthesis,
1 3
Distribution
1 4
Properties
i 5
Approximate
2.
Pseudo-Measures
2.1
Structure
2.2
Measures
2.3
Representation
2.4
Measure
3.
A Characterization
3.1
Introduction
3.2
Hyperdistributions
3.3
Riemann's
3.4
Pseudo-Function
4.
Independent
Sets
4.1
Independent
and
4.2
Examples
4.3
Arithmetic
4.4
Groups
5.
Kronecker's
Theorem
and
Kronecker
5.1
Dirichlet's
Theorem
and
Statements
5.2
The
5.3
Infinite
5.4
Wik
6.
Independent
6.1
Introduction
Notation
and
Analysis
Definitions
Arithmetic,
and
and
for
Inte~rati0n
Fourier
Uniqueness
Theory
Analysis
..... i
..........
Sets .....................
Theory ............................................ of
Identities
of
Totally
Associated
Disconnected with
Uniqueness
Arithmetic
Progressions
of
..............................
and
Non-Helson
Independent
Sets
and
..................
and
Independent
Sets of
Theorem
....................... Kronecker's
Theorem
and
Estimates
of
.....................................
Multiplicity
............................
...............................................
80 87
93
. . . . . 93
Sets ....................................................
Sets
69
S e t s . . . . . . 73
Sets ............
Related
59
69
Sets . . . . . . . . . . . . . . . . . . .
Symmetric
47 54
of U-Sets ....................
Progressions
40
46
........................
Progressions
29
46
Sets ................................
Kronecker's
Kronecker
...............
Principle ..............................
of A r i t h m e t i c
by
26
Sets .........................
Pseudo-Measures
Kronecker
Generated
20
...................
of P s e u d o - M e a s u r e s
Characterization
and
S e t s . . . . . . . . 20
........................
Distributions
Sets
16
Sets . . . . . . . . . . . . . . . . . . . . . . . .
of U n i q u e n e s s
Localization
Proof
Order
Properties
and
Disconnected
Distributions
of F i r s t
to
Totally
6
ii
.......................................
Supportedby
i
8
2~(F) ..........................................
Theoretic
Bohr
Fourier
OF
.. 103 112 120
124 124
VIII
6.2
Salem's
Theorem
6.3
The
7.
Helson
7.1
Equivalent
Definitions
7.2
Arithmetic
Properties
of
Helson
7.3
Uniqueness
Properties
of
Helson
7.4
Further
8.
Concludin~
Remarks
..........................................
159
A°
The
Process
..........................................
161
A.I
Probability
A.2
165
172
Existence
............................................. of
Rudin
Sets
.................................
Sets .................................................
Functional
Wiener
of
Helson
Analysis
Spaces
and
Sets
.......................
141
Sets
........................
141
Sets
........................
149
for
152
Criteria
Expectation
of
Helson
Random
Sets .........
Variables
.......
A
3
A
4
Gaussian
A
5
The
A
6
Homogeneous
Hilbert
Wiener
of
Chaos
A 7
The
A 8
Equivalence
A 9
Wiener
A
IO
Salem-Zygmund
A
Ii
Continuity Process
..........................................
Space
Measure
Variables
.....................
182
Wiener
Process
and
Homogeneous
Chaos ......
.............................................. Inequality
....................................
Non-Differentiability
a.e.
of
the
Malliavin's
Theorem
B.I
Malliavin's
Idea ............................................
B.2
Construction
of
B.3
The
Example
B.4
Tensor
B.5
Varopoulos'
Bibliography
Index
........................................
a Non-Spectral
Proof
Function
184 188 192
Wiener
B.
Algebras
177
..........................................
....................................................
Schwartz
169
179
the
and
161
...........................................
Process of
Gaussian
134
141
Independent Events .......................................... -c2x 2 e ...................................................... Variables
124
.....................
........................................
199
207 207 214 223
............................................
229
..........................................
239
....................................................
...........................................................
251
260
i.
Preliminaries
i.i
General Z is
plex
fined
and
the
of
ring
of
Analysis
Definitions integers,
respectively.
series
to
Fourier
Notation
fields,
vergent
from
is
numbers
L~(Z)
is
We
set
thus,
with
group
under
the
with
the
Banach
the
usual
space
compact
addition
quotient mod
2w
circle
ting
on
depending
measure
are
space
norm
of
the of
{a
real
and
com-
absolutely }ELI(z)
n
con-
is
de-
on
F
group.
on
B
group.
[0,2w)
. If
~:F÷~
is is
E
The
note
r I,@ j[
functions
right by on
F
hand
side
i ~ 2w < m ,@>
.
where
the
, E
group of
course,
As
such
we
shall
technical
(one
of t h e
Haar
norm
the
Banach
space
~
dual
Lebesgue Banach
@eLI(F)
is
f I@I ;
1
in
~(~,)d-~
JO
is t h e of
F
abelian
of is
~
and
identified
identified or t h e and
of) f I ,4 jr
integral
with
other
whim.
multiples
F
the
set-
Lebesgue Haar we
meas-
define
r2w
usual
LI(F)
with
t
2~
the
one
convenience
integrable i
on
also
use
constant
compact
subgroup
operation
is,
II@II
L~(F)
a locally
discrete
of
is
is
. F
JF the
LI(z)
a closed
r
where
of
EI2wz
I @ ~
and
~
Banach the
Analysis
Zla n I ;
dual
on is
a mixture
on
m
2wZ
~
~
topology
addition,
multiplicative
ure
and
Theory
be
F
is
the
Integration
Fourier
E
where
I{a n )11 1 and
for
and
LI(z)
complex
and
JF
of
Ll(r)
,
integral; space
of
defined
we Haar
to b e
also
de-
integrable
The
Fourier
series
of
ceLl(F)
is
^
¢(Y)~
[
¢(n)e InY
neZ where
¢(n) is the
{¢(n)}
C(F) tions
on
A(F)
is the
the
norm
set
where
Banach
the
Banach
of
12w J0 [2~¢ ( Y ) e - l n Y d y
of F o u r i e r
is the
F
z
CsA(r)
space
norm
space
coefficents
of
CsC(F)
of a b s o l u t e l y
is d e f i n e d
Banaeh
measures A'(F)
space
with
, the If
dual
norm
space
of
C(F)
~
continuous
is d e f i n e d
to be
convergent
Fourier
func-
series
where
[I~(n)I
is 2 ~ ( F ) ,
; and the 1 of p s e u d o - m e a s u r e s we
- valued
to be
II II
{a }ELl(z)
0.
of c o m p l e x
I I¢II A
The
of
;
define
the
Banach with
F({a
n
space
space
of b o u n d e d
dual
norm
I I I IA,
}) ~ ¢
where
of
A(F)
Radon is
n
¢(y)
~ [a
e inY
n
Therefore
we have
the
diagram
LI(z) ~ A(r) co
L (Z) ~'
where
F'
phisms,
is the and
adjoint
map
of
F
{a
A'(r)
,
F
and
F'
~
VTsA'(F) }> n
are
isometric
isomor-
where we
F({a
}) ~ ¢ . We d e s i g n a t e n the representation
define
F'TeL
E
as
(Z)
{T(n):neZ}
; and
thus
2~ZT(n)a -n
with
Fourier
coefficents
^ T(n)
since,
1 '"~;
~
~
LP(r
of
-iny>
formally,
-in7>
2W
tions
¢
(n)a
, for Y
on
¢~LP(r)
i<_2<~
with
is
=
-n
the
defined
y
,[a_ne
, is the property
to
Banach that
=
space
of
I¢IPsLI(F)
complex-valued and
g2~
Lq(r)
is t h e
ly, {a
n
{a
LP(z) :neZ}
, for with
}~LP(z)
Banach
is
space
l~A0<~
the
dual
, is
property
defined
the
norm
be
1 f i~(~)tpd~) ( i r t~t p) 1/p = (-77j0
l~llp
where
func-
the
of
LP(F
Banach
that
, where space
of
i/p
! + ~ = 1 . SimilarP q complex sequences
{a P : n s Z } e L I ( z ) n
and
where
the
norm
of
to be
n
l]{an}ll p Lq(~)
is t h e It
is
Banach clear
space
dual
~ of
([lanlP> 1/p LP(z)
where
1 1 -- + -- = 1 P q
that
and
A(r)C_ c(r) _CL~(r)C . . . ~ 2 ( r ) ~ _ Ll(r)q~(r)~_A'(r) the
inclusions
are
proper
and
indicate
not
only
set
theoretic
inclusion
but
also
that
are
norm
decreasing.
are
isometrics
norm of
the
to
respective
The
, whereas
decreasing. F'
natural
Also,
imbeddings
a subspace
imbeddings
c(r)~L~(F)
the
imbeddings
the
remaing
Plancherel
L2(F) is an i s o m e t r i c 2 L topologies.
It is c o n v e n i e n t
(from
also
to
theorem
into
and are says
isomorphism
the
Ll(r)g-~2~(r)
generally that
onto
space)
the
L2(Z)
strictly
restriction , with
the
define
oo
c (Z)
-
{fsL
llm
(Z)
0
f(n)
=
o}
,
T(n)
=
O}
,
lim
T(n)
=
O}
are
Banach
Inl+
~r(o(r)
~
{Te~?(r)
A'
z
{TeA'(r)
A(r)
and
(r)
lim
InI÷ :
0
As
is e a s y
spective maximal L=(Z) (on as
Z) such
to
see,
norms) ideal
under
space
is a B a n a c h , we
define
A'(F)
is o r d i n a r y
is
pointwise
of b o t h algebra the
multiplication
A(F) under
canonical
a Banach
convolution
More
C(F)
and
pointwise
algebra,
if
and
their
and
F . Similarly,
via
F'
canonical
measures
¢,?eA(F)
¢@~
is
multiplication
this
Radon
(with
of f u n c t i o n s ,
multiplication,
of b o u n d e d
specifically,
C(F)
algebras
when
the since
of f u n c t i o n s -i
, on
A'(r)
multiplication restricted
to
we h a v e
[c e inY n
where
c
= n
It
e a s y to
check
that
[ ~(p)~(q)= p+q=n T¢
[~(n-k)~(k) k
is w e l l - d e f i n e d
for
TeA'(F),@sA(F)
¥ ~A(r)
=
=
re-
, and
;
Finally, A'(F)
the
convolution
defined
T@ S,T,SsA'(F)
, is
a well-defined
element
of
as
^
where
S,¢>
= 2w
[ T(n)S(n)¢(-n)
,
¢sA(F)
For
E ~
F
closed
k(E)
we
set
--- { ¢ s A ( r )
: ¥¥sE,
~(~)
--O}
,
and
j(E)
Clearly, Banach
k(E)
- {¢ek(E)
is
: ¢ -= 0
a closed
ideal
ued
we
define
continuous
A(E)
and
where
each
= 0 T~E;
C(E)
F
to be
functions
~(E)
Closed
supp
in
a neighborhood
A(F)
and we
of
E}
consider
.
the
quotient
algebra
A(E)
also
on
is
F~ is
. A'(E)
F
E
is the
denotes
sup
space
the =
Banach is
of
the
space
;
dual
support
and with
A'(E)
norm
A~(E)
Banach
closed
obviously,
the
on
the
-: A ( F ) / k ( E )
algebra
the
Banach
of
C(E)
TEA'(F)(supp
property
of t h o s e
(A(F)/j(E))',
Ya
TEA'(F) where
complex-val-
space
T)
that
of
dual
if and for
A(F)/j(E)
F = ~
of
F
¥¢aj(F which has
the
),
quotient
topology.
Auspiciously
~2~(E)
A~(E)
a n & it is
clear
enough,
{Tc~(F)
=
= {TeA'(E)
: supp
T~__ E} ,
: ¥¢ek(E
,
supp
set
A' E) O
Synthesis,
Arithmetic,
Generally connected
subset
synthesis)
if
of S - s e t
Helson
set
is a set of
E
if
of
T~
E}
is,
of
is
and
~o(E)
~ {TC2~O(F)
:
or,
= C(E)
E
set,
set
if
A~(E)
for
all
closed
equivalently,
if
meaningful
resolution E
is a c l o s e d ,
(Wiener
equivalently,
, or,
and
Sets
notes,
is an S - s e t
course,
spectral; true
Uniqueness
these
F . E
of s p e c t r a l
l E
throughout
A(E)
(a set w i t h o u t
such
: supp
and
j iE) = k(E)
notion
set
~ {TeA'(F) o
E)
T CE}
1.2
E
;
that
~'~ (E)~A'(E)~A'
Finally,
= 0}
(Malliavin
is a set
pseudo-measures)
if
~(E)
of
totally
spectral
= A'(E) sets.
. The E
set
if
set)
if e v e r y
closed
of s t r o n g
spectral
resolution
~(E)
¥¢eC(E),I¢I
that
supI¢(y)-elnYI<e yeE
is a
= As(E)
= A'(E)
. E
is W i k
A'
is a K r o n e c k e r
dis-
;
5 1
, and
Ye>O,BneZ
sub-
if
and
E
is
a Dirichlet
set
if
Ve>O ~neZ
such
that
sup ll-elnTl
These
two
notions
of
Kronecker
if
for
any
are
and
motivated
Dirichlet,
finite
by
the
classical
respectively.
distinct
collection
We
approximation
say
that
yl,...,TneE
E
is
theorems independent
,
n
(l.l)
l[n'YJJ
implies finite
= 0
distinct
In for
njTj
similar
closed E
for
j
fashion
we
of
~
a U-set
; and
define
(Cantor
N÷~
is
all
ye
a U-set
E if
, we and
have only
E
is
str0n~
(i.I)
implies
independence
and
set
of u n i q u e n e s s )
~
c e
~TeA'
(E),T#O
c if
n
rnl <_N
= 0
for
A'(E)
=
all
= {0}
a strong
U-set
if
for
any
all
independence
n
, which
.
~TcA'(E)
......
l i m IT(n) [ Inl+~
^
=
if,
whenever
0
,
i'i'm I T ( n ) l > o
is
strong
for
iny
In I-*oo
E
n.J = 0
if
n
O
if
independent
.
lim
for
O,njcZ
yl , • ..Tn eE,
collection
subsets is
all
=
sup n
IT(n) I
We'll
see
is the
in
case
§ 3 that if
and
E
only
and
E
is
a U-set
these
uniqueness
ty
it
if
is not
1.3
in t h e
let
wide we
a U-set,
Distribution We
ly
notions
is not
a U-set
in t h e
that
E
it
is
~o(E)
a set
is
an M - s e t
of
strict
functions
, be and
the for
vector
space
O
we
k
II*II k
wise
As
such
each
multiplication
~
ck(r)
Corresponding (set
to
of m u l t i p l i c i -
multiplicity
of
if
it
k-times
i
¢(j)
the
continuous-
norm
II~
j=O
ck(F),O
functions
of
define
X -~711
~
C
ck(r)
{0}
Theory
ll*ll(k ) ~
on
=
sense.
ck(F),O~k~ ~
differentiable
if
say
and
wide
sense
, is
a Banach
algebra
C°(F) = c ( r )
and
under
Clearly
point-
C~(F) =
and we d e f i n e
l
j=o on
C (r)
that
C
(r)
functions; Banach D
(F)
x C (r) is also
algebra. ~ D(F)
of k - t h imbedding
order
. p
is
a metric
a Fr@chet the We
algebra
algebra let
is the
C
on
distributions is
of
C
under
(F)
(r)
and
pointwise
can
Dk(r),O~k~
space
cn(F)f.~cm(r)
2j l+ll~-~llIj)
not
, be
be
the
distributions,
, and
D°(r)
continuous
cn(r)
=
and
cm(r)
,
is
easy
to
check
multiplication
normed
strong D
=~(F)
it
so
as
dual
of
k(r),o<__k<~ For
all
of
to b e c o m e
a
ck(r)
, is the O<_m,n~
space the
whenever
m
. Thus
(1.2)
we
have
~(r)C_
the
imbeddings
Dm(r) ~ n n ( r ) C D ( r ) ,
m<_n .
Further
~.~Dm(F)
=
D(F)
.
m
The topology
strong
when
sO
dual
dealing
whenever For
any
topology
with
on
sequential
@~O,¢EC=(F)
open
set
U C
~
we
supp
TED(F) that
¢
is
if
zero
the
fined
notion
a notion
smallest
of
A'(F) CDI(F) i.I
mark
is
, and,
after
Theorem have
Theorem @(k)
= 0
Remark and,
i.i on
1. for
It
on
TED(r)
and
:
F
for
of
of w h i c h
It
is
then
T
(supp and (of
definition
of
distributional
is
prove supp
U is
easy
in
We m a k e
see,
Dm(F) § 2
T,TsA'(F)
one.
U
there
to
for
show
a well-de-
Similarly
support)
, we'll
to
on
T = 0
T). it
~ = 0
standard
T = 0
of w h i c h
Dm(F)
. Also
the the
= 0
open}
of
¢ ~ _ u}
outside
definition
D(F)
thus,
set
outside
support
the
supp
is
using is
that , given
a related
in re-
1.2. following
important
result
concerning
the
support
:
Let
T~Dm(F)
supp
T
is
each > = 0
m
set
: U
elements
not
for
0~k~m, .
and O
necessarily (k)
¢ n
lim
{U
the
with
the
distributions
(r)
(U),
V¢EC
, that
that
{¢cc
if
for
(1.2)
if
weak
define
closed
is
consistent
We of
F
Also,
the
TaM(F)
smallest
zero
support
with
~
closed
and
imbedding
consistent
§
U
_r
is
with
co
the
on
TED(F)
Thus,
the
is
coincides
convergence.
, then
oo
c (u)
where
D(F)
n + 0
let
@Ecm(F)
. Then
true
have
for
uniformly
= 0
TEDm(F) on
the
property
that
.
that
supp
T
if
{¢
n then
} C Cm(F) --
10
2. TeA'(F) which the
The
and
¢¢A(F),
problem
= 0 to
The
analogue ~=0
, where
determine
major
tion
can
he
tive
of
TeDm(F)
where
of
Theorem on
supp
supp
T~
S-sets
feature
it
be
generally
determine
and
~=0
on
those E
hold
for
sets
E
for
, is p r e c i s e l y
§ 1.2) theory
a well-defined
T'
: cm+l(F)
is t h a t
way.
In
any
fact,
distribu-
the
deriva-
is
, and
@ecm+l(F)
can
not
. To
in
T'
It
is
shown
that
÷ ~
,
is w e l l - d e f i n e d
easy
supp
Also
E
does
of d i s t r i b u t i o n
differentiated
T'eDm+I(F)
all
T
(e.g.
for
I.i
if
~ -
to
check
T' ~
¢'>
,
that
supp
TED(F)
by
for
TeDm(F)
,
T
and
T'eLI(F)
then
~Y T
= ° +
i
Y
an
absolutely
continuous
representation Now,
theorem for
each
function; is
an
just the
as we
did
for
corresponding
we'll
extension
TED(F)
^ T(n)
we
see
of
define
1 ~ --
2w
pseudo-measures; Fourier
'~'(~,)d~,
,
Jo
an
this
§ 1.4
that
the
result.
Fourier
coefficents
-inT> ,
e
thus,
,
for
each
TcD(F)
, we
series,
^
T ~ ~ T ( n ) e InY Y
The
following
neat
representation
Riesz
theorem
is
due
to
L.
Schwartz.
have
il
Theorem
1.2
a.
If
TsDm(F)
then
T(n)
=
O(Inlm),lnl
+
~
,
and ^
SNT
converges
in
D(F)
b.
to
If
K
[ T(n)e inY InI~ N
T .
{c
n
}~ ~ --
satisfies
:
e
O(lnlm),lnl
+oo
n
then
[ c e Inl~ N n
iny
converges
in
D(F)
to
TeD(F)
with the p r o p e r t i e s
^
that
T(n)
= c
for
n
Obviously thing we'll note
that
have
if
all
n
Theorem more
TeD(F)
and
TeDm+2(F)
1.2 gives
to speak
us an i m b e d d i n g
about
A'(F)C-~D(F)
in a d i f f e r e n t
has the p r o p e r t y
way
in
, some-
§ 2 . Also
that
sup N
TeL~(F)
then
and
IITII~ ~ sup
lls,Tti=
•
N
l.h
Properties
of
le~(F) ¢eC(F)
is p o s i t i v e ,
is n o n - n e g a t i v e
~O
for
solutely
continuous
CeL~(F)
(that
all
~(r)
TeD(F)
is,
on
and with
~
that
is,
F (recall
the
CEC~(F),¢~O respect
k>_O , if result
, then
to p o s i t i v e
is i n t e g r a b l e
with
CeC(F)
;F
~0 from
§ 1.3
Te2~(F)). Xg~(F)
respect
to
whenever that
~s~(F)
if there X )
if is ab-
is
such that
for
12
This
is
equivalent
<~l,@(l-y)> and
{b
to
saying
. ~s~(F)
}~__ ~
such
lim [ITT~-~ll. = 0 , where y -~ 0 1 discontinuous if t h e r e is { Y n : n
is
U,¢> Y -- i , . . . } C r
that
n
co
i
(1.3)
~
Now,
assume
=
¢
[ b n 1
is
~
y
on
C(F),~Ib
a Borel
measurable
I
n
n
function
{¢ } ~_ c(r)
and
n
has
the
properties:
sup n,y
I¢n(y) I <M
and
lim
¢ (y)
=
@(y),
for
each
y
n n + c o
Then
lim
<~,¢
>
exists
and
we
define
<~,¢>
as
n
(1.4)
lim
>
=
.
n n
(1.4)
follows
from
the
formulations
of t h i s
distribution
theory
we
Riesz
latter and
give
the
following
Theorem
1.3
Let
representation result,
the
fact
elegant
and,
that
there
considering
we
statement
theorem;
are of
our
really
are
several
outline
working
on
of
the
line,
it:
^
of bounded tion
on
variation
F
such
Because on
the
TeD(F),T'(O)=O.
Borel
(that
that of
algebra
is,
f = T
(1.4), ~
T's~(r) there a.e.
each of
XB
course from
one
the
is (not
usual ~2~(F)
the us
F
; in
measure is
~
characteristic right
now)
is
fact,
if T is f
to
a function
of b o u n d e d
Lebesgue
a well-defined
varia-
measure). set
function
YBe
<~ , XB>
of
establish
point if
only
respect
function
could
theoretic
continuous
with
and
a function
~2~(F)
~(B)
where
is
if
of
B
. At
this
integration
stage
of
theory
for
view.
YyEF,~({y})
= O
. Also,
for
each
13
~e~(F)
and
non-negative
There
is
ement
of
With
=
a natural
the
~(F)
;
notion
of
¢6C(F)
sup
I
uniquely I~I
total
variation
define
:
defined
is t h e
total
, we
and
,ec(r)
]@l!+)
extension
of
variation
measure
we
see
that
IPl
(1.3)
as
a positive
associated holds
if
el-
with
and
~.
only
if
I.I(4<~
and,
generally,
Related
to
~e~(F)
this
is
for
any
and
the
n
,
:
n
~e~(F)
=
, {yeF
definition
continuous
if
and
of
only
<1.1,
lim
X
XeT~(F) there
is
The space,
sums
sentation
Theorem
that
following up
and
1.4
such
the
X(B)
>
¥~e~(F
=
;
{y}>O} p
is
countable
we h a v e
that
,
o ,
function is
= 0
which
of
singular
and
Ip[(~
is t r u e
properties
Radon-Nikodym
a.
0
e,y
result,
major
vTer
x
is the c h a r a c t e r i s t i c £,y is p o s i t i v e , then ~s~(F) Be~
: I~I
=
continuous
if
e + 0 where
.... })
I
which
on
[y-e,
y+s]
with
respect
B) any
center
. Also, to
if
X
if
= 0 locally
about
the
compact Riesz
T
2
repre-
theorems.
,
I1.ti
= t.I(r) I
b. r
I I~IdI.I
!
II~ll~II.ll
JF c~
that
~.~(r
3 aI.l
measurable
function
~,1~1
~ 1,
such
¥@sC(F) f )F d.
such
(1.5)
¥
positive
~s~(F)
and
that a
s
¥~s ~ I ( F ) , 3
~a'Ps s2~(F)
'
i4
where
~
is
a
far w i t h
absolutely
respect
singular,
troduce
with
respect
to
k , ~
is
singu-
s
Because ous,
continuous
the
to
l , and the
of t h e
discontinuous,
following
~d(F)
previous
decomposition theorem
and
(1.5)
and the
absolutely
is u n i q u e .
importance
continuous
of
continu-
measures,
we
in-
notation:
~ {~c2~(F)
: ~ is d i s c o n t i n u o u s }
7~c(F ) z {~s~(F)
: ~ is
continuous}
,
,
(1.6) ~s(F)
ff { ~ e ~ c ( F ) :
~ is
~f~a(F)
~ {We~c(F):
~ is a b s o l u t e l y
The well
following
structural
It is a l s o we h a v e
not
~(G)
= [b n ~
respect
to m}
continuous
properties
of the
with
above
a.
~d(F)
is
a closed
subalgebra
b.
~c(r)
is
a closed
ideal
in
~7(r)
c.
~a(F)
is a c l o s e d
ideal
in
~(F)
d.
~
is
subspace
hard =
s
(r)
to
show
~d(G)
, ~ = ~c n 81 Yn
above
somehow the
with
,
respect
spaces
to m}.
are
known.
Theorem 1 . 5
The
singular
leads
elements
following
Theorem
that
for
G
if and o n l y
if
a ~d(F)
then
a locally G
¥~e~(F)
~
to
ask
from various
extension
1.6
one
of T h e o r e m
(possibly
compact
is d i s c r e t e . ~* ~ =
if a g i v e n of the
~(F)
~(r)
of
n
theorem
into
a closed
of
U
spaces
[ n,k
b
abelian
Further
group
, if
ck 6yn+l k n
can be (1.6).
decomposed In f a c t ,
we h a v e
1.4d:
F = F) t h e r e
is a u n i q u e
decomposition
~a + us + ~a' ~ d S ~ d (r)' ~s ~)ns(r), ~ a ~ T a (r ) '
15
where
and
supp
~d~_
tl~IlI
thus
This sitions ent
we
only then
tails
on
85,
pp.
pp.
tive
to
= II h l II
theorem
tinuous
¢
at m o s t
has
108
As
shall
and
the
only
see,
is
~ 1.2
measures. following
¢
is
many
for
all
the
of
classical
Theorem
a function
jump
various
1.3
decompo. At
of b o u n d e d
discontinuities. points
12;
32,
pp.
334
of
the
key
problems
pres-
of view
- 341;
57,
For
we
PP.
varide-
refer
to
34 - 45;
depend
These result
on
measures due
to
we
a knowledge are
of
rarely
Wiener:
encounter certain
tractable
Ws~(F)
is
relacon-
although
continuous
if
~e~c(F)
closed,
and
to
because
if
countably
several
in
N
if
that
Chapter
lim
Also,
,
] .
questions
singular
if
result
22,
W a C-- F
related
functions
decompositions
- 266;
we
course
variation
104-
do h a v e
of
the
255
we
is
, supp
+ II~ s II i + II~ a II i
recall
these
the
Ps C-- F
, supp
of b o u n d e d
ation
[i,
F
+
N Z
i 2N+I ~
and
^ 2 Iz(n)I
:
0
.
-N
X z {
: C e C ( F ) , I ¢ I ~ i}
, where
F ~
F
then
I
(1.7)
follows
rems,
and,
from
using
a judicious
a similar
II~II
=
use
of t h e
technique,
sup
{I<~
¢>I
we
Radon-Nikodym also
and
Lusin
theo-
have
: ¢~c(;),I~I
~
l}
1
for
each
~e~(F) The
concerns
the
final
(recall major
Theorem
result
approximation
of
1.4c).
concerning a given
measures
pE~(F)
by
which ~
we
with
mention supp
now
16
finite. Banach
We b e g i n spaces
is w e a k ~
dual
of a B a n a c h {T
skating
(and
ball
sequence
by
Helly
in p a r t i c u l a r
compact,
}
the
space
and has
in s u c h
or A l a o g l u
for ~ ( r )
so e v e r y a weak @
a space
norm
and
for
duals
A'(F)):
the
closed
bounded
convergent
is w e a k ~
theorem
infinite
subsystem
convergent
to
set
in
of unit the
. Further, T
if a n d
a
only
n
if
{T
} n given
the
separable that
{~
ly if
n
n
Banach with
bounded space.
the
and
÷ f,
for
trigonometric
}~_ ~(F)
for
{
Thus,
(resp.
{T
{JJ~n][l } (resp.,
converges f
is n o r m
each
polynomials
} ~ A'(F)) --
JITnJlA,)
meZ
f , f£LP(F), n
n
Also
lim n
->"
in the
inf
J I~
co
(resp.,
lim
is w e a k
that
inf
n
IJ
if
each
subset
~n (m)
~n
@
(resp. , A ( r ) )
~ convergent
and
weak @
1
for
as a d e n s e
is b o u n d e d
note
1 < p < ~) ---
(1.8)
,@>} is c o n v e r g e n t n example, since C(F)
from is
we h a v e if and on-
(resp. , Tn (m))
n topology
then
> J ]VJ ]
I Ifnl Ip ~
I Ifl Ip)
n + ~
With
an e s t i m a t e
Theorem such
1.7
that
similar
For supp
~
all C_
to
(1.8)
~e~(F) supp
and
there
V,
supp
1.5
Approximate Let
1
Identities
@eLl(r)
and
si~¢(y)
define % ( n ) e inY
-
In1<_N
argument
system
II~ ~ II 1
~ topology
and
1
- Banach
is a d i r e c t e d
W a finite,
in w e a k
a
an H a h n
,
{~
~ ll~ll 1
we h a v e
}~_2~(r) ,
17 N
~(~)
1 ~ s @(y N+I ~ n
The Dirichlet kernel is iny e
sin(N+~)7 sin y/2
=
Inl!N
t 0 mod 27 ;
and the Fej6r kernel is I
N l
][ D (~,)
~-I 0
n
y # o
"
F (y) N
N+I, y = 0 It is easy to check that
FN(Y) :
Straightforward
[ (1- n~)N+leiny - N+ll (sin sin y/2 In I~N
calculations
also yield
ON@(7) = F N ~ ¢(Y)
Recall that a sequence of functions
{K }
is an approximate ide,ntity
n
l
for
LI(p,],, if
supl IK I I <=, lim n I n
r
2w ]Kn = i , and
f l~ Y)IdY lim I n J i~ ~ n 6 <_._1Y
As is well known, if
{K }
= 0
is an approximate identity and
n
n
¥6s(0,w
n
1
@sLI(F) then
18
One
of F e j g r ' s
tity
for
important
LI(F)
functions
results
. Another
A (¥) £
was
that
approximate
{F
n
identiy
}
is an a p p r o x i m a t e
for
LI(F)
is
iden-
g i v e n by t h e
:
/
2~
-C
In d e a l i n g kernel
with
A(£)
(as o p p o s e d
to
LI(F))
the
de la V a l l 6 e
Poussin
,
v
(~)
~
1 2~
e
(~eA2
(~)
- ~A
(~)] e
is u s e f u l . We n o t e
that
generally
if
@cA(F)
and
¢(n)hO
for
all
neZ
then
I I$I IA
Using
this
fact
we
see t h a t
=
[ ~{n)
I IA e
II
IIvll
One
of the
the
corollary
S
sets),
key
properties that
is t h a t
A(F) if
of the has
(1.9)
%(0)
= 2F e
A
and
A ~ 3
de la V a l l 6 e
no p r i m a r y
~sA(F)
=
and
~(0)
lim
flV~li
Poussin
ideals = 0
A
kernel,
(and h e n c e
and with
points
are
then
=
o .
e+O
Since
¢-(l-V
identity
(not
)¢ = V ¢ , (1.9) in the
Generally,
sense
we h a v e
tells
of L I ( F ) ) the
us t h a t for
following
l-V
{~eA(F) important
is an a p p r o x i m a t e : ¢(0)
= O}
and n o n - t r i v i a l
re-
19
sult
concerning
Theorem
1.8
such that
y , then
approximate
identities
a.
If
K~r
@ = I
on
K
b.
If
CeA(F),
3 SeA(F)
is c o m p a c t
and
@ = 0
e>0,
and
on
ycF
A(F)
K~_V
C V
, open,
then
3
¢EA(r)
.
, and
V
is a n e i g h b o r h o o d
of
such that
~ = 0
and t h e r e
in
is a n e i g h b o r h o o d
W
on
of
- ~(y)
~V
y
such that
= ~
on
W
Notes §l.1
We refer
to
§1.2
We refer to Appendices
§1.3
[ 90, 53;
Chapter 22,
I, If,
We r e f e r to
[22,
r e g a r d to T h e o r e m for e x a m p l e ,
f'=6
Chapter
III;
o
-i
15.7;
43,
90, C h a p t e r s
Chapter 1.3
i ]
12;
and
57,
5, 7 ]
Chapter
8 cT~(r) Y
where
Chapters
f(y)=~
We r e f e r to
[ ii;
22,
Chapter
§ i.~
We refer
[ 22,
43,
85,
to
90 ]
12; .
98 ]
,¢>-¢(y))
With note that,
Y ' 1 iny e in
bounded variation. §1.4
• 1.7;
(<8
5, 9, ii and
31;
32 ]
is a f u n c t i o n
of
2.
Pseudo-Measures
2.1
Structure Let
nected;
in
least
one
tains
an
us
Totall[
Disconnected
first
that
fact,
note
if
component
0
by
of T o t a l l y
interval
measure
Supported
We
if
Disconnected
Sets
m(E)
= 0
totally
then
E
were
not
C
with
more
than
one
. Of
course,
we
have
m(E)>0
show
this
with
a
Sets
E
is t o t a l l y
disconnected point
standard
there
and
E
since
is
Cantor
not set
discon-
would C
be
then
necessarily example
in
at conof a
minute.
Remark
i.
nected;
Generally,
for 2.
for
all
hood out
Recall
if
if it
3. only
if
Example symmetric {~k
and
of
that
only
example,
xeX
U
(Lebesgue)
x
a line
that for
such
a
T2
a
segment
T2
every that
in
0 E
2
topological neighborhood
U~
space
measure
X
N is
and
U
compact
sets is
are
space
X
is
N
of
x
there
is
open
then
it
and
is
disconnected
(e.g.[35]
).
It
is
to
xCr
is
totally
of
Cantor
that
discon-
O-dimensional is
if
a neighbor-
closed.
It
O-dimensional
totally
see
totally
connected.
is
easy
not
turns if
disconnected
if
and
and
{~. X = F
2.1
We
set;
: ~k e ( O ' ~ ) }
now we
define
shall be
a certain
frequently
given.
type
use
such
sets
Set
E1
EIHE l
~
iV
2
where
~l
_:
[o,2~
i
12
:-
] I
[2~<1 - ~l ), 27]
Set 2
2
2
2
2
~ ~.lU~2 UE3U~, 4
for
set,
viz.,
examples.
a perfect Let
21
2 E2~EI E2 2~.EI El, 2 -- i' 3' E4 -- 2'
where
and
2 E1
[ o , 2 ~ 1 ~ 2]
=-
2
~3
~
[2~(i
- ~l),2~(1 - ~l ) + 2~1~ 2]
2
E~
We
continue
this
~
process
[2~(1 - ~1~2),27]
n o t i n g that
for each
k
2k
k
_J
j=l
k m(E.)j = 2W~l~ 2 . . . ~ k
and
. The p e r f e c t
symmetric
set
determined
by
{~k }
is E
E
is o b v i o u s l y
contains to
0
thus has
no
as E
closed.
interval k + ~
Also,
since
of l e n g t h
is t o t a l l y
greater
Finally,
=
F
Thus,
Let
FC
interval
--
for
D
r
be
-~
closed
of l e n g t h
=
o
k
for all
than
k
2~i...~ k contains
and b e c a u s e (which no
Ek
converges
intervals;
easy to check that
E
it is clear that
k
lim k
is an open
E~E
~,<~) we have that E J d i s c o n n e c t e d . It is e q u a l l y
m(E)
2.2
k
since
no i s o l a t e d p o i n t s .
Example
- ~E
e. J
(2~)2 ~l...~k . oo
and let
C
F =01.; o D
where
each
I. J
Then
.
.
.
.
22 n
Fn
we
-
~(yIj)
have
F
where
F
is
n
decreasing,
and
if
m(F
f% | IF
-=
m(F)
)
=
2.3
Let
E
be
= 0
[
n
Example
,
n
we h a v e
e.
j >__n+l
a perfect
J '
symmetric
set
determined
by
{~k } ,
where
~k 2k-1 ~k - 2 k + l
We h a v e
and
1 2
-
so
=
I.(2 2
-
2k~l...~ k
1.(2
2
-
1 2k+l
1)(2
3
22.2
i)<2
3
-
I).,.(2
k
-
-
1)...(2
k-I
2k+l
l)
k H
=
2.22.23'...'2 k
Hence,
since
infinite
~
products
positive
limit, Now,
of
E
E
~FCC~(F)
O~$F~l
gin,
p.
E
14]
~
as a t o p o l o g i c a l
that
~F ~ 1
of
we
with
support
characteristic
the
) that
be the space.
sets.
on
i)
:
1 (i
-
.)
j=l
2J
classical
properties
H(I
-
set
of c o m p a c t
For
~
is
each
I ) 2J
of
converges
E - F SF
to a
of
F
a subset
enough
open
subsets
for t h e
topolo-
it is e a s y
as a c o n v o l u t i o n
in a s m a l l function
(as
a basis
Fe~
a neighborhood
of
can w r i t e
by
-
.
let
on a n e i g h b o r h o o d
identity,
we h a v e
is, m ( E ) > O
an a l g e b r a
; in fact,
and the
(eg. [ 1 0 3 ,
a given
is
such
~F = 0
imate
for
and
converges
that
considered
gy on
F),
1 2j
k
find
(as a s u b s e t of of a
neighborhood
of an a p p r o p r i a t e
to
F ),
of
and
Ca(F) of t h e
neighborhood
approxorlof F.
23
We h a v e intervals;
@F
Also,
let
=
E C
can be ~
Proposition
formed be
o
2.1
a.
the
a.
Let
neighborhood
Let
I. J
have
the
just
@
this
=
in
Now,
let
and
{F.} 3
elements
on o n l y
in
~
such
many
I. . J
by
is c o u n t a b l e .
Borel
algebra.
~eC
(F)
such
that
O~@<_l,@ = 1
F , and
@ = 0
on a n e i g h b o r h o o d
O<@
for
such
that
I. J
F.e J F is
~
in
some
points
of
on
a
E - F
of
we h a v e
on o n l y
F , and
[ ~F. J
two
@F
'
Ik,~F " = 1 J
= 0
on
on a n e i g h b o r h o o d
a neighborhood
of
F. J
E
• J
an o p e n
cover
of
F
so t h a t
F
compact
implies
we
can k
F
by
F
,...,Fnk
; and,
consequently,
nl This
e.j ~ y.-k.jJ
finitely
and t a k e
O<@F. < 1 J
cover
b.
and
open
I. ; and adJ I. so t h a t @ = 0 on an o p e n i n t e r v a l of I. J J it r e t a i n s all its o t h e r p r o p e r t i e s .
on
each
where
Thus
~
of d i s j o i n t
[0,2~)
be t h e
generated
=
in
~
O<~F
is t h e
property
so t h a t
for
of
a family
l.o ~ ( k j , y j ) ~ _
~
?
is
set
o-algebra
~
{I.} J
r
but Doing
F
where
for w h i c h
Fs ~
of
in
we
[0,2w)).
b.
Proof.
~I. o J
notationally,
(considering that
~E
follows
since
~
is
a countable
set
~F = Z@F 1 n. J
basis. q.e.d.
A finite the supp
F. J T CE
decomposition
are m u t u a l l y , and
Fe~
disjoint , we
of E and
is
{F.E ~ J E = [JF. J
define
T(F)
-=
,
: j = l,...,n}
such
Now,
,
if
TeD(F)
that
24
oo
~FeC
(F)
i,...,
as b e f o r e .
n}
position
such of
A finite n
that
E,{F.)
T =
decomposition
Z T. j=l J
and
, satisfying
for w h i c h
supp
T. ~
J
Proposition defined, nique
2.2
and
for
finite
J
Let
TeD(F),
every
finite
decomposition
supp
of T
T
there
T.eD(F),
j =
J
is a f i n i t e
decom-
j
For
T~E
given
{T.: J J
F.
--
decomposition
of
is
all
Fe~,T(F)
{F.} of J T .j ~ T ~ F . J
by
E
is w e l l -
there
is
a u-
a neighborhood
of
%
Proof.
To
show
T(F)
well-defined.
Let
~F
= 1
on
%
F
an d
%
@F = 0
on
a
neighborhood
neighborhood
Given
{F.}
; to
of
show
{S.} J
be
<S.,¢>
a
where
<S.,¢@.> J
=
J
thus
@.cC
F
is the
unique of
T
[ <Sk,¢¢.>
(F),@.
on
then
CF- ~F = 0 = 0
finite and
=
J
~ i
;
k
J
on
E -
J decomposition
finite
=
J
and
{T~F .}
J
Let
E
of
decomposition CeC
(F)
=
J
of
F.
J
a neighborhood
a
.
let
a neighborhood
on
.
• J
of T
We h a v e
,¢>
, and
¢.
J
of
~ 0
J
E - F. J q.e.d.
For
given
T D(F),
supp
T ~E
, we
define
the
: Fe~}
,
following
two
semi-
norms:
(2.1)
IIT]I v
(2.2)
llTIIv,
Clearly
~ sup
I ITI Iv<~
{ [: ] T (_F j ) I
if and
~ then
the
topologies
It is a l s o II~II
{T on
immediate
; in fact, I
:
for
only
sup
{F.} J if
: TeD(F), ~
that
{IT(F)I
is
I ITI Iv,<~
supp
determined ~(E)C
a finite
a finite
~
; and
T~E,IITI by
(2.1)
and
decomposition
if
decomposition
if we
IV<~} and
{F.} J
set
,
(2.2)
~e~(E) of
of E}.
are then E
equivalent. II~Ilv, -<
we h a v e
25
is.
j
i
is.
where
has
J It finitely give
is
the
property
if
TcD(F),
I l@I I~ ~ 1
J
important
additive
conditions
set (in
to n o t e function
§ 2.2
on
and
~
; we
§ 2.4)
supp
T~E
exploit
that
, then
this
various
T
is
observation
TaA'(F)
are,
a to
in
fact,
measures. Further,
co
we
have
need
to
consider
the
following
space;
co
CE
~ {@~C
(F)
: ~
a neighborhood
N
of
E such
that
card
¢(N@)<~}.
o~
Proposition
Pro .....o...f .
2.3
CE
=
C~C (CF--) E
Also,
for
Thus,
in
First,
each
is yet
there
to
apply
sult,
we
need
where
the
l,y~E
.
clearly
order
let
C(F)
is
@eC E
- Weierstrass
only
prove
that
that must
E
~sC E
C
a compact
neighborhood
Then,
by
the
disconnectedness
such
that
y~E
and
Since If
a E - E
1
{ E I , E Y}
y,~E
let joint
Then
take 0
For
on
ycE,l~E C,
Thus
CE
a finite
U
be
compact on
U~JC
, let
there
# 0
theorem
. and
that
is
re-
this
is
a role.
# ~(~)
TIC
compact
= A
the
plays
¢(y)
that
get
points;
disconnected
such
U
decomposition
a compact
open
( [ 72,
Cy
and
be
of
neighborhood
neighborhoods
; set
a compact
separates
1
¢(Y)
adjoint.
p.
(in
E)
E
Y ) ; set
38]
y
is
Cy = 1
of
self
separates
such
E ~(C~E) Y
Y E
C
E is t o t a l l y
find
is
that
Stone
Take
total
such
and
the
fact ; we
a subalgebra
0
C
on
y UUC
E of
and
C
1
, and
we
define
E
disjoint of
1
y,l, = 2
@ z
~E
from
dis-
respectively. on
C1
and
¢ z
a compact
neighborhood
neighborhood of
k
. Let
of ¢ = 1
E
disjoint on
C and
from 0 on U.
points. q.e.d.
26
2.2
Measures
Theorem
2.1
II.-TIi
and
Proof. each For
Let v
T
Associated
:
is both
Fe ~
compact
is
~e~(E)
such
that
~ = T
on
o
finitely
open
additive
set
and closed,
and
sup
: He~,
function
T(A)
because
= 0 .
we define
v(T)(F)
E
Distributions
. There
is a r e g u l a r
Fc ~ each
Te~
with
~
{IT(H)[
H ~ _ F}
implies oo
(2.3)
Z T(F.) 1 J
whenever In fact,
(2.3)
=
U Fje ~,{F.}j
a disjoint
is A l e x a n d r o f f ' s
the total
variation
(2.4)
result
v(T)
using
=
collection.
and entails
of
~ v(T)(F.) D and then
T( U F.) 1 J
T
first
showing
that
satisfies
v(T)(U j ) F_
the b o u n d e d n e s s
11T11
of
and
(2.4)
to prove
V
(2.3); T
is r e g u l a r
both
calculations
if and only
and i m a g i n a r y Thus,
for the moment, T
countably
In the usual way
if the upper
parts
and lower
T ~ 0
additive
on
~
; further
where of each By the
~
X
=
(2.3),
inf
we
construct
F~ ~
Caratheodory
T*
by
(2.3)
we have
the
outer
measure
co
[Z T(F.) i
is a r b i t r a r y is a
of its real
on
eo
T*(X)
variation
are regular.
assume
of u s i n g
are easy.
in
:
XC~I ~ F j
Fj~}
I
E ; it is s t a n d a r d
measurable
set and that
that
TI(F)
each
element
= T(F)
for
. theorem
, the
set
of
T*
measurable
sets
is
a
27
a-algebra;
consequently,
It is s t r a i g h t f o r w a r d Thus,
for our
Therefore,
check that
complex-valued
measure all
to
~
Fe~
T~
defined
T
we
on
~
is c o u n t a b l y T~
additive
on
c
is r e g u l a r .
associate
a complex-valued
and such that
T(F)
regular
= ~(F)
for
.
since
~
is the
Borel
algebra,
Ue~(E)
, and
II~-TII
= 0
v
by d e f i n i t i o n . q.e.d.
Theorem
cE
2.2
and
For each
Te ~
there
is
ve~(E)
For
Tc ~
we c o n s i d e r the
fT
is o b v i o u s l y
Fixing
on
:
functional
CE
+
¢ ~
T
T = 9
Ir~-Tllv, = 0
Proof.
f
such that
CeC E
(2.5
well-defined. (see d e f i n i t i o n
and c o r r e s p o n d i n g
N¢
N
is open
~
x
N ~¢-l(x) ¢
for all
of
C E) ~
xe¢(E)
we have
,
(2.6 X
(2.6)
is clear
from the
definition
of
N X
For
2.5),
take
xE¢(E)
U~(¢(N¢)
- {X})
N,A,-I(u) By d e f i n i t i o n
Thus
{F
= A ; then
¢
there
are only
finitely
: xe~(E)
and
F
~ EAN
}
X
that
of P r o p .
is
open
many
N
x
T
such that and
and the
F
is a finite
2.2
is c l o s e d
we
decomposition
since
F
N
are
x
of
E
= E~¢-l(x) X
form the
canonical
finite
decomposition
"
of
x
X
X
Because
of
disjoint.
X
noting
N ¢ ~ ¢-I(u)
U
= N,f],-l(x)
of
mutually
and an open n e i g h b o r h o o d
{T } X
(that
is,
given
T
and
¢cC~
we
construct
{Tx}~(T~F
}) X
o
2S
Since
¢(y)
~ x
for
all
yEN
we h a v e X
(2.7)
[
xc¢(E) on the
other
,¢>
=
[
x
hand,
(2.8)
x
xs¢(E)
the
fT(¢)
lefthand
=
Z
;
x
side
x
xs¢(E)
of
(2.7)
is
fT(¢)
= 0 , and
and
so
,I> x
oo
Thus,
if
@~CE, fT(¢)
Once
we
and
fTs~(r)
if
supp
@~E
= A
, then
we t h e r e f o r e
get
supp
so
= 0
show
In fact,
TE~,
¢¢C(F)
supp
and
Cnf'~E = A
supp
¢(~E
= A
, for w h i c h
then
fT ~- E there
I I¢n-¢II~
+ 0
.
is
{$n}C
; this
C~
follows
co
I I~n-¢I I~ ÷ o, such
that
neighborhood
, from
on
of
E
Prop.
2.3,
a neighborhood
, and we
of
define
¢
then
that
fTs~(F) Ixl
we h a v e II
x~¢(E)
From
Prop.
IIT-~ll
and
I$(Y)I
~
have
fT
¢,$
~cC
(P
on
a
~ 0
-= ¢~ n
from
(2.8)
that
IfT(¢)
< T , ~ F >I <_ i I¢I Ico
vsE
2.3 we t h e r e f o r e ; hence
CsC ~ E
<__ sup
x
CcC(F)
Clearly
for
we t a k e
supp
n
To see
if
co
~nSCE
$ -= I
for
I IT
Iv ,
x
IfT(@)l
!
I ~II.
lITI1 v,
for
all
~ vs~(r)
= 0 V
v
q.e.d. In the
Proposition v
from
Proof. by
the
above
2.4
Theorem
This ~F'S
is
result
Let 2.2
clear
is d e n s e
Ts6~ Then
we
with
compute
corresponding
supp
v~
from
supp
T
.
Theorem
2.1
and
~ = v
since in
easily
C(F)
<~,@F > = by
Stone
, and the
- Weierstrass,
algebra as
generated
in Prop.
2.3.
q.e.d.
29
2.3
Representation
Proposition
Proof. suit
2.5
a.
The
of First
Order
Distributions
a.
ci(r)C_A(r)
b.
C (F)
C.
A'(r)C >i(r)Co(r)
fact
that
a continuous
,
is dense
in
imbedding.
A(F)
CI(F)CA(F)
is B e r n s t e i n ' s
classical
re-
(e.g.[3]).
To show the
continuity
of the
imbedding
ci(r)
(2.9)
the
where
domain
and the range We next
observe
$n' + @ By the
standard
space C 1 (F)
that
vergence;
space
calculus
the
if and
topology
the u s u a l
is c o m p l e t e
in fact,
uniformly
has
{$n}
4'
given
topology
with
the
identity
by
I1¢11A+11¢'11 ~
in
exists
(e.g.§
map
in this
I I I IA and
1.3).
I 1411 A +tl4'll~
the
is Cauchy
Cn ÷ ~
result
consider
ci(r)
÷
has
first
topology
as well
con then
as pointwise.
4' = n
~hus The
II~-~'ll~ identity
+ ll%-~ll map
the
(2.9)
A ÷ o .
is o b v i o u s l y
open m a p p i n g
theorem
a continuous
there
is
K>0
bijeetion
so that
such that
for
all
cEcl(r)
I1~11 A
in p a r t i c u l a r , b.
Let
CEA(F)
+
I1~'11~
CI(F)~A(F)
and define
the
:Kll~llc~
;
is a c o n t i n u o u s
approximate
imbedding.
identity
~
:
r~ s
Clearly,
¢ ~ @ E
c
is
2w-periodic
s
since
j
g --Tf
=
2~
by
•
30 tg ,~(2~+'y-.1,)~ l "~- e
i'e
e¢~
¢(V)=
1 *(Y-X)*~(X) ) -¢
dx =
(t)dl
= 9 ,,~,%(2Tf+'y)
rW
Since
$¢*
¢(Y)
Considering
=
I
¢(Y)~¢ (Y-k)dl'
A(F)C_A'(F)
we h a v e
~e ~ C e C ~ ( F )
and t h u s
@EA(F),
for
F({a
~e*
CsA(r)
}) = ¢ , n
* @,~>
<9
= 2w Z @ (n) ~ ( n ) a
C
Now,
Ye>O
and
it is e a s y
thus,
-n
Yn
(n) i =
Further,
g
i ]__~
fw
to see
since
y)e-inYdyl
that
@
e ln'(~C~(r)
£
<
÷ 2w6
f~
1
(y)d~
E ÷ 0
, as
=
, in
z
.
D(P)
; and,
we h a v e
^
(2.10)
Note
~ (n) + 1 as g
.
¢ ÷ 0
that
so t h a t
f r o m (2.10) and the L e b e s g u e d o m i n a t e d i for L (Z) we h a v e the d e s i r e d d e n s i t y .
orem c. D I ( F ) _ C D ( F )
was
a n d k. b y ~; on
mentioned In f a c t ,
further,
CI(p)
then
functional proof
on
in if
§ 1.3
and
S = T CI(F)
A ' ( F ) C_ D I ( F )
TEA'(F),
it is w e l l - d e f i n e d , by k. and the
Thus,
convergence
is d e f i n e d for T
if
is c l e a r for
SeA'(F)
follows
from
all
cool(p)
and
S = T
is a w e l l - d e f i n e d
continuity
the-
since
linear by the
of a
iIz lITlIA, 11~11A zKl]~llcl q.e.d.
3~
Proposition
2.6
a.
For
each
k>O
Dk(F)
the
+
Dk+i(F)
T ~.~
maps
Dk(r)
function
T'
onto
^
X
b.
-
{sgDk+I(F)
For
all
: S(O)
=
O}
.
TEDk(F)
T
=
c
m
+
S
k
O
w~ere
T
=
c
o
, m
c.
is
Haar
For
all
T
=
measure
, and
TeA'(F)
c
m
+
SeA'(F)
,
f'
on
C
i
(F)
,
O
^
where
T(0)
=
c
o
, feL2(F)
and
T(n)
-
for
all
n
#
r2w I nf(y)e
i
0
inTdy
2~ JO
Proof. Now,
a. if
Obviously
SeX
we
T'(0)
have
(e.g.§
S~fc
=
=
-
=
O
+
~
for
TeDk(F)
1.3)
e inY,
c
n
:
O(Inlk+l),
Inl
n
Setting
T%~
gives
d
=
O(Inlk),Inl
n
Distributionally,
T'
=
S
'
d
n
e
÷
iny
~
c
, d
,
n
and
n - -in
thus
,
TsDk(F)
•
32
b •
Letting
TeD
k
(F )
we
have
T~c
We
+
o
[
c
einy
=
n
c
+
o
T
.
I% j
+
set c
S%[
n
n
e iny
(in) k
k
Clearly Also,
S
=
for
¢eC
T k
distributionally. (F)
=
c
o
+
=
c
o
k and
so
we
have
T
=
c m+T
on
C
(F)
O
c.
For
TeA'(F)
T~c
+
[
C
O
e
iny
, {c
n
}eL
(Z)
n
Letting
Cn iny .--- e in
f~I C
we
have
{.--~n}sL2(Z)
so
that
by
the
Plancherel
theorem
in
Thus
T
~
c m+f'
on
CI(F)
, where
fsL2(F)
, since
0
For
n
#
0
e
n
-
i 2~
-iny>
and C
n in
hence
-
i 2w
-iny>
,"
TsDI(F)
faL2(F).
33
T(n)
C
f2"~ [ nf(y)e-inYdy
i 2~
-
n
•
JO q.e.d.
We Let
now
introduce
X,ye[O,2w),l
~ {~cck(r)
Hk
-
Let
f~Ll([o,l])
l,y
2.7.1
terminology
for
the
next
proposition.
Set
Ck
Lemma
some
: B ¢ eck
ck
{~¢ k,y
¢ ~ _ (l,y)}
: supp
be
for
,y
,
which
real-valued
@'
= 4}
and
assume
Then
that
fl
I f~--o JO for
all
Proof. Take
Assume open
G
X+
the
result
C [0,I] e --
the Let
which
continuous
Beppo
standard
measure
are Define
i
from
on C+
of
C+
4_(i)
Set
:
¢ ~ 4++4_
theory
there
and
m(X+-C+)
C+,4+
disjoint
from
¢(0)
f
f
I
f~ + I J o
g
i
and
let
e>0
Ifl<E -G
+
)>0
on
is
a.e.
a consequence
of
K
(measurable
or
are
; this
f = 0
m(X
)>0
C + ~- - X+,
C
sets)
(or C
so
that
by
both).
X
such
of
C
that
C + ,C --
= 0 = m(X_-C_) some
= -i
on
C_
neighborhood C
,~
; further
= 0
on
disjoint
some
take
4+(0)
:
I
neighborhood
= 4+(1)
= 4_(0) =
.
f
--rO,ll-~
0 ~
Then
J
and
{tsGE:f(t)<0} m(X
If~-J
theorem. ~'~
; define
o
true
0
f 1 j
either
compact
~+
not
at
that
~ {t~Gs:f(t)>O},X_~ hypothesis
By
is
such Levi
-
vanish
+
+
= ¢(i) f
f
I
+ T
J x -c +
= 0
f
J c
+
and
-
+
I
J ~ -~
f
f
f¢ +
J
r o , ~ -o ~ . ,
C
i
Ill.
J cuc +
-
34
r If¢ J
Thus
=
where
I Ifl +r J c + U c_
Irl<e
; note
~6>0
such
Vg>O
that
the
t corresponding
C + ,C -
have
the
I
property
Ifl>~
•
J
c+Uc_ Hence
I f¢
# O
, the
desired
contradiction.
J
q.e.d.
Proposition assume a.e.
2.7
R = O
on
on
(4,¥)
•
Hk k,y
Proof
f e L i (F)
Let
(k,y)
, and,
. Then
obviously
cause k C~,y
= -
R = 0 -
represent
on
and
~e C kk , y
we
s 5
since since o
I¢
j
supp
~,
~(t)
= 0
,9'
= h
some
(2.13)
CsC
fixed
and
h
-- @ - ¢ o S e H k
supp
~o
C (k,y),¢(t) --
when
t<~
, and
representation
for
all
f = c
SsH
k ,Y
k
and
,Y
= 0
, be-
k ¢oeCk,y
- H
k
In
,
¢'¢
[t _= I ( @ - s ¢ ) 70 o
is
,Y
such
that
¢ s+h o
the
and
right
of
,y
t>y hand
f
the
, and
,y
that
= 0
take
by side
fact
the of
0
s~C
in
definition (2.12)
,y k ck,y of
is
C
oj
Further,
C
such
as
@ =
where
c
a constant
of
distributionally,
(k,¥)
k HX, Y # A
any
for
<2.12)
and
is
a subspace
since
We n o w
there
k is
(2.11)
Clearly,
R = f,
letting
oj
(2.12)
is
unique.
(Sl-S2)¢ ° + (hl-h 2) = 0 .
To
see
this
we
assume
s
3S
We
have
the
cases:
i.
If
sI # s2
this
ii.
Thus
for
If
is
then
h2-hl - Sl-S2
Go
a contradiction
sI = s2
then
k eH k,Y k
since
hI = h2
because
,¥
;
k HA
-
in
h~gH
(2.13).
k
every
¢EC k
= s
> +
= sc
-
I C¢
where
the
second
=
,
JF
O
equality
follows
by
letting
c ~
>
and
0
Note
k h~H k,y
because k that C
C
is
((l,y))
(so t h a t
dense
~
C
(2.11)
we
have
= 0).
in
{¢:[k,y]
~:~
+
0
where
by
((k,y))
has
the
continuous,
sup
~(~)
=
¢(y)
=
O}
norm.
0
Obviously
the
restriction
of
f
to
(k,y),
f
, is
a continuous
linear
r
functional
on
C
((k,y))
so
that
since
f -c
0
dense
subspace
we
annihilates
a
r
have
f
= c
a.e.
by
Lemma
2.7.1
.
r
q.e.d. For Proposition
convenience 2.8
we
recall
(Hausdorff-Young)
a
~f ¢~LP(r), <~(n)}sLq(z)
b.
If
{a
}eLP(z),
~a
n
which
Theorem
Let
II¢I
e
in¥
I<~_p_ <2,
an~
-i- + - -i= i . P q
lJ~IIqDr¢llp
converges
in
Lq(r)
¢
for
qAll{an}II p
2.3
Let
T£A'(E),
re(E)=0,
T%[Cn einY , ~E=UIj, 1
¥o eE , Y o # ~ j , V j a.
to
n
for
T = 2~c
all
j
, and
6
+ [k.(6k
o Yo
i J
ej~yj-kj
-6
j
)
~j
on
. Then C I (F)
Ijz(kj,Vj),
36
b.
S - Zk.(6~
-6
1 a
s(
with
n)
=
c
-e n
Remark
C.
Vr
We
have
e
)
j
-in-~
extends
to
a
unique
element
of
A'(F)
J
o
o
<
i
~
~ e 1 1 T J l A,
j=1
Chosen
¥
sE
rlkj
I
J
in
Theorem
2.3
though
this
is
done
for
0 convenience The
since
advantage
we
of
assume
taking
generally
7
eE
is
that
that
our
the
sets
E
are
representation
in
uncountable. a
above
has
o its
support
Proof.
in
From
E
Prop.
2.6
we
have
T
c m + g ' , T (^0 ) o c
=
=
,g e L 2 ( F ) , g '
c o
n
tributional For
y
derivative we
of
g
, and
(n)
-
in
for
n
#
0
form
o .
-iny
-in7
o. )e I n ¥ ,
S%}{C~" - C e n o
b
e n
c
-c n
e
o
o
^
so Letting
that
S a A ' (F)
and
S(0)
=
0
.
,
¢¢A(F)
in(y-y ) <S,¢>
=
-
Z
c
<e
o
,¢>
o
Thus,
because
i
-iny>
2~c
6 o
we
(n)
=
c
<6 o
y
in(y-Y
[
0)
<e
/ / ~
,¢>
=
o
(2.15)
o
have
c
and
,e y
•
6 (n)emnY,<~> o 'y o
hence
T
:
S
+
2wc
8 oy
on 0
A(F)
,
.
the
dis-
37
Further,
supp
SC
E
since
y ~E
--
above,
supp
and
T ~
E
and,
as
mentioned
0
S(O)
= 0
and
SsA'(F)
Letting b f%~
n
iny
in
we
recall b O 2 f~L (F)
as Supp
S C_ E
impies
f ~ k. J m(E) = 0
Since
and
= 0
on
S = O
on
we
have
each
I.
distributionally
that
by
Prop.
2. 7
we
as w e l l
take
,
f('r)
kj×i.(¥)
= X 1
for
, so
3
= S
I. 3
(2.16)
Now,
f'
¢ecl(F)
a.e.
3
,
r~j <X'I.,¢>
= -<XI
a
j
Distributionally,
'@'>
= -j)
~' ( X ) d ~
a.
= ¢(~.)-¢(y.)3 a
= <6k - 8 y . , ¢ >
a
a
= X kj XI' " I j
f'
on
C
1
(F)
and
a
hence
co
{. 2 . 1 7 )
Recall
S = X kj(6 x.-SY.) 1 O J that
primitives
(2.16) Combining follows
is
and
since
SEA'(F)
element
because c first
set
(2.17)
of
el(r) d
{b
n
= b
}~L~(Z)
and
for
all
(F)
differ
by
a constant
so
that
a.
hence
the
, viz.,
S
representation itself.
The
(2.16) extension
/(in),n
# 0
, so
that
since
f(n)
n
we
p>l,{d
is
= d
have
n
= o{]~-JJ,lnl
÷ ~
l n l
, and,
as
we
saw,
f~L2(r)
to
unique
and n
}~LP(z) n
extends
= A(r)
d
Thus,
gives
A'(F)
n
--
distributions
C I
well-defined.
(2.15)
an
For
of
on
3$
By
the
Hausdorff-Young q<~
Id
theorem,
; that
is,
fELq(F)
q~l/q
r
~ (till j
since
2~q< ~
ll{dn}tl
_<
we
from
,
,
can
, for
all
,
p
([
<
--
take
d
la n IP) 1
= 0
/p
'
ICo I
+
(2.16)
and
n
(because
i/p
~
c
Inl
b
<~ P
= 0).
Also
we
'
can
take
O
or
for
21ITIIA , 2~q<~
,
llfllq ~ (3 I[ kjx I (Y)IqaY] t/q = ( o
i
K(Z -7---~,p)
i
f2~
(2.18)
Lq(F)
in
. Further
O
IITIIA'
K z that
converges
so
Ilfllq
Note
iny
for
tlfllq and
e n
j
=
t
Xzj
j=
i/q
Ikal%~)
i.
J m
in
fact,
for
all
m
and
m
IZ kjxi.(Tllq 1
have
×i(xllk 3 Iq
[
=
3
1
acteristic we
m,I [ k j X l (y)l q _< I f l q E L l ( r ) i j (since
we
are
dealing
functions),
so
that
by
Lebesgue
dominated
(2.18).
Thus
Ilfllq q
Z Ikjl q~.a
~
< Cqp' ~P + !q = 1 ,
2~q<~
n=l
Now
we
observe
that
c q < (2K) q (q-l) 2~q<~ p -2 q "
(2.19)
to
see
this
note
I 2
with
char-
J
that
for
~>i
;
,
1 < ] dt - t n e -- 1 t e i-~
I
1
1 a-i
'
.
convergence
39
and
1 c~-1 P q
so w i t h
we h a v e
o0
2 X
I
2
this
<
2
n p-
q q-1
_ -
2
1
1
implies
K([
C
t
1
P
)l/p < K 2 I/p ql/p
Inl p
and h e n c e
C q < K q 2q/p p --
which Choose
r>0
gives
such
qq/p
= (2K)} 2
q (q-l) , 2 < q < ~
,
(2.19).
that
(2.20)
r <
1
1
2Ke
4eIITfl
\
A'
(rC)q
Then
q !P
~ q=O
converges
since
(rC)q
and by the
l im
test
.
usual
vergent
rlkjl j=l
ratio
(q+i)'
by the
e
ql
(2rK)q+l(q+l)q
q
Thus
[ q=O
~ e. = ~ J j=l
~ ~ q=O
of
q-i
[ (2rK)q q=O
q'• (2rK)qqq-i
properties series
~
< 2
= 2rK
lim q
interchanging
qq!
(q+l)q-i qq-i
sums
= 2rKe
in a b s o l u t e l y
.
con-
we h a v e
(rlkjl)q~j = q!
~ ~
~ ~
(rlkjl)qa.
q=O
j=l
ql
~ J < ~ -- q=O
(rCp)q q!
q.e.d.
40
^
Theorem
2.4
Let
p~(E),
m(E)
f =
and
f
is
Proof.
a function
Clear
from
= O,
~(O)
~ kj× I a.e., 1 j
of
bounded
Riesz
= 0
~E
=
Then
U
lj
p = f'
where
,
1
variation.
representation
theorem
and
Theorem
2.3.
q.e.d.
2.4
Measure
Theoretic
Properties
of
Pseudo-Measures
co
Proposition
Proof. If
2.9
Let
T~[c
e
If
m(E)
TEA'(F)
iny
and
have
fact,
2~c
=
then
the
C E = A(F)
property
n iny = [ p cr-- e in
f(Y)
n In
= 0
= 0
since
a.e.
I~C E
= 0
then
C If¢ ' = 0 j
and
for
all
SsC E
when
¢s
Cco E
thus
O
0 = -
Now To
we do
Let
are
in
this f
a position
first
be
real
take
to
an
proceed
as
interval
(without
=
loss
~2~ I f¢' JO
in
(X,¥)
of
Lemma
2.7.1
disjoint
generality)
from
and
assume
e>O
and
E f#Oa.e,
on
(~,y) If
Y~
(k,y) let
has
the
open
G
property
that
(X,y)
satisfy
C
m(Y)
g
= | J
Ifl>O
on
Y
,
IfI
(~,~)-G Then
form
X+ , X- , C+ ,
Next
define
@sC
of Set
@(t)
C+
(F) , and
gt - I ,~(x)dx JX
and such
C
on
of
E
and
so
CcC E
Lemma
2.7.1
@~(X,y),@
. Then
. = i
a neighborhood
tc[X,2w+X) oo
hood
in
supp
that
@ = -i for
as
of
@(t)
on
a neighborhood
C = O
on
a neighbor-
41
Now,
from
Thus
Lemma
f = 0
r If@ # 0 , a c o n t r a d i c t i o n . J (X,y) so t h a t s i n c e m(E)
2.7.1
a.e.
on
= 0
we
argue
that
f = 0 a.e. This
means
that
T = 0
so
that
by
the
Hahn-Banach
theorem
we
have
co
c
-- A ( r )
E
q.e.d.
Theorem
2.5
Proof. (÷)
Let
(÷)
If
m(E)
= O
and
TeA'(E)
. Te~÷+
Te~(E)
Clear.
Te~
we
apply
~e~(E)
either
such
desired
that
Theorem
2.1
T = ~
or
on
CE
Theorem
. Then
2.2
Prop.
to
2.9
find
gives
the
result.
q.e.d.
More
Theorem
2.6
generally
Let
we
have
TeD(F),
supp
T C
E
, be
of t h e
form
T = c m + f',
--
feLl(F)
, on
Proof. (÷)
CI(F)
(÷)
Again,
and
let
m(E)
2.1,
there
o
= 0
Then
Te~
~÷
TeN(E)
Clear.
from
and
Theorem
I ISI Iv = 0
where
is
psi(E)
T-p
~ S
= 0
such
that
T = ~
on
^
Note
that
Also
S(0)
~e~(E)
Now,
h
is
any we
Now
•
Since Also,
for
C i (F),
on
Prop.
any
j
= O 1 , choose
m(E)
of
representation
a function
theorem,
of b o u n d e d
that
~
variation.
g e L I (F) (~,y)
primitive
we
Riesz
, we
have
g = [ k j x i . a.e. 1 j
on
U
I,J
•
k
I IS11 v = 0 because
the
I.j =
2.7
other
take
by
derivative
= __UIj, 1
by If
first
S = g' if ~ E
since
implies,
the Thus,
= 0
S,
h-g
is
a constant;
in p a r t i c u l a r ,
. ~F
have
= 0
of
,
such <S,@F>
that = 0
O<~F
only
on
I1
and
I,j
is
42
<S,~F>
Generally,
=-
=-!([kjx
I.(Y))9~(Y)dY J
I (Zkj× I ( ~ ) ) ~ 9 ( ~ ) d ~ : Zk~l ~9 by L e b e s g u e ; l vergence; and b e c a u s e of the w a y w e ' v e d e f i n e d
j
(2.21)
0 = <S,¢F>
kj
= kI
, which
holds are
con-
implies
for
each
j
equal
a.e.
and
that
g = 0 a.e.
; consequently, hence
@F
we
have
= -+ ( k l - k .j)
I
Thus
dominated
j;Ij
T = ~
I
on U I j
the
since
primitives
of
(2.21) ~
and
T
.
q.e.d.
Remark
i.
then is
to
~A(r)
- C E~
viously
that
; thus
by
2.
There
Prop.
E,
are
of
dense
than
, such
is
due
to
of
strong
in
all
fact,
TeA'(E),m(E)=O if
2.6
we
Prop.
have
for
¥SEE
2. 7
, and
(2.22)
For
any
YTEA'(F)
E>O which
there
is
e>r(~)
so t h e r e some
C E = A(F)
. Ob-
[55].
and was resolution
= r>O
such
satisfy
leieeinY-eimyI
< r for
all
yc
have
lei@T(n)-T<m)
I ~
el IT11 A,
it
2.9.
Varop0ulos spectral
, and
# 0
C E = A(r)
that
C E # A(F)
A(F)/j(E)
and
Theorem
result
for
2. 9 . In
CE/~(E)
and
m(E)>O
sets
Prop.
is not
2.3
result
I I T I I v <~
if
on
weaker
are
sets
use
T = 0
a much
following
the
that
used
to
[104,
prove pp
3831-
].
Theorem
we
that
is
see
CE/~(E )
such
Kronecker
to
that
this
The
easy without
check
T~A' (E)
3834
is
C E = A(F)
easy
is
It
supp
T
,
that
Ym,
neZ,
43
Proof.
Given
e>O
Let
6
5
8(e)>O
and
¢
z
@
satisfy
cA(F) E
_<
II¢II A
e, l-eiY-¢(y) = 0,¥ye[-6,6],
mod 27
This can be done by Theorem 1.8 b Set
r(e)
~ min {a,ll-eiS/2 I }
and let
m, n, 8, and T satisfy
(2.22).
Define
~(y)
= e
i(ny+e)
@(my-ny-8), ye*
~(y) = e i0 e iny - e Z"m Y - ~ ( y ) ,
and
T = my-ny-SsF
ycF
,
,
.
Therefore (l-eiT-¢(x))e i(n7+8)
and this is By hypothesis,
if
7a supp T
= e i(ny+e)_e-i(ny+8) eimYei(ny+8)-~(y)
,
~(y)
ys
supp T
we have
then
Tel- ~2, ~]' mod 27 . In fact, if
II-ei~l ! r ~ Ii-ei6/2 I
by (2.22)
For example,
1 e i6/2
ei6/21~
~
-
l-e
From this we get
~ = 0
on a neighborhood
observe that since then
Tel-- ~2' ~6]
Also for a given
y s supp T
that not only does
~(y) implies
=
of
it
supp T. To see this first
(l_ei~ - ¢( ~))e i(ny+e)
when ys supp T
l--eIT--¢(T) = 0
the corresponding $(y) = 0
but
T
~ = 0
is in
[- ~, ~ ] ,
so
in an open set about
44
¥
which
since
the
Hence
= 0
On
other
hand
the
is
the
map
inverse
y g~,~¢ T
image
is
6 + --) - we 2-
6 (T- ~,
of
can
do
this
continuous.
.
i
^
)_<
: e eT(n)-T(m
T,¢>
Consequently
teiST(n)-T(m)t
: t ITI tA, 1 t~l I A
Now,
It'll A
and Because
leiel
:
Ilei=~ll A I I ~ ( m y - n y - e ) l l A
I1~11 A
so _<
II¢II A
s
II¢((m-n)y-e)ll
A
I1¢ll A
:
we
:
are
done.
q.e .d.
Notes §
2.1
For we
§2.2
§ 2,$
the
measure
refer
to
[4;
For
Theorem
rem
2.1
The
elementary
and
Theorem
duce 2.3
we
some was
ment
2.2
5; we
refer
used
to
2.3
2.9
a key
lemma
of
strong
is
for
in
[ 5;
104,
pp.3831-3834~
given
H.
between
and
Varopoulos
in
part
Pollard
104,
pp
3831-3834].
and
[77]
various E
Theorem to
resolution;
E
2.6
prove it
is
is
that
it w a s
the
guess
and
are
technique and
to
are
~ 2.3
where
The
enough
by
of
[ 74]
results.
supported
spectral
first
is n a t u r a l
[55]
[ 55;
developments)
used
synthesis
theorem
further
was
in the
by
(and
for
Theo-
.
first
earlier
in
to
[ 4]
was
Relations
Prop.
. C~ E
results
of p s e u d o - m e a s u r e s §2.4
6]
properties
refer
spectral
of the
theory.
theoretic
at
in
in [ 4]
Kronecker
essentially
used
to
de-
of T h e o r e m
general in
properties
given
standard,
distribution of
[6;
state-
primitives
7]
•
. Theorem sets
2. 7 was
are
a result
sets
relating
45
the
differences
given
a sup norm,
sociated
vergent
the
Taylor
that
series
in some
cently,
also b e e n
(e.g.
general
§ 8);
groups.
Wik
on
sets
its o r i g i n s
[ 523
coefficents
s h o w e d that
F
In
to
[14]
also have
classical
this
research
effectively
of p s e u d o - m e a s u r e s
A(F)-norm,
set of r e s t r i c t i o n s
Dirichlet
has
Fourier
not n e c e s s a r i l y
exponentials.
is p r e c i s e l y
2.7,
between
if
E E
relation
is H e l s o n
it is shown, property.
con-
using
Theorem
Theorem
2.7 has
on u n i q u e n e s s
easily
as-
then A(E)
of a b s o l u t e l y
u s e d by D r u r y
it is, of course,
between
sets
[19]
generalized
and,
re-
and K a h a n e to more
3.
A Characterization
3.1
Introduction
to
A natural relative
to
introduce U-sets for
the
sets
which
this
set
questions
we
gave
this
and
study
of
not
easy
Proof • that
which
tells
3.1
If
us
E~
and
m(E)>O
~P ~
E,
= UI., I. open J J t i o n of P
it
not
did
P
is
finite
number.
matter
sets was
W.H.
and the
P
Young
Clearly,
xsL
(P)
all
only
worry
positive
are
about
Haar
and
the the
~c
e
P~_ iny
; in
in
an
Fourier
Fourier E
X
we
sets
measure
to be
× = 0
the
Fourier
fact,
on
in
series of
sets
are
for h i s
1908
- these
U-sets. have
the
following
of measure
then
E
0
is
.
an M - s e t .
c.J # 0
j
and
containing
the
, such
characteristic
I. , f o r a l l J of × converges
series
given
interval
series
, the be
the
and
m(P)>O
Take
Therefore
to of
closed,
background
showed
angle
intervals.
; and
ysl.j
iation
Let
was
then
a different
perfect
is
definition
countable
immediate
from
has
E
certain
sets
P
arise,
arbitrary.
denumerable
need
synthesis,
that
series
if
all
we
harmonic
dimension
trigonometric
co
Thus,
of
In t h e
research
things
problem
We f i n d a t r i g o n o m e t r i c series ~c n e inY , s o m e iny [Cne converges pointwise to 0 in P-E .
Since
CP
at
E~
- that
the
arithmetic
§ 1.2
that
Sets
Sets
approach of
§ 3.1,
showed
Looking
Theorem
in
and
are
result
to
1872;
theory
things
means
section,
in
Uniqueness
of u n i q u e n e s s .
Cantor U-sets
of Uniqueness
yelj y
and
, ×
is
so b y
to
func-
0
for
of bounded
Jordan's
var-
test
converges. X
converges
to
Fourier
series
of
X
Fourier
series
of
×
> 0 " i.e.
co
# 0
0
on
converges
P to
so t h a t , 0
on
since
P-E
n
1 o
o
-
r2~
2~
I
JO
m(P) x(y)dy
=
2~
q.e.d.
In 0
and
1916
Ranchman
Menchoff showed
showed in
1922
that that
there there
are are
perfect large
M sets
classes
of
of measure perfect
47
U-sets
(of
sets.
The
measure triadic
In which
we
result
3.2
this
refer
in
, F
is
erties
to
as
we
the
and
a U-set
space
of
we
I
thickens
result
closed
of
though
not
Rajchman's
U-sets
pA,eudo-functions;
in
we
the
work).
terms
of
mentioned
A'(E) o this
Pseudo-Measures
if
on
F is
on
{s:Isl
{s:IsI>l}
refer
to
that
topology
on
plot
(a
characterize
and
on
. Note
induced
V
is
the
.
analytic
of ~
= O
I ~
chapter
§ 1.2
tributions
H
set
Thus
A hyperdistribution H + } where F is a n a l y t i c
finity,
the
Cantor
Hyperdistributions
+ {F
O-naturally).
[3] F
from
3 V C
~
. We
for
let
viewed
. Letting
, open,
~
be
as I~
and
F
of and
a leisurely
, when ~
a pair
analytic F
functions
, vanishing
the
space
explication the
unit
of of
circle
F ~
be
open
analytic
in
V
at
hyperdisthe
in (in
such
in-
propT
F)
, has we
say
that
and
< F+(s) if s~VN{s:Isl
might
be
(3.l)
It
is
expected
){-supp
easy
to
check
we
H
if
define
the
~ U { I C
that
s~V~{s:Isl>l}
F
(3.1)
is
support
: I is
of
open,
~ V~B-
He~
to
be
H =
0 on
I}
well-defined.
+
Proposition
3.1
The
following
a.
Vi ~ " F
b.
~-supp
are
equivalent
, open,
H =
0
for
on
H = {F ,
F
}~2':
I
H = A
+ c.
Proof.
a ÷÷
b
and
F
and
c ÷
a
F
are
identically
are
trivial.
F-}
. The
0
+ To
show
a
÷ c.
Let
H =
{F +,
hypothesis
in a
says
that
F
and
48
-
+
are
F
F
to
same
entire
and
"F(~)
= O"
F
constant
entire
orem Thus
restrictions
F+
and
function
is
F-
are
--
B
F
and
B
vanishing
imply
F
and
, respectively,
identically
O
the
at
bounded "F(~)
of
=
so O"
that
by
implies
Liouville's F
~ 0
the-
.
.
q.e.d. +
Given
H
=
{F
, F-}ej)~'
and
set
co +
(3.2)
F
n
(s)
=
[
c
s
,
Isl<~
,
Isi>i
,
n o
-F(s)
=
-i [
n
o
-co
(3.2) -F R>r>O
makes
sense
analytic , and
from
for
the
Isl>l
Laurent we
s n
series
let
Cr,
theorem.
CR
be
To
see
concentric
this,
given
circles
of
radii
calculate co
-i
--
-F
n
(~)
~
[
as
[b
+
s
n
n
n
where
[
1 an
thus
a
part
of
n
=
0
(3.2)
for
all
where
n c
=
-
by
2~i
Cauchy's
= b n
F ({)
]CR
d~
~n+l
theorem
Because
of
and
(3.2)
we
we
have
the
second
write
n co
H~
(3.3)
~
c e
iny
n
for
Hc~/
lytically
; here
we
continued
have across
s
~ e F
iy
EF
, the
. Thus
if
Fourier
function
F + ( e iY )
F - ( e iY )
F
+
series
and of
F the
-
can
be
continuous
ana-
49
is
given
by
(3.3);
is
clear
by
the
usual
The
key
representation
Theorem
3.2
a.
the
fact
that
in this
calculation
of the
theorem
(3.4)
Ic
b° iny
that
Assume
(3.4)
n
we
get
Laurent
is
H~[c n einye ~
Given
case
a Fourier
series
series
coefficents,
-
. Then
Ve>0
3N
such
s>O
• Then
that
¥1nl>N
I < e
holds
for
all
3He 2
such
H~Cne
Proof.
a.
Given
s>O
and
consider
the
Zcnsn. Isl
function
o By
the
Hadamard
radius
of
convergence
lim
so Thus,
~N
that such
since that
sup
Rhl
we
Yn>N,l c
theorem
ICn II/n
have
lim
l~(l+e)n
of
E
follows on
(0,~)
positive),
Also,
b.
F
and
, analytic way,
with
radius
{c
n
}
only
(clearly
sup
(3.4).
Ic
II/n
for
the all
radius ~>0
,
of and
at
derivative
; this
with
infinity,
dealt
it
as
with
a function
in-
function
respect
l+e<e C
is
last
increasing
= i - hence
to
in
E is
the
of
~ = i/s
for
all
l~l
Thus
for
< lim
convergence hence
IcnIl/n~l
a strictly
considering
n
Consequently,
its
e -0
convergence
satisfying
is
o
vanishing first
of
lim
e -e
because
and
same
Given
since
sup
= en log(l+e)<een
n equality
= I/R
[cs n o
sup
R of n
is
c
n
, n~0
, and
e>0
ee = e
[c
n
s
n
analytic
satisfies for
Isl
Rhl/(e ~ )
50
Again,
we get the
trivial
corresponding
result
-i ~CnS~L
for
by first
making
a
modification. q.e.d.
Now
H%[c n e in¥ , J%[d n e i n y e ~
given
c
d
=
0(e~
we have
,,in1
1 n ,1) ,
+
~
,
nn
and
so
H @ J~Cndn
is a w e l l - d e f i n e d Hadamard's ments
element
of k
mu!~iplication
The
theorem
einY
following
[103,
is a t r a n s l a t i o n
pp.
157-159]
~-supp
H*J~
in terms
of of ele-
of
Proposition For
3.2
H, Jc ~
Let
analytic
~(s)
• Then
~-supp
n
=
~b s , r < I s ] < R , ic(r,R)
H+~/-supp
J.
, we set
n
n
Also,
from T h e o r e m Our next
sistency For
between
Theorem
3.2,
aim
we have
(Theorem
the notion
3.3 we need the
-n
the n a t u r a l
imbedding
3.3 and T h e o r e m
of
supp
3.5)
T and y - s u p p
following
preliminary
D(F) C
is to prove
~
. the
con-
T for TeA'(F) results
on P o i s s o n
sums, For
~>0
define
P (y) =
where
the
By T h e o r e m
series
on the
3.2 we have
right that
~e-@lnle inY
converges YHe~,
uniformly
¥~>0,
Hg#P
and thus exists
P cA(r) C A ' ( r ) .
, and
for
51
Hm[c
e inY n oo
i~P~
Looking
at t h i n g s
slightly
[Cn e-~Inl ein7
differently, -i [CnS
--
-F
(s)
=
if
H = {F +, F-}
, then
for
n
• Isl>l
,
-oo
we have
_F_(e~+iy)
noting we
that
have
e >i
Isl
=
and
ISl
so
= -i[ C n e - ~ I n l e inY
- le~+iYl>l
. Similarly,
for
-~+iy s = e
and
e
F
+ e-~+iy) (
= [c n
e-elnle iny
0
Since
c
,,inl ÷
= O(een),il
n
.
for all
, co
-~Inl 10
converges
uniformly
(and
F+(e-a+iY)
(3.5)
n
e
e
absolutely,
_ F-(e ~+iY)
~
,
then
in~
of
course)
~ ~)a(y)
= -co
is a c o n t i n u o u s ries
of
(3.5)
Proposition
Proof.
(3.6)
Let
function is the
3.3
on
Fourier
¥Tsi'(r),
T%[cne
F
iny
(in
series
¥¢~A(F),
, *(y)
= ~ane
-
fact of
c e n
an e l e m e n t F+(e -a+iY)
and
so,
-~Inl
for
e
all
and the
- F-(e e+iY)
Then
= 2~ ~ C n a _ n ( e - ~ I n l - l )
,
iny
of A(F))
,¢> +
~>0
as a + 0
se-
52
Hence,
by
Lebesgue since ~Ic
dominated ~la
n
I< ~
convergence and
c
a (1-e-a[nl)l<~) n -n
as
~
for
= 0(i),
n
the
right
series
(which
In I + ~
hand
we
can
use
, implies
side
of
(3.6)
tends
to
0
0
÷
q.e.d.
Proposition
3.4
a. K G
I,~
Let
If
÷ O,
b.
I~_
F
~ ÷ 0
If
Hs~
.
is
open
and
, uniformly
cELl(r)
H = 0 on
on
and
K
¢ = 0
I
then
for
all
compact
.
in
a neighborhood
of
~-supp
H
then r lim
Proof.
a.
borhood
V
H = 0 on of
and Observe
I
I
such
implies that
a continuation
that
given
K~
I
do t h i s
a finite est For
all
take
cover
"radius
~>0
is F-
-~+iy
,
there
to
e
0
is
V~B
~8>0
~+i7
neighborhoods by
=
F
a n a l y t i c on a n e i g h + of F to V~B-
a continuation
compact,
compactness
in the
~V
+ such
~-direction"
Vas(0,6),
VeiY~K
-
about and
that
each
e
iy
determining of
these
eK
thus 8
as
getting the
neighborhoods.
,
(T)
(3.7)
as
(T)¢(y)dy
that
F of
e
to
I ~'
in
= F+(e -~+iY)
_ F-(e ~+iY)
,
(3.5).
Also,
(3.8)
lim o. ->- 0
F(e - ~ + i ¥ )
=
" F( e e + i T ) = F(e IY)
lim ~
~
0
uniformly
on
K
;
small-
,
53
we must V
prove
imply
let
that
C ~
V
From b.
(3.7) Let
In fact,
for each
be
is u n i f o r m l y form
(3.8).
yeK
compact
KC I C V and F continuous on -~+i~ F(e ) ÷ F(eIY), ~ + 0 ; further,
such that
continuous
on
C
and this
and
(3.8),
¢~LI(F),
p~
~ 0
@ = 0
on
I ~())@
uniformly ~-supp
clearly
(~)d
where
the
--
gives
F
the uni-
÷ 0 , for
lIFe ~ l > 2 e
1
that a>a --
¢(~)6 ° (~)d~ (~
H
follows if not
by a. 3 s>0
such
; but by properties
that
of the
¥~o 3 a > ~ 0 Lebesgue
for
integral,
r I @ ~ : 1 $~a ' ~K(~__ ~ - s u p p H, compact, JF J r @~-supp H r IIK¢ ~ I > e ; hence, taking ao for which IIjK~ ~a]
and since
such
K
,
convergence
13 F ¢ ~ I which
on
1
g g2supp
if
C ~_ V ; then
H . Then
~X-supp
Therefore
C ~
convergence.
; F
Thus ,
K ~_ int
we get the
desired
contradiction.
o
q.e.d. Theorem
Proof.
3.3
Let
Given
NOW,
T~P~
For
$eA(r),
~
TeA'(r)
Ts ~ @~
supp
so that
and
where ~ ~
. Then
let T~Cne
T = 0 iny
U , we have
by Prop.
3.4
Jr
supp
,
T~_~g-supp
on open and
$ = 0
U ~
~ (y)=
T .
r
in the sense -~Inl iny [Cne e
on a n e i g h b o r h o o d
of
of~/
~-supp
.
T
54
But
by
Prop
.
3.3
Consequently Thus
if
E
,
,¢>
= 0
z~/-supp
T
+
,
; and
hence
then
supp
~ + 0
T = 0 T C_E
on
.
q.e.d.
In ization
3.3
§ 3.3
principle,
Riemann's
I C_
say
[0,2w)
if
3.4
sented
shall that
show
that
(Theorem
~/-supp
Localization
We
Theorem
we
T~
3-5),
supp
(Riemann
, for
the
Riemann
local-
TEA'(F)
Principle
a hyperdistribution
T =" 0
T
using
on
U
T = 0
on
an
open
interval
~ {eiY :yEl}
Localization
principle)
Let
TeA'(F)
be
repre-
as
iny T~
and
~Cne
set y2
(3.9)
Given ~/÷÷
I~ F
Proof.
[0,2~) is
Let
linear
F(y)
- c
an
open
interval.
on
I
w = y+iv
and
F
for
all
ye~
~c
and
I[ ~ne n=l
- [
2
c n ~
'
T = 0
e
iny
on
I
in
the
sense
.
define
F+(w)
Clearly,
o
forms
w2 ~ Cn inw 2 - [ n--2 e 1
- Co
-
the
-1 c n ~ ~
(w)
---
for
all
inYe-nV
e
v>O,
inw
F+(w)
~Ict
n
I<[--
1
--7
-nv e
converges
since
of
55
and Similarly,
{c
if
}aL~(Z)
n
ye~
and
ic I ~ ~ -~
Thus,
F+(w)
is
and
v<0
n
defined
both
,
iny
e
for
e
-nv
~
Im w > O
functions
are
Ic
I
-n n2
I < ~ -1
and
e
nv
<
F-(w)
analytic
is
in t h e i r
defined
for
respective
Im w < O
regions
of
convergence. Consequently,
we
calculate
= [ c e inw n
d2F+(w) dw 2
~ G+(w)
for
Im w > 0
,
0
d2F-(w ) -I inw dw 2 = ~c n e
Further,
by
definition
ist
and
are
of
F(y)
(÷)
Assume
Let
V
be the
F(y) the
= ay+b
open
=
F+(w)
F
of
(3.10),
lytic By
a standard
in
H
vr~{wa~
complex
analytic
is
in
, we
have
the
line
I
Im w < 0
that
w = y
converge
both
since
uniformly;
the
ex-
series
- F-(y)
. plane
on
V
with
diameter
:
weV
and
Im w > 0
(w)+aw+b
for
weV
and
for
: Im w
Im w < 0
Tel
continuous
in
> o}
variable
trick
V
[91,
(e.g.
functions
and
for
ay+b
Because
F
= F+(y)
in the
function
I
H(w)
on
disk
following
on
obviously
(3.10)
for
and
continuous
representations
Define
F+
- G-(w)
V and
using p.
316]
; and, in
V
Morera's ).
of
course,
{we~
H
: Im w
theorem
we
is
ana-
< O} have
H
;
56
Hence,
not
only
does
H(w)
and
vice-versa,
to
G
Therefore
(w)
since G + ( e iz ),
but
and
T
is
extend also
F+(w)
through
I
to
d 2 Hd(ww2)
extends
G+(w)
F-(w)+aw+b through
I
vice-versa. the
G- (z)
hyperdistribution
~ G - ( e iz ) , a n d
, G
H"
is
an
} , where
analytic
z)
continuation
+
through
I
sense (+)
Let
of
T = 0
analytic
in
of
so
I
: Im w
all
V
we
have
in
the
G
is
that
V~{we~ of
and
G
, we
have
T = O
on
I
in
the
~/ on
V
G
> O}
. Note
simply
sense
of ~
, V ~
a continuation
(and
that
V~{we~
since
transformed
of
open G+
: Im w
T = 0 the
I
on
(and
< 0}, U~
definition
in
, and
G-)
G
from
respectively)
F, of
~
U
to
z {eiY:yel},
hyperdistribution
+
to
be
and
pairs
lower
tional Hence
the
of
analytic
functions
half-planes,
ation
primitive of
G-,
(e.g.
of
say)
, G
respectively,
transformations
second
G
G
must
by
[15,
the
Chapter
(considered differ
defined
as
from
usual
4] the
F+(w)
on
the
linear
upper frac-
). analytic
by
continu-
a linear
function
+ on
that
part
of
V
where
F
is
defined.
+
Thus,
F
- F
is
linear
on
I -that
is,
from
(3.10),F
is
linear
on
I.
q . e .d.
Theorem the
3.5
sense
Proof. Note
Take Let
Let
of
A'(F)
Without that
TeA'(F)
if
, then
loss
of
¢eA(F),
@eA(F),
supp
and
consider
~el ¢eA(F)
look
and
I~
T = 0
[0,2~), on
generality
supp
¢ C
¢@C_
I
all
in t h e
take
I
, we
and
so
h>O
I
open.
I
to
have
T¢
such
that
If
sense
be
an
= O in
= O
T = 0
by
[h-h,
I i.e.
, in
of
open
interval.
fact,
for
all
hypothesis. k+h]
G
I
#(y)=
i h2 Y -
_
1 h2 k
I
like
_ k-h
on
l+h
1 y + --~
+ h,ye
[
] k,l+h
.
57
We
calculate
the
Fourier
coefficents
for
¢~[a
e in7
and
find
the
right-hand
n
a
a
Obviously,
Now,
for
-
n
f n__k) sin 2' 2 , n # 0 nh 2
-ink e
(-y)
i 2w
-
o
1 2w
T%[c e inY n
and
with
on
T = 0
F
as
in
(3.9)
0
for
we
sin2(nh/2) (nh/2) 2
h2
F(k+h)
by
direct
substitution
of
(3.12)
is
any
k¢I,
sense
of
ink e
A'(F)
,
ink e
= 0
have
l[
-
sin2(nh/2) (nh/2) 2
IC n
in the
ZC n
(3.12)
Hence
=
I
(3.11)
With
ICna_n
so
Thus,
= 2w
the
h>O
+ F(k-h)
into
left-hand
, such
F(k+h)
-
(3.9); side
that
+ F(k-h)
in
of
[k-h,
2F(k)
]
fact,
side
(3.11). k+h]
= 2F(X)
C
I
,
;
and,consequently,
F(X)
which ff.;
is 2,
= F(k-h)
precisely p.188
of
the Volume
+
F(l+h)
condition I;
Lemma
2
F(t-h)
(e.g. [ 3, 3.7.1
of
Chapter § 3.4]
2;
8,
) that
pp.142 contin-
58
uous
F
be
linear
F(k-h)
Therefore
we is
apply the
on
I
(e.g.
I
Theorem
sense
of
l
I
i
~-h
k
l+h
3.4
directly
and
)
conclude
that
T = 0
on
I
)~
q.e.d.
Corollary
Proof.
3.>.i
This
Let
is
TeA'(F)
immediate
; then
from
supp
Theorem
3.3
T = ~-supp
T
and
3.5.
Theorem
q.e.d.
Remark for
i.
every
and by
¢,
the
2. port
ume
TeA'(F)
supp
usual
We
; then
¢CU,
functional
refer
T = 0 = 0
. This
analysis
to [3,
on open is
B] for
F
immediate
definition
Chapter
U~
of
if and from
only
Cor.
if
3.5.1
support.
further
results
on the
sup-
of p s e u d o - m e a s u r e s .
3. ment
Let
of the I]
With
regard
Riemann
to T h e o r e m
localization
3.4 we
recall
principle
the
(e.g. [ 2,
classical pp.
state-
195-201
of Vol-
):
Let
F I and F 2
nometric these
series
functions
perhaps,
be
with are
if t h e i r
(a,b),
then
series
is
(a,b)
and,
the
the
Riemann
coefflcents equal
difference
moreover,
tending
on some
difference
a series
functions
of the
uniformly
to
to
interval
is a l i n e a r
convergent
for
given zero
in
two zero;
(a,b)
function
if or, in
trigonometric
everywhere
any
trigo-
interval
in
59
[~,B] ~_ (a,h)
3.4
Pseudo-Function
Lemma
3.6.1
Both
Characterization
the
real
and
of
U-Sets
imaginary
parts
of
a trigonometric
se-
ries e~
iny
(3.13)
can
be
~Cne
written
in
the
following
l,
2
(3.14)
a
+
o
(3.15)
where
formally
[(a n 1
[r n o r2n =
a2n + b2n
for
cos
cos
n>l,_
ny
(ny
rn_>O
equivalent
+ b
+
¢
for
n
sin
n
all
n>O,_
To
show
(3.13)
implies
(3.14)
set
a
and
to
show
(3.14)
implies
(3.13)
set
Cn
Now,
given
(3.15),
Conversely,
let
given
a
(3,14),
¢
and
~ r
n
cos
n
we u s e
r
0
, b
n
these
> 0 0
¢
so
= c
n
formulas
that
a
--
,
CnS~,
an,
+ c
, b
n
-n
bneE
.
~ i(c
n
n
-c
); -n
1 ~ ~(an-ibn)
= -r
n
ny)
) ,
Proof. thus
ways:
sin
n
(for
= 2r 0
n
n>l),
cos
and
define
¢
0
0
q.e.d.
The (without ized
by
Theorem on
a set
TeA'(F) O
following
the
use
Lebesgue
3.6 X C such
of for
theorem
was
integration sets
of
theory
positive
(Cantor-Lebesgue) F
, with
Haar
first
proved - e.g.
[Cne
Let
in~
m(X)>O
measure
n
1 2~
, e 7
-iny>
Cantor
[43,
(Lebesgue)
that
c
by
p.
for
55]
intervals
) and
general-
measure.
converge
. Then
there
pointwise is
to
a unique
zero
60
Proof.
From
Lemma
3.6.1
we
show
lim
r
=
O
.
n By
hypothesis,
Assuming
lim n
there
is
all
j
r
cos(nv+¢
) =
n
0
for
all
vEX
•
n
a sequence
{n.:j J
=
i,
2,...}
6>0
and
such
that
for
,
-r~<j.±ol ~
>
r
6>0
,
n.
J
we From
shall
get
a
contradiction.
(3.16)
(3.17)
for
(3.17)
all
yeX
lim j
cos(n.y+¢ j
lim
cos2(n.y+¢
from
Lebesgue
(3.18)
show
that
the
our
value
j
dominated
lim
t I
j
Jx
of
n.j
) =
0
convergence
cos2(njY+¢n
)dy
we
=
0
have
.
j
the
left-hand
side
of
(3.18)
is
non-zero
for
contradiction.
that
i
(3.19)
J
j
i -2 the
f
[ cos2(n ~+~n )dr = T J (1 + cos 2
On
0
implies
and
Note
) =
.
j
We
n. J
other
hand,
x
m(X) + i [ Z~x
taking
XX
cos
Jx
2
(njv+¢ n
)dy
•
j
• the
r
I
j
characteristic
function
of
t 2~
cos
2(n
J
V+¢n.)d
j
Y
=
cos
2¢n j ] 0
Xx(Y)
cos
2 n .Jy d v
-
X
•
61
f2~
2qbnj J 0
sin
so t h a t
by
the
j Combining
(3.19)
and
2njydy
n
)dT
= 0
j.
we
have
~x
J
contradicts
,
theorem
Jx
I cos2( n Y + ~ n
j which
f I cos 2 ( n j T + ¢
(3.20)
lim
sin
Riemann-Lebesgue
i lim 2
(3.20)
XX(Y)
)dY
= 2
m(X)>O
j
(3.18).
q.e.d.
Notation for
For
any
¥eU
any
continuous
h
Lemma I C
is
chosen
3.7.1
F , and
Let
G
assume
G
Proof.
is
For
on
all
c>O
where We
prove
an
open
set
U~
F
p
be
enough
a real
so
+ G(T-h)
that
continuous
- 2G(y)
and
[y-h,
,
y+h]
function
~
on
U
an o p e n
interval
that
linear
o (~r)
- G(y+h)
small
A2G(y,h) h2
lim h ÷ 0
then
on
, define
A2G(y,h)
where
G
function
I
for
each
Tsl
;
.
define
~ o(~r)
[a,b]
- 0
-
C
o(a)
I
~(b)-GIa)
b-a
(y-a)
+ c(y-~)(y-b)
,
62
G
(3.21)
and
G
C
then,
(y)
£:
< O,
(y)
for
[a,b]
yc
,
setting
~ G(y)
- G(b)-G(a) b-a
- G(a)
(y-a)
-
e(y-a)(y-b)
,
(y)>_O,
ya[a,b]
,b
we
Thus,
indicate
in
a similar
that
G
g
for
since
c(y-a)(b-y)
we
G(a)-
IG(y)-
and
since
(3.22)
so
that
is
arbitrary,
back
to
G
G
is
(b)
=
= G(a)
linear G
(3.21)
true
ye[a,b]
,
G(b)
G(a)
G (y)>O g
for
some
e>0
[a,b]
we
;
conclude
(y-a)
,
Consequently,
because
that
G
(a)
_ ~ - , b ,~- _n# ,a, b-a
there
= O
and
(h-a)
is
+
0
=
yoe(a,b)
0
, an
maximum. We
now
claim
that
there
exists A2Gc(Yo,h)
(3.23)
To
prove
lim h + 0
(3.23),
first
h2
note
A2G (~ S
h2
h(Yo-b)
0
that
,h)
~ D2G
by
direct
A2G(Y
g
(y) o
< O --
.
substitution
) 0
-
[a,b]
i
[a,b]
ye
all
on
first
-
< e(b-a) 2
- G(b)-G(@) b-a
linear
note
for
on
e
if
for
O(b)-G(a)(y-a)l b-a
is
G(y)
Getting
< e ( b - a ) 2,
have
(3.22)
Thus
fashion
h2
+ h2 + h2 -
g
+ -1~
(h(y
h(Yo-a ) -
0
-a)
+
h(Yo-b))
;
absolute
.
63
thus A2G(Yo,h)
A2G
h2
and,
by
+ 2e
D2G
(Y) E
D2G
g
(y
y
h2
,
hypothesis,
(3.24)
Since
(Yo,h)
-
2~
=
o
) e x i s t s f r o m ( 3 . 2 4 ) , and b e c a u s e G has a m a x i m u m at o g we h a v e (as a g e n e r a l p r o p e r t y of the S c h w a r z s e c o n d d e r i v -
o ative
D 2)
that
D2G
(3.25)
(y) g
this
is
clear
since
< 0 --
o
h2>O
and
;
G
(y
) , a maximum
, implies
o A2G g (Yo,h)
and
(3.24
G
-<-
(3.25) c
(y)
0
give
> 0
.
the
for
desired
some
contradiction
ye [ a , b ]
; and
to
thus
the
(3.21)
hypothesis
that
holds.
q.e.d.
Theorem zero
3.7
Let
E C
Let
T~[c
be
closed.
E
is
M set
an
÷+
there
is
a non-
TeA'(E) o
Proof.
(÷)
pointwise Without
to
0
loss
of
be
for
the
n
e iny sA'(E) o
each
generality
clearly
pointwise Recall
F
we
sufficient
kernel
prove
DN(Y)
~
that
to
[cn e inY
converges
= 0 assuming 0~E ; t h i s w i l l n g u a r a n t e e t h a t we h a v e the r e q u i r e d
for
(e.g.
[c
all
[
e
Inl<_~
and
N
y~E
§ 1. 5 )
iny (3.26)
show
y£E
convergence
Dirichlet
. We m u s t
sin
(N+~)y
=
, y # 0 sin
y/2
64
The
right-hand
(3.27)
side
of
N T cos(7/2 sin(x/2)
sin
(3.26)
)
is
+ cos
N Y sin(7,,,(,,,2.),- cos sin(y/2)
Ny
+ sin
N~
cot(y/2).
Further,
(3.28)
[ c n = Inl<j
by
definition
of
the
scalar
product
on
the
right-hand
side
of
(3.28).
Now,
Ic n
Also,
1
÷
]n I
as
0
CeA(r)
taking C
of
Hence
that A T@(n)
supp
T
from
¢(y)
with
TeA'(F)
and
cos
, we
N7>
Since
Ny>
CeA(F)
in
imply
(3.27)
E
is
0
as
N
a closed
sin
÷
neighborhood
Ny>
(e.g.
TceA'(F)
[3]
).
O
O
+ 0
+
have
+
Inl
~ as
and
Y
~ ~
@(y),
, and,
sin
Ny>
an
M
consequently,
+
0
as
N +
(3.28),
C + O in i<_N n
(+)
cos
,
= cot(T/2)
O{C
=
Thus,
implies
~
where
Recall
÷
set
there
as
[cn e
is
N -~ ~
in¥
such
that
some
e
# 0
and
n
~c
e
converges
pointwise
to
0
U Ik
on
n
{I k} By
Theorem
3.6
is
a countable
there
is
set
TeA'(F)
of
open
for
¢
, where
E =
intervals.
which
0
e
Hence,
we
need
only
prove
n
--
supp
i 2w
T ~
E
y
,e
-iny>
. As
72
F(y)
~ co
2
in
(3.9)
, en
-
[
77
in~f
e
we
let
U Ik
and
65
be
the
Riemann
function
(3.29)
for
lim v ÷ 0
for
all
and
therefore
tells by
yel#ljv
us
Cor.
' Thus
the
T = 0 3.5.1,
by
Riemann on
We
shall
v2
- 0
Lemma
3.7.1
(for E
all
, and
is
linear
principle
j) so
show
F
localization
I. J T C
supp
T
in we
the are
each
(Theorem
sense done
on
of #
once
we
I.
3.4) . Hence, prove
(3.29). We
have
A2F(y,2h)
=
c
o
(y+2h)2
2
' cn
-X
inYe2inh
+°o
C
A2F(Y
' 4h 2
C
t
Therefore,
C y2 o
-
+ 2['
~'~ n e iny
that
2h)
+ I
O
c o [ - 4h-~ 4h2+2hy-2hy
e
n in¥ 2h2n 2 e
for
w(x)
[
1-cos
(sin
]
' + [
2nh
]
x)/x
A2F(y'2h) 4h 2
(3.30)
c n iny [ -2inh 2inh)] 4h2n 2 e 2-(e +e
' iny + ~ Cne (2
= Co
Z
, ~ ~CnW(nh)e
-
iny
Define
iny TN( Y ) ~
[
InI~N Then,
T
0
(¥)
= c
0
, and
for
all
c e n
N
TN(Y) - TN_I(y) = c N e Consequently,
2
e
' ~-~ n e iny e - 2 i n h
so
(7-2h) 2
7~ e
iNy
+ c_Ne
-iN7
sin2nh)/(2n2h2).
=
66 oo
(3.31)
co
ICnW(nh)elnY -~
this of
follows
(3.30)
w(O)
= i
is
-
from
- TN_I(Y))w(Nh)
(-sin(Nh)) 2
convergent
the
right-hand
[ TN_I(Y)w(Nh) N=I
(3.30)
A2F(y,2h) 4h 2
(3.32)
I (TN(Y) N=I
(sin(-Nh)) 2 =
absolutely
so t h a t
[ w(Nh)TN(Y) N=0
and
since
+
(so
;
and
that
there
(3.31)
is
the is
series
no
problem
rearrangements).
about NOW,
= To (¥)
and
side
=
(3.31)
X TN(Y)(w(Nh) N=0
we
have
N-I I T (~) [ w ( n h ) n n=O
-
of
for
- w(h(N+l))]
any
N>I
- w(h(n+l))]
+
co
I Tn(Y) n=N Note
that
w'(x)
[w(nh)
= 2w(x)
w(x)
[
cos
- w(h(n+l))]
[ C O ~x X
x
sinx2 x ]
sin
x
x
x2
]
so t h a t
x cos =
= IN +
IN "
since
x - sin
x
x2
w(x)
and
x cos lim x ÷ 0
we
have
x - sin x2
x
cos =
lim w'(x) x + 0
lim x + O
= 0
and
x - x sin 2x
1 lw' (x) Idx
x - cos
x =
0
- C<~
lO
Thus, (n+l)h
co
f
I
Iw ( n h
- w(h(n+l))
Z I
I :
n=0
n=0
Since
(3.33)
C
is
independent
of
fIN1
h
(>O)
F~
I
'(x)dxt <_I lw'(x)Idx = C
J nh
,
< C sup n>N
ITn(Y) I
J0
;
67
Now
let
Then
yeUl
there
~
is
be N
fixed such
and
given
g>O
.
that
O
(3.34)
! ITn(Y) I < C
sup n>N --
since
[ c e in 1<_N n
in¥
O
- T
(y)
converges
to
0
as
by
N ÷
hy-
n
pothesis. On
the
other
hand
(3.35)
IN
lim h~O
because and
w(nh)
by
the
and
= 0
o
w(h(n+l))
definition
of
both
converge
to
i
as
h ÷ 0
IN O
Combining
(3.33),
(3.34)
and
(3.35)
gives
us
A2F(y,2h) (3.36)
lim h + 0 in
fact,
from
1
4h 2
1 <_ ~
(3.32),
A2F(y,2h) lim h ~ 0
;
1
N
4h 2
I =
lim h + 0
+1°I
I~ N
<_
o N
N O
lim h + 0
(fIN
I +
I~
O
lim h + 0
Since
(3.36)
is
=
I) <__
true
for
If N
lim h ~ 0
1 + O
all
~>O
IZN E ÷ o
lim h + O
tim h ÷
I[ °I <_ O
e =
,
A2F(~,2h) 2h
and
thus
with
v
lim ~ 0
~ 2h
4h 2
we
have
-
0
,
(3.29).
q.e.d.
68
Note s The
problem
dates
back
ization
of u n i q u e n e s s to R i e m a n n ;
of u n c o u n t a b l e
structure
of the
sight.
refer
sion
We
of what
set,
and
today,
U sets there
to B a r y ' s
is k n o w n
of r e p r e s e n t a t i o n
and
is
is no
although intimately such
for
it is k n o w n related
adequate
two
volume
for
references.
trigonometric
treatise
that
to the
series a character-
arithmetic
characterization
[2]
for
a detailed
in discus-
4.
Independent
Sets
4.1
Independent
and
It vious
that
strong
is
clear
the
and
Kronecker that
converse
independent.
Arithmetic
Sets
strong is n o t
Similarly,
of b o u n d e d
order
then
E
take
E
; in f a c t ,
if
YI''''' point
Yk aE
set
" then
{y}
is
independent
but
P r0position
4.1
dependent
÷+
Proof.
We
{0}~
E
is
is
not
leE
F
strong
E
independent
contain
is
strong
only
show
no
y
of
and
but
contains (for
order,
every order,
bounded
order.
order.
not ele-
, and
infinite
finite
ob-
example,
n>O
Trivially,
has
elements
and
independent
of b o u n d e d
if
independent
independent
independent
strong
is
are
is
nl
Let
need
true.
sets
+ ZOO. = 0 and n # 0 J strong independent if y has
not
it
independent
if
ments
~ {l,w})
Progressions
one and
E
is
is
in-
independent.
(÷)
Let
~ m . y . = O , a f i n i t e sum. B y h y p o t h e s i s J J Since y. is n o t of f i n i t e o r d e r m. = 0 . J J
m.y. J
= 0
.
J
q.e.d.
Theorem
4.1
a.
If
E
is
a finite
b.
If
E
is K r o n e c k e r
strong
independent
set
then
E
is
Kronecker.
Proof. Set
E
a.
Let
~ {YI''''' with
@eC(E),I@ I ~ i Yk }
and
integral
let
, and
Then
k y = ~n.y., i J O Further,
the
independent n. = m. J J
for
representation - for
if
all
j
E
is
strong
independent.
~>0
H = <E>
coefficents.
($.l)
Define
then
, the
group
each
ycH
generated is w r i t t e n
by
E
and
as
n.eZ J
(4.1) is u n i q u e s i n c e E k k y = Zn.y.jJ then 0 = _~(n--m.)Yjj J I i
is
strong
, and
so
70
:
H
{z~
+
:
Lzl
i}
=
r
=
by k
k H 1
@(y) ~ @(Zn.y.) 1 J J
Note
that
~(yj)
= ¢(yj)
and
that
k X = ~mjyj
if
k
n. J
[@(yj)]
then
( m . + n .)
=
~(~)~(~)
i
Thus
: H +
F
(i.e. to
is
a homomorphism,
~yeF,
Wn
# O,
and
keF
because
such
that
F nk
is
= y)
a divisible we
can
group
extend
a homomorphism
: F -+ F e
< [90, ~ 2.5.z] I. ^
Let
Fd
be
P
with
group Since
~
is
e
F
discrete
~eEZ
on
P
; recall
fundamental
topology
and
set
Z
E Fd
, the
it
~e(Yj)
property
of
the
is =
a continuous ¢(yj)
Bohr
for
character j = i,.°.,
compactification
on k
F
d
-
.
( [90,
~
Z C
dual
d
a character
i.e.
By t h e
of
the
§ 1.8.2]), O
Z
(a
continuous
imbedding)
and
Z
is
dense
in
Z
.
Now
m
N
is
an
- {f~z
open
: )f(Yj)
- ~e(Yj)I
neighborhood
of
@
by
j = 1 .....
the
k}
definition
of
the
dual
e
Since
topology
on
Z
from
compact
sets
of
F
7 = ~Z
we
have
F
d
and
because
the
finite
sets
are
the
d
N~Z
sup
l!j!k
# A
[e
Therefore iny. J -
there
~(yj) [ < e
is
.
n~Z
for
which
71
and
b.
Let
Tl'''''
¥ k CE
We
write
(n,y)
~ e inY
eral For
neZ
groups,
we
have,
#sC(E),
I$I
the
(y. ,n)
~[njYj
is
standard
homomorphic
n. i =
k ~
for
some
notation
for
By
nature
( n . y . ,n) 1
i=l
~ i
= 0
=
of
'
characters
on
gen-
( [ n . y . ,n)
1
hypothesis
characters,
1
there
lim
sup yeE
-~
l(m
,y)
= 1
1
is
{m
}~Z n
n
n.EZ3
§ B).
1
i=l
Let
(this e.g.
by
k H
(~.2)
assume
- ¢(Y)l
such
that
--
= 0
n
Then k
n.
i=l in
fact,
for
each
i
, n. 1
lim
(y. ,m 1
)
n. 1
=
¢(y.)
n
,
I
n
and
so k
lim
H
n
1
n.
(y.,m) 1
thus Therefore,
(4.3)
since
(4.3) n. 1
i.e.
follows
Hz.
= 1
n
from
holds , we
for
any
have
n.
The
n.
give the
= 0
1
k-tuple
in
= 0
for
= i,
etc.
{(Zl,...,
all
i
. For
Zk):Izjl~l} example,
-
take
i
and
zI
l
1
(4.2).
1
any
n.
I = gC(y.)
z2 =
. . . = Zk
strong
independence. q.e.d.
Remark set
1
in ~ ;
We in
~l
didn't fact,
....
we
use
the
only
, ~k ~E,
~,
full
strength
of
the
definition
needed:
I~1
- 1 on ~ l . . . . .
Yk'3<mn
)~z
of
Kronecker
72
such
(on m
k
E) ÷
that
2.
If
E
for
any
n
-~
, such
for
is
is
mk
with ¢
(4.4), such
only is
a
as
by
characters
there
to
is
that
sup yeE
whether
p
n
~
=
I¢l
@(yj)
~ 1 , and
¢(')
: mkeZ , k
=
i,...},
(Y,mk)]
0
.
# (n,.) mk + ~
or
=
is
Kronecker
this
0
, because
we
I = this
contradicts
means
are
our
that
really
there
dealing
hypothesis
that
Kronecker
sets
E
2 the
uniformly
where
such
-
E
, and
Remark
can
,.),
I¢(Y)
since
mk
on
of
we
(m ÷
{ mk
I¢(y)-(Y,mk)
many
Because
E
( m n , Y j)
CeC(E), is
sup yeE
not;
character
3.
there
lim + ~
assume
finitely
not
lira
that
k
prove
j,
Kronecker,
, then
(4.4)
To
each
m
question
arises
approximate +
n
~
(or
m
(on
+
n
for E)
-~)
a given
In
character
fact,
we
show
that
that
n
(4.5)
To
lim n ÷ ~
prove
Remark
(4.5), 2 take
first m
sup TeE
take
eZ,
m
n
I(P
@eC(E),
÷ ~
,Y)
-
ii
=
0
.
n
(say),
]$I
z i,
such
that
¢(.)
#
(n,.)
on
E
n
tim
sup
l(m ,~)
-
¢(~)J
=
o
.
n
n ÷
Let
E>O
and
let
~
N>O
TeE
have
the
I(mk,Y)
Thus,
Yn,m>N
and
property
-
@(Y)I
that
<
2
Yk>_N,
YyeE
,
"
YysE
i
] ( m n , Y ) - ( m m ' ¥ ) [ -< I ( m n ' Y ) - @ ( Y ) I + I ¢ ( ¥ ) - ( m m 'Y) ] < e ;
that
is,
Yn,m>_N
and
YyeE
. Then
by
73
[(m
Consequently,
Ym
take
N
n
-m
m
,y)
and
m
-
Pm
that
~ m.-mj k
Pm+l>Pm
4.
and
Pm+l
Pm
Theorem
measurable define
; for
~ 1
can
consider
uncountably
bly
many
we
(of
A)
E
can
take
indicates
on
]¢I
-
for
l[
< -1-
which
;
m
the
corresponding
such
~,k>N_ m + l
~
4.1a
characters CeC(E),
÷
we
<
j,k>N -- m
](mj-mk,¥)
set
i]
F
In
, such
a way fact,
that
many
for @(y)
such
consider
to
¢ -
get
any yeF set E imy # e for any
and
uncountably
uncountably
hence
many
for
non-
{y}
and
~
each
CeC(E))
many
m
(thus
of
uncounta-
Now,
the
were
it
we
e
be
corresponding
continuous
to
- by
this
contradicts
fact
that
the
the
~
@
is
usual
not
properties
hypothesis
extends
measurable,
that
of
for
if
Cauchy's imy # e
@(y)
it
functional for
any
would
equation;
m
and
the
¢
e
5. study
this
properties
4.2
Theorem result of
Examples In
non-Helson part
of
addition
in
of
sets
mod
is
more
shall
2w
introduce
(naturally),
G
that
§ 5,
as
theorem.
well
~
we
{kS
and
the
arithmetic
we
(4.6)
in
Kronecker's
Progressions see
involves
section
of
as
We
studying
shall further
sets.
Arithmetic we
a form
detail
independent
§ 4.3
this
4.1a
most
Independent natural
progressions. the
notation
define,
: k~Z,
~/w
for
Sets
means Thus
we'll O
for
need
to
construct
the in
first
§ 4.3.
,
irrational}
B +
(4.6+)
G
We'll = 8
see F
(kS
B
in
§ 5,
from
, and
thus
G
: k
=
a very =
F
.
1,2,...,
weak
form
B/w
irrational}
of
Kronecker's
theorem,
that
With
74
Next
define,
for
(4.7)
F
Proposition
Proof. Now,
4.2
Let for
F
8 =
positive
P~, q
e
i(q+q)8
e
: kgl,
a finite
(p,q)
integer
=
,
~ {kS
8
is
8,
Since
0~8~2w
set.
= I
and
multiples
28,...,
i2wp
we
8/we Q }
p/q
< 2
8,
mod
of
qS,
2w
, we
( q + l ) 8 .... ( q + q ) 8
have
(2q+I)8
=
8
have
•
again,
and
thus
F
has 8
a finite
number
of
terms.
q . e .d.
+ E {~+J8
~ {a+J8
: a,
8eF,
: ~,
j = O,
BeF,
+ - i,
arithmetic + + we w r i t e E 8 £ E0 ,P + for E 8 (resp., ES)
(resp.,two-sided = 0 things avoid
too
much
j = i,...,
N}
+ + - 2 .... , - N } ) progression); (resp.,
(resp., is
we
an
can
E
arithmetic have
"N
progression
= ~''
and
when
E
~ E ). W e s h a l l s a y a n d p r o v e 8 O,B + are true for E (resp. , E ), t o a,8 a,8
, which
notation. +
If
E
(resp. , E
contains
), k = i , . . . ,
countably and
many
arithmetic
progressions
E
if
8k
8k +
lim k (resp.,
then
we
two-sided tive)
if
elements,
say
that
infinite there and
is {N k
E
contains p rosressions
a sequence : k = 1,...,
card
E
=
I~k lim k
card
= ~)
,
Bk
infinite ).
E
E
progressions is
additive
(resp., (resp.,
{ y k } _C--E , c o n s i s t i n g Nk~N}
lim
N
k
such
=
that
of
E
contains
two-sided
pairwise
addi-
distinct
75
and
(4.8)
Yk,
{Yk,2Yk,...
where
the
finite
The
Proposition it
is
only
sets
+ + + { O , - Y k , - 2 7 k ..... - N k Y k } ~
(resp.,
in
(4.8)
also
consist
following
is
immediate.
4.3
a.
E
contains
infinite
b.
E
contains
two-sided
of p a i r w i s e
E)
distinct
progressions
if
and
,
elements.
only
if
additive.
if
it
is
Example
4.1
taking
Yk
two-sided
a.
E
F,
E
as
in
Observe shall
1 [
and
Note
progressions
if
and
: k = i,...}
Nk = k
that
F
is
additive.
This
is
clear
by
.
is
two-sided
additive
since,
for
example,
a.
that
see
that
for
no
need
to
really
infinite
additive.
~ {0,
= i/k!eE
b. E~
,NkYk}~ - E
two-sided
the
additive
purpose
of
distinguish
sets
are
constructing
between
additive.
In
non-Helson
additive
and
§ 4.3 we
sets
two-sided
there
is
additive
sets. We of
now
arithmetic
a geometric
Example
4.2
[0,2~)
by
E
is
vals
in
our
attention
progressions, procedure
We'll countably
strong At
turn
to
first many
to w h a t
viz.,
is
independent
construct
perfect
sets.
strong
construct
a general
steps
then
and
essentially
antithesis
We b e g i n
by
independent
type
choose
the
the
of
giving
sets.
Cantor
set
parameters
E
in
so t h a t
independent.
the
[0,2~)
first
;
at
step the
we
take
seoond
I 1l'
step
I2 1
we
' two
take
closed
--
disjoint
i'
12
inter2 1
3' 14~-- 12
'
2 w h e r e the I. are c l o s e d d i s j o i n t i n t e r v a l s ; at the k - t h s t e p we t a k e J ik k 2k-1 l''''' i k ' closed disjoint intervals, w h e r e for e a c h j = 1,..., 2 k
12j-i'
Ik
C
2j -
Ik-I
j
76
Ijk
such
Also
choose
the
easy
to
intuitively;
see
(4.9)
lim k÷~
that
l ik m
m ( I ~J)
precisely,
we
= 0 - this
are
max
{m(l~):j=l,..., J
and
set
asking
2 k}
is
easily
done
and
that
= 0
.
k
Next
let
E
k
2 q_]
=
j=l
k
i. J
(~.lO)
= 6
E
k
.
k=l
clearly
so
E
is p e r f e c t .
We
now
E
is
that
j=l,...,
2k
choose
the
independent.
(in
the
k I. of the a b o v e c o n s t r u c t i o n J In f a c t , w h a t w e ' l l do is to
appropriate
Ik-I
) such
in
such
choose
a way k I., J
that
n
%;7., 0 for
k y.~l., J J which
j=l,...,
Injl~k
2
k
and
, and
not
Yn.eZ, O
all
the
j=l,...,
n. J
are
2
0
k
, we
have
2k
(4.ll)
[n.~.#O j=l
Let's cally. vals we
Thus
such
have
do t h i s
given
that,
F
at the we
nlYl+n2x2
# 0
first
want
whenever
J J
to
and
choose
Inil
step
see
II i'
II 2
Inll+In21>0,
we
must
show
what ' n.z
that
it m e a n s
disjoint
closed
integers, 1 I.
geometri-
exist
and such
inter¥.eI~,l 1 that
1
the
block
I1 1 x I1 2
(4.12)
Note
that
in
2
nlYl+n2T2
the
equations
does
= 0,
of
not
Inil
(4.12)
intersect
~ i,
yield
any
Inll+]n21
the
four
of
the
hyperplanes
> 0
distinct
hyperplanes
77
Y2
>
Y
i
h
in
the
yi-Y2
axis
system.
Therefore
We
assume
that
closed
intervals
it
is
trivial
to
find
the
required
i I. i
ed
as
now
disjoint
i.
Yj=l .....
2 k-2
ii.
YYjelk-l'o •
Yn.sZo
whenever
Define
the
k-i I. ~ j=l,..., J satisfying the
the
k-i
I2j_l
,
j=l,
Inj l<_k-i
ik-I
,
. .
.
2
~ have
following
2j
~- -
,
2k-l,
and
k-i
I
been
construct-
properties:
k-2 j
}Tnjyj.#0
Zlnj I>O
•
block k
ik-i
z I
k-i i
x
ik-i i
Ik-i 2
x
x I
k-i 2
x...x
I
k-I k-i 2
x I
k-I k-i 2
~ • --
2
Take
k
I
k
~ II x
ik
2 x..
.I k
C
k --
lk-i
ik
'
2j-l'
k
k-i
12j ~I'--j
2 so t h a t
Ik
doesn't
k 2 Z n.y.=O
(4.13)
j=la Since
intersect
I
k
doesn't
any
, where
of
the
hyperplanes
~Injl>O,
Injl
.
j
intersect
any
of the
hyperplanes
of
(4.13)
we
have
78
for
j # m
; for
if n o t ,
then
yel k~l
3
k
n.y
and
so
7
belongs
check
have
the
is
k
Assume erty
E
property such
o
in
that
now
that
least
this
way
that
The
is,
of
that
ik
by
We
make
this
joint formed which
a set closed
of
k
have
(4.11)
Take
the
y.
are
in
3
n. = 0 3
above
for
thus,
of
disjoint
in
certain
from
any
in
above
that
of
any
was
on
the
# 0
k l.j
there
of
E
we
prop-
required
choice
where
of given
we
of
found
- tuples"
"totally
k
independent.
collection
k
n.aZ S
hyperplanes 4-5
need
further
, the
is t h e
the
we
let
the
strong
example 2
and
with
in P r o p .
"independent
anything
is
. Thus
components
k>k -- o ~nj~ E
rigorous
the
k
Y m aE
example
d o e s n 't i n t e r s e c t
intervals;
each
distinct
, and
elements
a contradiction.
hypothesis
geometrical
procedure
,
Y]'''''_
our
independent
neighborhoods were
hyperplane,
. By
each
-n
m
= O
the
SO
struct
=
n. # 0 . T a k i n g 3 ..... m} ~ k we h a v e
Ikq~ ik-1 (4,13).
-i
of t h e
that
crux
we
=
3
_~njYj
Yk>k -- o
one
one
n.
independent.
max{Injl:j=l
contradiction;
Remark
at
is
that
that
+ n y = O, m
to
Proceeding only
3
implies
m
J
con-
k dis-
blocks E 2k
in
dependent",
which and
i.e.,
hyperplanes. The
Proposition
following
4.4
well-known
Let
G~
result
F, m ( G ) > O
is
due
. Then
to
G-G
Steinhaus.
is
a neighborhood
of
O.
t
Set
Proof.
f(;~) =-
]XG(X)XG(k+y)dx J
Then ¢
f(X)
f
is
continuous
by
HSlder's
=
I
jGXG
(X+x)dv
inequality
and
the
fact
that
I I~ g-gl n a
as
a ~
O
, when
gcLl(F)
and
~
indicates a
Also,
if
f(1)>O
then
k~G-G
; this
is
clear,
for
if
translation by r I XG(I+~)dy>O
JG
÷ 0
i
a.
79
then Now,
f(O)
there
is
= m(G)>0
borhood
yeG ~ (X+G)
so t h a t ,
of
0
, and
and
since this
hence
f
is
does
l = y-z,
y,
continuous,
zeG
f>0
in
a neigh-
it.
q.e.d.
Proposition of
4.5
F . Then
Let
3
GI,... , G k
Yj.£Gj,
j=l,...,
be
k
non-empty
, such
disjoint
that
open
{Yl'''''
Yk }
subsets is
strong
independent.
Proof.
Take r
yl=~r, Now,
any
YiCGl
irrational
assume
that
which
and
such
is
of
that
{¥i'" "" 'Yj-I }
infinite
YIEGI
is
order;
in
fact,
just
take
y. sG. 1 1
and
.
strong
independent
where
j<_k Let
H. J
be
the
countable
ed by For
each
subgroup
I~H. J
and
(this is
~,p
Further
is
closed in
each
non-zero
the
for
that
that int
F
if
y
int -
,P
k,p
defining
the
property
of
y EF ~ y,p
, then
and
py
= A
. To
see
is
,P
divisible
= k
a
this
category
pact
regular
sets
is
G.
theorem
space
which
we
a countable
generat-
F
of
~
Prop.
if we
= 0
we
are
.
from
0
py
, since
and
4.4
assume so
every
a contradiction. us
that
in
a locally
intersection
of
open
dense
C
, properly,
comsub-
dense.
X0pC
Fk, p
is
is
open
and
yj.eG j - ~ -D .F
let
each
have
dense
and
CGj
consequently
~ xVpFx, p
C G
is not
ZniY i = 0 i
since
dense.
J i,p
Thus,
¥p
Z-{O},
p y .j~ H j
j Now
+ p¥
observe
V
tells
J Take
of
groups).
py for
a neighborhood
F
Baire
Therefore
2w)
define
group,
F F
pEZ
# A ; b u t t h e n if ysV we ~,P e l e m e n t of V is of b o u n d e d o r d e r ,
Recall
mod
~ {yeH.:py=l} j
÷ y,
a
a topological
note
addition
{YI'" " " 'Yj-I } "
F
F
(with
j-i . If
n. # 0 J
then
nj~j
=
-
[ i
niY
i
, a contradiction
gO
since
n.y.lH. J
strong
J
. Hence,
n.
J
= 0
, and
since
{YI'''''
Yj-i }
is
O
independent
we
have
n.
= 0
for
all
l
; that
is
1 {~i'
•.
"
' Yj
.}
is s t r o n g
independent.
q.e.d.
4.3
Arithmetic
Proposition
Prosressions
4.6
Let
~N
Then,
~
[
8eF,
Sets
B # 0 , and
c 8
define
Co=O , e . = i / j
for
j#O
YN
Proof.
¥n
we
calculate
^ ( ] >N "n" -
Note
~,
and N o n - H e l s o n
~'
i 2~
lJi<_~
-in¥ > T1 <~ J a+jB 'e
-
e -ins
1
~'
2~
lJliN
e
-ij~n
J
that
T
cos(-~Sn)
I j l<__~ It is w e l l
N
[
=
J known
COS
jSn
+
J
j =l
-1 [
cos
iBn J
j =-~
=
N
[ _
1
N
[ Cos
1
,iSn
J
=
O.
that N
sinj
t~
(4.15)
~71
<__ " ~ + i
j=l Thus,
from
(4.14),
^ I~N(n) I ~
1 2n
t
[
' (COS ~ n
i sin
j
J~n]T', < 2 w + 2 2~
w+l
since -i sin - N~
jBn J
N = -7 sin l~
,~n J
q.e.d.
81
Observe
that
if
6 has
infinite
order
{a+jS:j=0,
then
+i,...,
+N}
N
is
a set
Cor.
of
2N+l
4.6.1
consisting
Remark
distinct
If E c o n t a i n s of
distinct
1.
There
2.
zf
an
convergence
find
TsA'(E)
then
slight
such
progression
E is not
on
we
{a+jB:j=O,
+i,...}
Helson.
to
E
. Thus
(4.15),
have
by
e.g.
[57,
(4.15)
P.
and
22]
the
.
point-
3
[ sin. ~y J 1
of
I I~NI Ii = 2[ i i J
refinements
~ X s,,,,!nj~ 1
wise
and
arithmetic
points
are
f(~)
points
that
that
T = f
feL~(E)
, and,
in
. Thus
fact,
T
it
is p o s s i b l e
to
is
an u n b o u n d e d
Radon
measure.
Proposition
4.7
If
Proof.
prove
that
We
E
is
two-sided
YM>0
there
(4.16)
Given
is
then
W~(E)
E
such
is not
Helson.
that
II~il I > MII.II,, and
{N k }
{¥k } , f r o m
a constant From
additive
Prop.
4.6
3
the
definition
of t w o - s i d e d
additive,
and
M>O
k
o
and
{e
j
:IjI~N k
such
}
that
O
(4.17)
~ t c j t > M/2w IjI£N k
1
supt---~-w
and
n
~ I j I < N- -
c.e
t < 1 ;
k
O
in
-ijy k n o
O
fact,
each
k,
thus,
3
let
d.=i/j, j
{0,
~Yk'''''
k
for
j#O,
c =0, o
~NkYk}
which
is
(4.17)
c =--d j 7+1 a set
j of
'
and
note
distinct
that
for
elements;
holds.
O
(4.17)
implies
(4.16)
since
for
given
=
M
we
choose
c.6.
J J~k
lJ
o
o
q.e.d.
82
Remark
Kahane
constant
K
and
Salem
associated
with
N terms
contains
at m o s t
sets
do
not
this
more
than
sequences (and and
N k)
have
hence
{N k}
and
E
if
In
N terms
contains
Nk
more
if
is
Helson
is
a
progression
of
E
Notice
that
additive
is,
additive
sets
contain
is
additive
with
associated
E is
given,
; thus
by
hypothesis,
K log
there
arithmetic
K>O
than
then
any
that
fact,
{yk } , and K log
E
so t h a t
"log-property";
N points
that
that
E
K log
K log
such
proved
then
N k elements
we
choose
of
k
{Yk'''" ' NkTk} ~
of t h e
E
arithmetic
progression. We result,
shall
and,
later
for
now,
prove
a more
content
general
ourselves
form
with
the
of t h i s
Kahane-Salem
following
special
ver-
sion.
Theorem
4.2
If
E
such
Y I , . - . , Y k cE
has
that
the
distinct
Proof. tinct
property
Yk points
Given
k
we
in
E
and
appropriate
of
E
find
and
[ I~ k
, then
of
that
for
e. = O, O
i
each
k
there
are
2 k points
k y = [ e.y., j=l 0 0
(4.18)
are
the
E
the
is not
Helson.
YI''''' form
,
Yk
' let
Ek
be
the
set
of
dis-
(4.18).
~k~(Ek)
and
Ii = i
. This,
nk of
such course,
that
~k
tells
us
÷ O, that
I lUkl IA , ! ~k' E
is not
, the
largest
Helson. %
By
(4.18)
OeE k
• Also,
integer
less
for
than
given or
k
equal
we
set
k
~ [k/2]
k/2
to
%
Yj
= l,
2,...,
k
we
form
1
v2j-i ~ ~ (~ o +~ ~2j-1 +~ Y2j -~ (¥2j-l+Y2j) ) Obviously, We
now
11~2j_lLl I : i
and
supp ~2j-1
{0,~2j_I,Y2j,V2j_I+Y2j}~E.
prove 2w
i
83
For
nsZ
and
j=l,...,
^
k , an easy t r i g o n o m e t r i c
= I--~ e x p { ( - i n Y 2 j _ l ) / 2 }
[cos{(n72j_l)/2}
i exp(-inY2j)sin{(nY2j_l)/2}]] 1 4~
(Ic°s{(nY2a'-i)/2}
the "1/2 angle"
!
+ Isin{(nY2j-l)/2}l)
formulas
I sinai
/1-cos
o
a I
/l+cos
I cos ~ I =
/F
I
Therefore
Note
that
/1-cos
(1-cos
Since
CL
Isin ~l+lcos
Isin ~]+Icos
~] =
a + /l+cos
a < 2
a)+(l+cos
and hence
C~
~1 <_ ,,'7 we
we've
/1-cos
proved
a + /l+cos /~
because
~)+2/i-cos2a
= 2(l+sin
e) < 4
have
^ ' ' ~ ' ~ 2 . j - 1( n ) l
,/F <- 4~ -
1 2/7.~
(4.19).
Define
~k ~ V l ~ v3
"'" ~ v 2k-i
Then
(4.20)
gives
1
Iv2j_l(n)l
Recall
calculation
k
2w - nk
m=l
+
84
Since
supp(~
%# 68 ) =
~+e
,
k (4.21)
supp
the
set
supp Expanding
~k
~ {0, j--1
of
~k
~k
=
all
= Ek
(4.21)
Y2j-l'
sums
Y2j'
with
one
Y2j-I+'~2j } =
element
X { }j--1 J
,
from
each
{
}. D
6 +8
, we
have
; thus
"
' using
and
k
the
fact
that
~ ~6
e =
from
(4.18)
that
i ~k
-
k
~ TeE
2
hence Therefore,
e 6 , a = +i Y Y Y --
;
k
I l~kl Ii = 1
by
(4.20),
we
are
done.
q.e.d.
Remark
i.
Note
Helson.
In
fact,
the
elements
ratio
of
71=1/3,
for
of
Y2=i/32
applying
set
given
2.
any
by
4.2
k
are (of
we
can
constant k
constructing
an
easy
proof
that
F
is
not
choose
y., j = l , . . . , k , s u c h t h a t J taking powers of t h e d i s s e c t i o n
by
ratio);
. Another
for
proof
example,
that
a two-sided
F
additive
for is
any
not
set
k
Helson
in
r
let is and
4.7
Since
F
is
not
Helson
we've
shown
that
the
total
variation
Fourier
trans-
map
(4.22)
is
gives
distinct
Yk= i / 3
'''''
Prop.
Theorem
(4.18)
a Cantor
analogously
form
that
not
range (4.22) Fourier
~(r)
open space is
when has
not
the the
open
transform
domain sup
we
space
norm.
get
+ L~(Z)
has
the
Obviously,
A(F)
@ C(F)
if w e . To
see
map
LI(z)
÷ A(F)
,
start this
with
norm the
observe
and fact
that
the that the
85
where and
A(F)
such
open
that
the
A(F)
sup
p.
259]
Related
to
the
of
Theorem
c ,..., o
norm,
= C(F)
(e.g. [ 9 0 ,
proof and
has
4.2
CNe~
i-i,
. Thus,
),
continuous,
if
A(F)
that
gives
the
(4.22)
following
and
To
prove
~5
~ ~ ( 6 +6 L+6 -6 4 ) ... * o 2* 25 2 +25 ' '
we
do
in t h e
(4.20);
3. of
{yk }
E
not
Helson
÷ my
has
note
the
but
sup TeE
that
F
for
as
1~ 2
~c e n 0
map
have
we
also
note
result:
the
way
E
is
if that
To
m~ N
once
see since
E
in
again we
(4.22),
(4.22)
that
Ve>O
the ~ N>O
we
the
and
Yk.
~
J closed
Y and
as
Uk
and
I l~kI ll = 1 v. J
some
irrational
let
- 612)'
define
get
defined
is
is
I < £
~(E)
additive
y/w this
-iny
i ~ ~(6o+64+68
v3
elements
that by
property
E =
; mM~E
note
follows
Further
y
mYk.
and
(4.23)
we'd
classical
v I ~ ~(60+61+62-63),
theorem
thus,
open
N
form
dual
that
~I c I = 1 n 0
(4.23)
= C(F)
is
N
(4.231
with
a contradiction.
fact
such
is
limit
then
point
not
so t h a t
only
is
VmeZ,
additive.
Now,
by
J the is
weak dense
form in
F
With ization
of
tinct Without
Kronecker's and
elements
If
any
of
in P r o p .
J -
Nk-i 2
employed
in
§ 4.2,
{my:m=l,
2,...}
E = F
additive
E
k,
of
E
is
Nk
sets
we
on
4. 7 we
+ l,
additive
, and
. Also,
generality
measure As
to
theorem
get
the
following
slight
general-
4.7
4.8
Given
loss
thus
regard
Prop.
Proposition
Proof.
of
then
Yk
let
assume
E
; then
is n o n - H e l s o n .
{Yk'''''
NkY k} ~ E
are
dis-
M>O N
k
is
odd,
for,
if
not
we
place
our
{yk,...,(Nk-l)Yk} define
c.
j
-
{c.:j=l,..., j
~(~+i)
if
N
k
}
j = 1,...
such
'
that
Nk-i - - , 2
and
c. = 0 j
c.
j
-
for
4(~+i)
86
if
Nk-1 2
J -
Consequently
there
+ 2,...,
is
k
Nk
, again
o
as
in
Prop.
4.7,
for
which
MII~ k tl A, i li~ k El i o
o
where N
Uk
=
k
o
~ c.6. j=l J 3Yk
o
o q.e.d.
Example the
For
h.~
any
finite
sequence
Then
IIUnIll
there
is
= i
the
a subsequence
weak
~ i
that
~
then
~
if
topology. <~,¢>
F
=
{i,
n
y # 0
of
points
define
n
{i},
Example
h.h
it
Helson
is
open
as
there
are
many
U
Prop.
~.8
that
it
U
Further,
is
and
it
is
are
also
not
to
=
1 ~},
1
By
n ~(F)
the
Alaog!u
such ~ 0
and
and
clearly
theorem
that
F =
{0,
; in
~
m
~ n
fact,
I I~I I1 _< 1
that
because to
i,
~,...}
; consequently,
get
F3 =
whether
sets
and
= i
the
1 {~,
same
A 1 5' 6 } '
Helson
every
sets which aren't 1 {0, ~ : n = l , . . . } is
U
= ~F
}
then
again
proved
difficult
perfect
u({O})
can
1 {~,
F2 =
F
IlUl I1 = 1
'~>
~}
if
y
if
. Note
n
3'''''
we
F1 =
<~m
i
disjoint
n
n
2'
pairwise
{~
n
i
y = O,
and
distinct
n
. Let
Further
= lim n
; if
yeF
> 0 Wn --
and
m
if
n
of
n
measure
n
in
F
Helson.
. If
of
situation.
etc.,
we
have
are
set
is
For
U
the
but,
we
since
F
In 6
the
= 0 are
n
fact,
.
wide
other showed
sense,
hand in
is
countable,
!:n=l,...} is c o u n t a b l e n construct true pseudo-measures on
it
is
which
Helson,
in
example,
the
~ =
U-sets . On
~({y})
it
E =
not
U = 6
type
sets
Helson
and
{0,
aren't
Helson
[86]
From
what
E we've
S There done
,
87
just
take
fect E
Em
U
j
E
and
m
+mk
fact
m
~E
m,j
km
~ E m --
,
we
m)
- E
are
of U - s e t s .
let
that
that
y
decreases
m
to
O
and
place
m
per-
., j = l , . . . , m , in (Ym-l' Ym ) in s u c h a w a y that m,j c o n t a i n s at l e a s t m p o i n t s in a r i t h m e t i c progression. Then
union
fact,
such
E
{O}~J(UE
countable in
}C F --
m
U sets
-
m
{y
and and
U-sets
By
~m
our
have
therefore
possibly
have
by
construction
the
property
choose
an
Bary's
a
N
of
involved
on
E
that (of
m
result ,
the E
closed
is
additive;
~m+km , am+2km,...'
§ 4.2)
to be
doesn't
affect
another
such
m
The
the
utility
m
of
ProD.
4.8
and
4.4
Groups
Generated
by
The
following
result
H ~ <E>
Then
Proof. Note Fix
Let
that k
[ 90,
p.
FI
but
O
of
definition
of
for
ses.
Thus
E~
show
F
that
. By
which
) ,
V~
X to be uct
the with
set
true
be
for
strong
where
F3
Sets all
independent
independent
~ F 2 +F i ,...,
m ( F k)
Prop. V C
2k points
of
2k+2
note
sets.
and
F
= 0
4.4
Fk+l
for
all
k
V
be
an
, let
~ Fk+FI, ....
open
neighbor-
- F k
k
and
F2k
there
are
= F2k_I+FI
k-terms
= F2k_2+( FI+FI)=
inside
the
parenthe-
F
--
Take
Symmetric
actually
Fk-F k = Fk+F k
Fk+(FI+...+FI
and
F 2 ~ FI+FI,
m(Fk)>O
FI,
example
Helson.
--
By
As
.
We
0
is
Let
= 0
Fk
not
Independent
K E~(-E),
1 assume
hood
U
iO8]
m(H)
H =
and
is
is n o n - H e l s o n .
the
4.3
set
E
that
Theorem
Cantor
thus
(YI'''" ' Y 2 k + 2 ) e F x . . . x F
coordinates)
such
that
YieE
~ F 2k+2
(a p r o d -
and
2k+2
X sj~j~v i
For
each
for
some
set
of
for
some
choice
of
E.j = ±i,
(~1'
s. = ±i . 3 l<j<2k+2_ _ , with
f
the
Y2k+2 )cF 2k+2 , ¥jsE
.. " "
g
:
r
2k+2
+
F
property
, we
that
define
_~cjyj~V
the
function
88
(~l ..... f
g
's e x i s t
Clearly Hence,
f
since is
g
for
for
any
yeE,
continuous
each
e
~2k+2)""
(for
X~j~-j
y-y+...-y
since which
we
are
there
is
(with dealing an
f
), ~
empty,
and
is
contained
in
2k+2
terms)
is
OeV
with topological -i f (V) is open,
groups. non-
E
X .
Further,
x
[_Jf-l(v)
=
E
and Let
so
(YI'''''
X
is n o n - e m p t y
Y 2 k + 2 )CX
' Thus
and
there
open. is
{~.:j=l,..., D
2k+2,
E.=±I} J
such
2k+2 that
Therefore
~ I
~yj~F2k
, 2k+2
Z
p
~j
= X~-~., l J
l
where i#j We'll We
show
assume
we
have
and
that
assumed,
l~i, Yj
jJ/0
= iYi
skip Therefore,
the
if
S
that
no
next is n
is
two
the
a
~.~,
6.~z
J
J
some a set
loss
of
that l~i,
of
generality,
l~jl~2k j~2k+2
distinct
,
for
that
all
k.#k. , j z
l~j~0
. elements
(if n o t
we
can
and
p
such
and
so
steps).
set
is
without
Observe
for
{±y~:l~j<_2k+2}
1Lv~2k,
J
of y. j
integers , we
n
between
i
have
2k+2
X njYj i
By
the
Hence
way
we've
formed
things
-
all
X ~.x. J .3
--
o
j~S
the
¥~'
~i
are
distinct
n. = 0 for all j . But t h i s is a c o n t r a d i c t i o n since the J number of ~. of the f o r m y is l e s s t h a n 2 k + l ; so at l e a s t J P t w o of t h e ~. are of t h e f o r m ~. and thus non-zero. J J weVve shown that (YI'''" ' Y2k+2 ) as an e l e m e n t in F2k+2 belongs to
the
hyperplane
(in
F2k+2)
defined
by
the
equation
y_±y. = O. i J
89
Therefore
Finally
X
is
contained
Yj±Yi
= 0
F 2k+2
respectively)
we
shall
X° = A
see
any
union
of
i being
the
j-th
the
interior
desired
(which
fiEF,
njsZ,
g(k I,
and
that
, the
m(Fk)>O Take
(j
in the
all
of
and
j = i,
2
X to , and
12 ) = h i l l + n 2 1 2
, is
i-th
be
to
of
the
form
coordinates
a hyperplane
contradiction
implied
hyperplanes
is
of
empty
the
assumption
the
map
and
thus
that
open)
note
that
continuous;
g
assume
: r x F ~
that
F
,
nI
is
get
a con-
non-zero, g
I(0)
We
is
closed -i( g O)
assume
and
Fix
{(ll,
contains
tradiction; is
thus
and
a non-empty
thus
the
and
set
~
= n212
o
For
each
interior
set
open.
VIXV 2
and
of
the
hyperplane
is
a neighborhood
nlil+n212=O
~ Vl-V I ysW
so t h a t we
but
because
borhood
since
have
=
of
there
are
0
have
W
of
0
.
that
nlYl ,l-ni¥1
we
i
m(Vl)>O,
¥ 1 , 1.~V i
for
nlY
. V~eV
= - n 2 k 2 = -X o
nix
W
open
is
empty.
12sV 2
Let
X2) : n l X l + n 2 1 2 # O }
,2 =
elements the
X
o- X o =
of
0
;
infinite
desired
order
in
every
neigh-
contradiction.
q.e.d.
Remark
i.
It
2.
From
pendent
sets
subsets
of
basis then
used
(e.g. not
over
§ 6)
closed
is
(§ F
the this
easy
to
Theorem 4.2),
which
and, under
from
4.3
Rudin are
rationals fact
see
to
and
if
the
showed bases
but
finite
this,
he
that was
m(Fk)>O
fact
that
not
prove
complex
that
able
conjugation
with are
to
(that
some
there are
is not
sums
there
that
there
(this
for
k
are
perfect
perfect
meant
integer
independent
demonstrate is, ~ o ( F ) ~
then
H = F.
inde-
independent
to be
a Hamel
coefficents). non-Helson that
~
is not
He sets
(F) is o symmetric)
90
[87;
89] It
is n a t u r a l
to
with
the
property
that
Theorem
4.4
[43,
E
ed by
{~k
Proof.
Let
H
can
also
be
q
n
k
103]
Let
Y ~
f n k Y k, keS
be
fact
we
a perfect If
perfect
have
-
symmetric
~k + 0
sets
then
set
determin-
m(<E>)
= 0
.
of p o i n t s
S~
as the
q
non-independent
In
§ 2).
set
and
H
E
(e.g.
are
= 0
the
¢Z
write
if t h e r e
m(<E>)
: ~k ~ ( 0 ' I / 2 ) }
where We
p.
ask
I )n k ~ q ' keS
Yk ¢E
Z
is
a finite
set
of
elements
'
set.
q
(4.24)
where
the
that
the
any
n k >0
if we
Y =
I CkYk ' k=l
Yk
in
(4.24)
elements
n k<0
e k = O,
of
H
and write , write
aren't have
q
as
YkCE
,
necessarily the
as
nkY k
nkY k
±l,
distinct.
representation
¥k+...+yk
-¥k-...-yk
To
(4.24)
n k times;
nk
times.
prove take
similarly
Since
XlnkI~q
have P y = ~ CkYk • P<_q , k=l
and We
prove
let
m(H
ek = 0
) = 0
q
ously, Now
E = ~E k
<E>
k
for
for
each
-- U
H
where
the q
each
of
closed
k E., J
have
q
fixed
intervals.
Further,
J=l,...,
and
gives
; and
2
k
for
the
E k. j
terms.
m(<E>)
= 0
{ Ek: j = l , . . . , J for
given
k
= 2~l...~k
moment
also
keep
k
since,
2k }
, is
k m(E.) 0
We
. This
q-p
qq0k
Ek =
j=l
tion
remaining
fixed.
is
, the
obvi-
a colleclength
of
91
Consider
the
set
F~~ 'q
of p o i n t s
of the
form
q (h.25)
y =
[ si~ i, ¥ i ~ E k i=l Ji
where
e.=O, ±i and j.e{l,..., 1 1 q - t u p l e of Ji need not consist fixed form
Define
the
q-tuples (4.25)
of
as
j.'s 1 goes
y. z
2k},
i=l,...,
of d i s t i n c t
and
e.'s we 1 through E~ j. i
map
f
,
:
Fq
+
q
. Thus
elements,
consider
all
a given and
y's
for
of the
F q
(YI' • . "' Yq ) ' ~ Z ~ i y 1 Since
f
is a c o n t i n u o u s have
We
also
F k'q
observe
function
connected,
i
and
E k• x . . . x E k is c o n n e c t e d , J1 Jq t h e r e f o r e , an i n t e r v a l .
and,
that
(h.26)
m ( F ~ 'q) < .. J,e -- 2 ~ q ~ l "~k
This
we
is c l e a r
geometrically.
.
In f a c t ,
take
k {c2Y2:Y2eE.
}
and note
that
the
endpoints
{elYl:YleE~
} +
Jl of t h i s
k,~
inter-
02 val
are
precisely
the
sums
e l Y l , p + e 2 Y 2 , p'
w h e r e , for k E.Jl Thus manner Further,
for j
Note
that
any and
for vals
example, if
c i'
Yl,p c2=i'
(4.26)
becomes
k
y
, if
SlYl,n+e2Y2,n
is the n-~=2~l
left
endpoint
.. ' ~ k + 2 ~ l
of the
. "'~k
,
and
interval in this
transparent.
is w r i t t e n
as
(4.24)
then
y e F ~ 'q j,e
for
some
£ .
fixed
q
and
of the
form
a given F~ 'q J,e
k
there
are
at m o s t
3q(2k) q inter-
92
Thus
because
q -- .
for
,g
we
any
have
from
(4.26)
that
m ( H q ) -< 3q(2k)q2~q~l...~k
Consequently,
m(H
q
(2k)q~
and
~k
) = 0
..
l"
÷ O
since
=
~k
..
2kq~l"
=
~k
(2q~l)'''(2q~k)
.
q.e.d.
Remark
Recall
dissection clever
one
q = 2
of
for
±K ~)
= T
, m(K
proof
the
KI/4
the
for
measure
in f a c t
we
of
, we
-K
case,
m(H
if
we
that
a set
of
1/3
for
of q
) = 0
measure
[8,
4.4,
at
each
written
w k ~(3/4)
at
iiO]
and,
of
ternary
Thus,
if
though
the
(in
Steinhaus
forming
in t e r m s
of
the
step
k-th
a
(2k)q~l...~ k =
even
step
),
of
~2q
adapting
of
is
since
whenever
By
p.
rate
; there
in t e r m s
; therefore,
calculate be
constant
~ ~ < 1/2
example,
~ < 1/4
can't
with
(e.g.
Theorem
~ < 1/4
) = 0
case,
if
set
Steinhaus
proof
have
Cantor
= F
whenever
[0,2w)
have
the
to
1/3
in t h e
= O
KI/3-KI/3
due
the
that
~. ~ ~ J
, m(K
is K -K
fact
prove
Note
K
then
this
can
expansions. (2q~) k
if
~ < 1/2
proof
course,
that
KI/4)
KI/4-KI/4 and
;
so
oo
m(K1/~-K1/4)
= 2~- -~2 Z (3/4) k = 0 k=O
It t u r n s
out
that
T
. Sarat
and
basic
proved
m(K
-K
) = 0
whenever
~ < 1/3 [9]
Notes The to
theorems Kahane
and much
Rudin,
and
Salem,
results
beginning
particularly
related
work
in t h e
the
for with
the
their
important
1940's.
notions
discussed
Comptes
note [89]
Rendus • Of
in
§ 4 are
notes
course,
of
Salem
due
1956, did
•
5.
Kronecker's
Theorem
and
Kronecker
Sets
5.1
Dirichlet's
Theorem
and
Statement
of
Dirichlet's tion
theorems
which
Given I¥-ql<e lated and that
any
; this
but
and
ye [ 0 ,
leads
to
Ye>O
conclusion
(5.2)
Ye>O,
we
and
the
know
of
that
Diophantine
in h a r m o n i c
YE>0, ~qeQ are
dense
approximasynthesis.
, such
in the
that
reals.
A re-
given
strong
theorem
has
ful
when
q
r s
in
&s
y/2w
~
theorem
observe
2w),
= i .....
shall
that
the
irrational.
in
has
To
see
(5.2).
k
As
in we
question this,
rl 2~
ql
=
-
and
ms
and
see
,
we
the
prove
ap-
hypoth-
that
these
really y 2w
l I >--Imsl
two
Kronecker's
Kronecker's
assume
nsm__ --I
(5.1),
in B o h r ' s
is
YN
(5.2).
in w h i c h
origin
that
2w) I < E
significance
manner
and
shall
for
relate,
result
2w),
then I
.
deeper
~k e [ 0 ,
necessary
its
above
much
that
I(ny.-e.)(mod J J
special The
, such
2w) I < s
j=l,...,
is
the
work
k
that
neN
the
motivate,
analysis. our
is
for
independence
out
2w),~
¥ ~ i ' " "'"
uniformly
we
says
Iny.(mod J
uniformly
point
to
is
Yj
reals
Fourier
applies
Let's
0
follows
as w e l l
especially
Remark
that
linear
In w h a t
which
= i .... ~ k,
Kronecker's
such
then
theorem
V y 1 , . . . , yk E [ 0 ,
any
results,
question
are
useful
rationals
Yyl,... , 7kS [0,
approximate
proximate
--
since
quite
we
Dirichlet's
of
n>N
=
2w)
sophisticated
Yj
esis
find
theorems
Theorem
presents itself: given c>0 r ye [ O , 2w) w e w a n t to k n o w h o w s i m p l e q = -- can be c h o s e n so s s . The s t u d y of t h i s IY- [sI < e - t h a t is, h o w s m a l l can w e c h o o s e
(5.1)
Thus
shall
follows
more
question
The
we
Kronecker's
Kronecker's
theorem
research.
only
meaning-
= -n ; thus, m
if
94
the to
inequality our
follows
hypothesis.
for
Hence,
if
ns
if we
= mr
take
we'd have q = y/2~ , contrary i E -2 t h e n we can o n l y get S
I--Y-- - q l < e 2~
when --
--
<
s<m
2w
-ql
Ires I < l-~-
proximate
y/2~ The
it t e l l s
us
it
simple.
by
that
a
if
We
us
also
if
means
q # y/2~ result
within
tells
fact,
• This
following
approximated above,
; in
s>m --
that
is
the
E
a rational;
the
explicitly
irrational
then and,
approximating see
very
small
1-dimensional
is
that
for
i < -I s -- m
--
and
so
we
can
~
not
ap-
.
7/2~ by
, then
that
Dirichlet
theorem
any
_X__ 2w
for
in
fact,
rational
(5.1)
is
~>0, apropos
is
our
relatively
equivalent
and
can be question (to
e) of
to
this
type
r/ss [0,
2w),
(r,s)=l,
approximation.
Proposition
5.1
a. for
Let
There
y/2w
are
be
irrational
infinitely
x
Proof. (where,
b.
¥e>0,3
a.
Let
we
- ~I
We
all
-~-) 2~
2w)
s2
,
many
rationals
i.e.
Iys(mod
Inv(moa
that
consider
operations
: O
the are
2 - ~2~ -),
[22,
,
1
such
QeN
{ny(mod
<
_
heN
repeat, [0,
•
are
.
•
2w).
2w Thus,
3 s I,
s2~Q,
Sl~S 2,
sieN
, and
2w
(5.3)
n-~-- ~ s i Y ( m o d
s2>s I
Consequently,
and
set
there
s ~ s2-s I is
r
such
mod
for
that
~ n~Q-i
2~
; then
s!Q
that
]sy-2~rl
<
2=
Q
s
'
numbers
0,
y,
and
Q
intervals
the
2~ )--~--, 2=)
2=)<(n+l)-~,
;
2__z
~
2~) I < E .
2~)
distinct
t h e n t h e r e is r such r , a contradiction. n-m
2=)I
Q+I
, [(Q-1
n # m
(5,4)
[0,
which
1 2=
Let
in
if
i=l,
= 2wr
which
2
3y...,Qy
elements
n 7 = my(mod
(n-m)y
, for
. The
27,
2w)
which
of for
implies
95
to
see
(5.4)
note
n - ~2w - ! s2Y
from
(5.3)
2~
< (n+l 1--~--, 2w -(n+Z)
2w Q
hence,
and,
that
< sy
27 Q
<
2~
a
< -SlY
• (5.4)
<-n
is,
Q
of
course,
obvious
intuitively. From
(5.4)
and
since
sLQ
(5 5)
,
i y 2~
•
We
now
assume
there
exists
_ -
ri <
~
1
s
Qs -<
s ~
only
a finite
•
number
of
i = l .... , m Y 2w
Because
is
,
for
which
irrational
I Y
there
,
1
i
- --~ll < s. -l QeN such
is
r./s.
rationals
1
r.
9
s ~.
l that
for
i=l,...,m,
all
r.
I~ --~Is
>
For
this
Q
we
first so
repeat
the
inequality r --
that
is
above of
not
•
1
Q
argument
(5.5). any
--
to
Since
one
of
the
S
Thus,
there
can
b.
Given
e>O
(r,s)
i < ~ we
= i
m
the
~I i I Y2~ - s I < -Q
have
i=l,...,
and
.
S.
not
clusion
get
i --~ r. 1 --,
be
only
a finite
1 of
number
r/s
for
which
the
con-
a holds.
there
is
r/s
with
the
properties
2w/s<s
and
I Y
- !l~ 7
"i s2
, because
of ~
Therefore
take
n
S
~ s
q.e.d.
The proved
in
k-dimensional
a manner
Dirichlet
similar
to
theorem
Prop.
5.1
as
using
indicated
in
Qk
and
boxes
(5.1) Qk+l
is points
in k - s p a c e . We izes
(in
any
Ic [ 0 ,
rem
an
(which
ficult
than
now
prove
easy
way)
2w)
by
is
our
the
the the ny
main
1-dimensional Dirichlet not
just
concern)
1-dimensional
is
Kronecker
theorem 0
. The
in t h a t
as
we
different
opposed
to
this
now
k-dlmensional
conceptually
version,
theorem;
the
general-
approximate
Kronecker and
more
relation
theodifbe-
96
tween
the
k-dimensional
Proposition n=l,
5.2
Let
is
dense
2,...}
[0,
2w)
and
XE [ 0 ,
By
5.1,
there
is
{kny:k=O,
i,
Thus,
there
is
[ O,
in
Let
Consider
1-dimensional
y/2wE
Proof. Prop.
and
nsN
for
E>O such
irrational.
Then
{ny(mod
2w):
.
[ny(mod 2~)l<e
that
m}
where
which
IknT-~l
be
theorem.
2~)
2,...,
0
2w)
Dirichlet
mny<2~(m+l)ny
%~ [ X n y , ( k + l ) n y )
Inv(mod 2
<
)1
, and
hence
.
<
q.e .d.
Remark
i.
Observe
e i m Y # e inY
In
wise
7 2w
tion
that
in
[0,
in
gives
therefore
an
y/2w
it we
Prop. . We
is
elementary,
irrational have
m#n
tells
Prop.
5.1
the
"weak
is
note
that
form
Hardy
thorough,
and
1916)
able
then
ry#2~k
hypothesis;
straightforward
get
also
we
our
when
5.2
is
r=m-n>0
contradicting
, and
§ 4.2
if
if
e i m Y # e inY
2. to
fact,
_ k r
2w)
limit;
that
for
thus that
to
show
of
Kronecker's
lovely
neZ,
all
ksZ
erY#l
us
and
Ym,
{my} that
Wright's
The
has 0
-otherobserva-
limit
is
points
one
such
theorem"
Theory
treatment
m#n,
referred
of N u m b e r s
of K r o n e c k e r ' s
theo-
rem. 3. as
Weyl
a special
case
(we
define
this
the
1920's
(e.g.
ent
proofs
of
for
our
his
prove
on
uniform
a minute).
H.Bohr
Collected
Works,
theoretic
setting
in F o u r i e r
Bohr's
one
work,
of W e y l ' s
refer
although,
results
to [ 1 3 ;
Chapter
in his
for
[ O,
2~)
out
the
gave to be
sake
sequences dating
several quite As
(5.2)
from
differ-
important
such
we
shall
of p e r s p e c t i v e ,
Prop.
a more
of
research
analysis.
which
8]
theorem
distribution
turn
for
from
Kronecker's
Vol.lll)
which
now
Weyl's
see
in
results
to
theorem
on
We
his
was
Kronecker's
concentrate
corollary.
of
notion
measure
present
(in
5.2
is
thorough
an
we
immediate
treatment
of
contributions. Let
{X.} C J
--
[0,
2~)
I C ,
__
an
open
interval
and
NI
97
the
cardinality
of
{hi'''" , k N } ~ l N
lim
I
If,
i
-
N N
-~
for all
I ,
m(1)
2w
o~
then
{I.} is u n i f o r m l y d i s t r i b u t e d . C l e a r l y a u n i f o r m l y d i s t r i b u t e d J s e q u e n c e is dense. Thus Prop. 5.2 is i m m e d i a t e f r o m the f o l l o w i n g :
Proposition
5.3
tegrable
[ O, 2w)
on
a.
Let
y/(2w)
be
irrational.
Then
Vg
Riemann
in-
we have N
(5.6)
lim N
+
b.
~ ~
If
¥/(2~)
is u n i f o r m l y
distributed.
Proof.
convenience
b.
For
Let
IC
[0,
2w)
~ g(ny(mod n=l
we
be
2w))=
g(O)
is i r r a t i o n a l
drop
"mod 27"
an open
then
{ny(mod
2~):n=l,...}
in the proof.
interval
and set
g = X
N
Then
.~ T g ~-n ~ l
Thus,
i from
:
N
I ^
a and the
fact that
g(O)
-
I m(1) 27
--
, we have
{ny} u n i f o r m -
ly d i s t r i b u t e d .
a.
Let
By the
mcZ
be n o n - z e r o .
standard
trigonometric
inequalities
N I ~ eimn¥1
<_
n=l
Since
y/(2~)
is i r r a t i o n a l ,
a contradiction. m~ I < I CSC Consequently '
'
(my)/2
m#O
we have
holds.
2
1
# jw - for if it were,
y/2~
= j/mcQ,
and hence
2
N i imny lim ~ ~ e N ÷ ~ n=l
(5.7)
Since
Isin
(5.6)
for
g(~)~e
=
O,
imk
m~Z-{O}
If
m=O
g~l
and again
(5.6)
98
Hence,
by
linearity, by
the
we
get
(5.6)
Weierstrass
for
all
approximation
trigonometric
polynomials;
theorem
holds
(5.6)
for
and
all
g~c(r) Now,
if
g
is R i e m a n n
there
are
integrable
gl'
g2 ~C(F)
gl -< g -< g2
From
this
it
is
easy
to
see
and for
we
have
that
for
all
s>O
for
all
N>N
which
I r --27 j ( g 2 - g l )
and
that
real,
there
is
N
! < 6
such
"
that
O
i
~ j
i- -27 Thus
(5.6)
holds
ing
the
for
g
real
real
-
g
and
--
0
1 N ~g(ny) I < e N i
, and
complex
the
complex
cases
case
follows
by
consider-
separately.
q.e .do
Since linear
the
statement
independence
remarks
on
we
5.2)
shall
independence
make
before
only
follows
a number
giving
of
careful
with
some
hypothesis
of
supplementary
(to
statements
Kronecker's
of
§ 4)
theorem.
Remark if
i.
xeS
Let
SC
choose
E
and
n eZ
consider
such
that
SC
F
y ~ x+2wn
X
(5.8)
If
{w,
Xl,... , x
in
~
. Also,
in
E
then
independent
first
then,
by
part
of
(5.8)
hypothesis, Conversely,we
(5.9)
Let
the
E [0,
obvious
27)
way;
. Note
that
is,
that
X
m
}
{7+2Wko, X l + 2 W k l , . . .
The
in
is
each
if
is
{7,
r
is
strong
X l , . . . , Xm}
is
in
of
then
independent
strong
Xm}, as a s u b s e t
E
independent
F , is
strong
.
obvious. n.=O J
independent
, Xm+2Wkm}
{Xl,..., in
strong
and
For
the
this
second
does
part
let
In.x. = 2wn; J J
it.
have
Xl,...,
XmCE , yj~xj+2wn
x. J
~ [0,
2w)
in
~
. If
{y.} j
99
To
prove
which also
(5.9)
strong
independent
is
strong
independent.
let
nw+[njx.3
= 0
gives
~2n.x. = 0 in J O n = 0 . As a s p e c i a l
get
x I, . . . , x e [ O, m Also, we
is
have
dense
that
in
E
27 ) C with
if
SC
E
to
~
so
then
that
{7,
n.=O for J (5.9) we c o u l d
of
the
start.
the
relation
is u n b o u n d e d
between
then
{x:xS
Xl,...,
Xm}CE_
[2n.x.j0 = - 2 n 7
implying
case
from
regard
in
F
each
j
; thus
obviously
sets
in
in
E• we
take
E
and
F
is
dense
mod
2~}
that
{log
p:p
prime}
,
is
.
2.
One
a linearly
•
£
in
of
Bohr's
independent
key
observations
subset
of
E
; in
was fact,
is
if
m
In.log 1
p;
n.
we
have
by
the
apply
n
log
Hp. J = O and a fundamental theorem
Kronecker's
theorem
= o
J .
hence
Hp. J = I - this means a arithmetic. Bohr used this
of to
the
Riemann
that
each
n.=O 3 observation to
( function:
co
~(s)
3. elements
-- Z
1
1
n
We
know
of b o u n d e d
strong
independent.
is n o t
Kronecker.
and
- H(l+e s
¢6C(E)
had
from
Theorem
order
then
Thus For
the
s log
p+e-2S
log
p+...),
p prime
closed
EC
if
form
E
0
example,
4.1b
that
is n o t
or
7
if
Kronecker
belong
explicitly,
Czc#±l
; then
to
since E~
suppose for
[0, wsE
any
{m
,w) I = I c - c o s n
Further
note
pendent
then
n
w+~njyj
that
= 0,
if
EU{w} yjsE
n
O
or
or , then
o which
w
EU{o}
is
2wn
n.--O a
(since
it
is
not
2w)
then
E
el~sE~
F)
Z - -
,Ic+ll}
in
E
and
independent;
+~2n.y.
2n.=O) J
contains
> 0
--
are n o t
1 implies
m wl > m i n { I c - l l
F
(i.e.,
}C n
l%(w)-(m
.
p
= 0
hence
for
so t h a t
O 0 and
E
is
strong
example, Z2n.7. 3
n
o
~=0
0
= 0
indeif in
F
100
4. strong mable
As we
shall
independent
then
by
result
exponentials
gives
the
every
sufficient @
ing
that
limits
the
definition
on
% : E ÷
E
those
sets
is u n i f o r m l y
approximable
set
E
E
tells
in
and
the
fact
gives
the
necessary
are
every
us
that
is u n i f o r m l y
is f i n i t e ;
fact,
that
continuous
is n a t u r a l l y
on w h i c h by
theorem {z:IzI=l}
of e x p o n e n t i a l s
of K r o n e c k e r
following
if
condition
infinite
characterize
The
Kronecker's
if and o n l y
a discontinuous uniform
see,
motivated continuous
if
E
approxi-
Kronecker's
there
is a l w a y s
conditions, functions. by
is
not-
Thus,
seeking
to
¢ : E + {z:Izl=l}
exponentials.
results
are
four
} C--~
be
statements
of K r o n e c k e r ' s
theo-
rem.
Theorem
5.1
{el,...,
Let
Cn } C ~ _
{YI '''''
Yn
x~E Let
V{al'''''
an}C N_
V j = l .....
n
{Yl''''' and
any
any
Yn } ~ E
i be
strong
independent.
3 { m I .... , m n } C Z _
and
Then
3XEE
such
that
,
5.3
Let
{~'
e }g" ~ n --
(5.12)
YI' • "'' Yn } C-- E
5.4
and
Let
{w,
¥{~i ..... ~n} C:E_ Vj=l,..., n
and
be
strong
independent.
Then
for
we h a v e
sup m~Z
Finally,
(5.13)
for
Ixy.-a.-2wm. I < s O $ O
{el,...,
Theorem
i J
VE>0,
(5.11)
Theorem
Then
n iy.x n l~c.e 8 1 = ~ICj 1
sup
5.2
independent.
we h a v e
(5.lO)
Theorem
strong
this
n imy. I [ c.e Jl j=l 8
is
(5.2),
" ~ Yl' .. "' Yn } g-VE>0,
n = [Icjl i
3 {ml,...,
Im~j-~j-2wmjl
be m
n
< s
strong
independent.
}~
and
--
Z
m~Z
such
Then that
101
Remark
i.
We h a v e ,
as we
Kronecker's
theorem,
Theorem
and is
(5.9)
the
strong
the
hypothesis
independent
conclusion
Kronecker.
see
imy.
so t h a t
be w r i t t e n
Theorem {w,
ia.
5.2'
(resp.
3 x~E
Theorem
quivalent;
(resp., ~meZ)
give
Bohr's
for
by
close
that
only
Theorem
from
if
{w}UE~E
4.1a).
{Yl'''''
(5.8)
Further,
Yn }
is
i(my.-~.-2wm.)
if and
Theorem
>.2
5.4')
Let
=
le
e
only
a
if
le
e -ll
,
imy. i~. e-e eI
(resp.,Theorem
{YI'''''
independent. such
that
Then
5.4)
Vj=I,...,
< e (resp.,
proof
le
(via F e k e t e )
yn} CE _
V{~I''''' n
imy.
Jl
>.i
some
intimated now
÷
prove
is small.
can t h e r e f o r e
(resp., an}~ E--
and
,
i~.
J-e
e I < s)
of K r o n e c k e r ' s
theorem
,
the
to n for
for
this
x
b
Theorem
that
Theorem
5.1
- 5.4
are
somehow
5.2
if
By R e m a r k
~i=0
i we w a n t
to
. This
5.2
ixy. i~. e-e Jl
trigonometric
some
x and
gives
÷
e-
show
x
Ie
(5.10)
above it.
Theorem
(5.14)
but
the
i~.
e-e
We've let's
Theorem
that
if and
from
fact
. In fact,
of
5.1
2.
a.
holds
notation
e -zl
is s m a l l that
4.1a
a proof
that
e
strong
ixy.
§ 5.2 we
le
Theorem
be
Ie
In
note
given
as
Yl .... ' Yn })
Ya>O,
the
4.1a
i(my.-e.)
el =
Imy.-~.-2wm.l e o J to t h i s we see
Related
of T h e o r e m
just
already
in T h e o r e m
is p r e c i s e l y
this
J-e
le
earlier,
5.4,
(E b e i n g
(5.13)
To
noted
thus
(5.14)
Theorem
5.1
< ~,
j=l,...,
n
;
n-i~. iy.x 11+Ze Je e I is very polynomial 2 -ia. i y . x l+e Je J m u s t be v e r y c l o s e to 2 which
Let
gives
(5.11).
c.=r.e J
J
J
r .j->O
, and
let
E>O
102
By T h e o r e m j=l,...,
5.2'
n
(our
there
is
xeE
given
reals
such that
in this
case
le
are
ixy. -i~. J-e JI<£/(Zrj),
-~i'''''
-~n)"
Consequent-
ly ixy . Ic.ej O-rj I =
for
c.
j=l .....
The proof
identical
n ; and t h e r e f o r e
of the
to the
f r o m Theore m 5.2 can be
chosen 3.
{YI''''"
in
Note
Yn }
)c~l Ie
same
case
type
IZc.e J
ixy. J-~rj[
of e q u i v a l e n c e
of (5.10)
, say,
ixy . -is J-e J l < er-/(Zr k ) j
and
since
(5.11).
leiWX-ll<¢
of
,
which
(5.12)
Further,
gives
and
(5.13)
Theorem
by T h e o r e m
5.2'
(5.10).
is
5.h follows
and hence
x
Z . that
Theorem
be s t r o n g
5.2 is p r e c i s e l y
independent
in
~
equivalent
and let
{c
to:
,...,
let c }C
O
~ ;
n
then n
ixyj
sup ic +Zc e x~
We have
the
analogous
E U{O}
is Wik then ~.
tion
that
prove
5.4
. Thus,
E C
if
and
r
independent.
proved
seemingly a neat
such that
theorem
for the p r o b l e m s
the
converse
Z ~T ( nA) e i n
complements
the
p>l
Z¢(n)e i n Y -
is of course
to K r o n e c k e r ' s
theorem
(sic)
fields.
theorem
(1926)
[ 57, PP.
everywhere.
Carleson
converges
a.e.
to
¢
and,
in fact,
theorem
series
: there
latter
for
ap-
sev-
we only men-
of Fourier
59-61]
This
(and Hunt)
for d i o p h a n t i n e
At p r e s e n t
of it in the t h e o r y
diverges
important
crucial
we are d i s c u s s i n g ,
independent
application
Kolmogoroff's
, then
we've
Kronecker's
proximation, other
for T h e o r e m
h.lb
5.
eral
0
is strong
Obviously,
in T h e o r e m
o 1 J
situation E
n
i = [Icjl
result (1966):
is
is to
@iLl(F)
perfectly if ¢~LP(F),
103 5.2
The Bohr P r o o f The n o t i o n
nique,
and w i t h
ditions;
of K r o n e c k e r ' s of Riesz
that
idea
product
plays
are a s s o c i a t e d
so, let us b e g i n w i t h A Riesz-product
Theorem
has
and R e l a t e d a crucial
several
Estimates
role
in Bohr's
natural
arithmetic
techcon-
some d e f i n i t i o n s .
the
form n
(5.15)
where cos>-i
Rn(X)
on t h e i r
particularly singular
properties
useful
and let
represent
set
have b e e n in [ 5 7 ;
and each
¢.sF. Since J e x t e n s i v e l y and
studied
i16,
Fourier-Sieltjes
[0, y. J
(where [0, 2 ~ ) C ~ )
finite
set of reals
Volume
I] . They
transforms
are
of c e r t a i n
measures.
{vj:j=l .... }C_
x.=y. J J
~
are given
to c o m p u t e
(continuous) Let
ery
H ..(l+c°s(xjx+@j)) j=l
{x.:j=l,..., n}~E is a given J we have R >0 . Riesz p r o d u c t s
reports
group
-
be the
w h e n we
.
{Xl,... , Xn}
2w)
We
say
accessible
points
of
consider
it as an e l e m e n t
that
is
E
the f o l l o w i n g
i
in~
conditions
if
E~_F of the
for
ev-
hold:
n
(5.16)
- S.Xj=0, ~ 1 J
where
e.eZ J
..lejl!N,
and
implies
s'=O'0 j = l , . . . ,
n
;
n
(5.17)
3~e'xj=x k, w h e r e 1
Remark
From
(5.9)
{YI'''''
Yn}
is s t r o n g
clearly,
if
E
is
Motivated diophantine
l
and the d e f i n i t i o n
IN by the
lejl
of
independent
then
in
it is
F
then
definition
bound
of i n d e p e n d e n c e
~ inf
{
of
IN
implies
£.=Oj for j#k
set we have
{¥i''''' IN_ I
7n} in
is
that IN
if
; and
,
F
I
sets we also define i for the finite set {¥i'''''
the ~n } CF -
to be n
(5.18)
d
n
w h e r e we e x c l u d e For and r e c a l l
[ (~ x +xm) l:~j=0, -+l, m--0,.., j=l
all
zero sums
convenience that
J j
k, ~ --0} , '
in the r i g h t - h a n d
we state
in t r i g o n o m e t r i c
the
calculation
polynomials
o
side in
of the
of
(5.18).
(5.19) form
as a lemma,
104
m Z
P(x)
the
q. J
Lemma
are
frequencies
5.5.1
(5.19)
Given
R n (x)
of
~
c.e
j=l
O
when
a(O,...,o)=l it d o e s ,
if
R (x) n
a (e 1,... ,e
series
of the
)exp
in
,
[as in
(5.15))
ix(~iXl+...+~nXn
e. O
(5.19)
has
are n o n - z e r o
la
(5.20)
o
Then
)
n
; the k
c. #
,
J
product
e.=O,±l J where
J
P
a Riesz
=
ixq.
L
3n t e r m s
generally,
and,
then
i/2k
I =
(e I ..... a n ) Proof.
2 cos(x.x+¢.) J
Thus,
for
= exp
i(xjx+¢j)
O
• 101
e
ixx
- I•¢ I -
way
also
i(¢I+ ¢ 2 )
+
e
we
see
then
that
when
with
by t h i s
the
i@2 2
ixx 2 e
i(¢i-¢ 2) +
R
n
we
+
+
ix(-xl-x2 ) e
4
(x)
2
ix(xl-x 2 )
e i(_¢i-¢2 ) +
expand
-xx
e
4
ix(x2-xl)
we
-i~2 e + - - e 2
e
e
4
(5.19)
clear
e
i(¢2-¢i)
+
e
= i +
ix(xl+x2)
4
-ixxl
form It is
e
e
2
In t h i s
i
e
2
-
J
example
(l+oos(x!x+¢1)) (l+oos(x2x+¢2))
e
+ exp{-i(x.x+¢.)}
O
get
a series
of the
3n terms. calculation
corresponding
that
k
if
of t h e satisfies
a ( e l , . . . ,Cn)
a. are J (5.20).
0
q.e.d.
The where
we
analysis
note
step
the
problems
Proposition X.=y. J J
key
5.5
is the
in the
estimate with
Given
proof on
various
N
in
(5.22)
diophantine
{YI'''''
corresponding
of T h e o r e m
Yn } ~ -
element
of
is the
since
following
it r e l a t e s
result
our F o u r i e r
problems.
[O, E
1.1
2z)
) and
~ F
an
I 1 set
{ c . g ~ : j = l , .... J
n}
(where . Then
105
a.
<
¥s
1
3N
2
such
s
that
n (5.21)
n
i
1
For
N
any
ixx .
n
n
Zlcjl
1
1
2
> 3
n+2 d
n n
(5.22)
s
-- i
,
xgkO, °~)
N
al > [Icjl
J
sup IZc.e
b .
c.
ixx.
I c.e
sup x~ [ O , N ]
>
¥N
ixx.
1
JI
s,-u p "I I~ c.eJ xetO,NJ 1
n
>__~ ~.I c
jl
1
i¢. Proof.
Let
Also,
J
c .=r .e J J
, r.>O,a_ ~J~ [o, 2~)
, for
j=l,...,
n
.
define n ix.x ~ Zc.e J
P(x)
i
,
J
n
Rn(X)
,
--- I I ( l + c o s ( x x j + ¢ j ) ) 1
R
(x)
n,o
=
R
n
(x)-i
,
and
i(xx+¢) R
(5.23)
Note
that
if
n,j
.(x)
j=l,...,
~ 2R
n
n
J
(x)e
J
-i,
j=l,...,
n
.
then ixx.
(5.24)
Since
Rn(X)c.ej
YI''" " ' Yn term
in
is the
the
series
and
so
we
Ii
' we
series (5.19)
apply
J = !2 rj ( l + R n , j ( x ) )
see
(5.19) is
(5.16)
from is
constant
(5.16)
that
a(o,...,O
)
if
directly.
and
only
the ; in if
only fact,
constant a term
~ x +...+s x 1 1 n n
in =
0
106
Thus,
the
frequencies are
of
the
of
R
n,O
(x)
(when
written
as
a series,
of
course)
form n
~¢.x., s.=O, i J where We
now
From
at
least
one
give
the
frequencies
(5.19)
and
(5.24)
c
(5 • 25)
ix
2 - -m e
x
m R
(x)
rm
the
~( - ~k)~k
1 if
constant =
Sk=O
for
all
r
Therefore,
by
the
"-i"
(5.24),
term
m
a
m
is
the
a(
..
~i'
~exp .,e n ,
C ix X m m 2 -- e R (x) r n m (5.17) this latter "
of
by '
2c
where
e. is a l w a y s n o n - z e r o • J of R (x) , m > l n,m
, and
Ym
-+i
J
c [ 2-- m ~.=0,±i rm J
=
n
Consequently, n
J
k#m
Hence
the
in
the
m-th
constant
= i
of
R
(x)
term are
of
R
(el,...,~n)
of
where,again, Next
we
define
the
the
mean
, ej=O,
"-i"
value
is of
in
equality term
holds
the
±i,
form
the
except m-th
a function
is
is
0
n Xa.y.+y i J J m
; and
the
for
all
(0, .... 0 , - i , 0 , . . . , 0 ) ,
coordinate. F
: E +
~
on
[O,N]
~N 1
and We
we
estimate
MN(Rn,m)
1 F(t)dt
,
, m~l
have 1
(5.26)
rN I R 0 n,m
(t)dt
i =
N
CN I0
only
'
(x)
n,m n-tuples
when
coordinate.
n,m frequencies
occurs
constant
( 0 , . . . , 0 , - i , 0 , . . . ,0)
n +[¢,x~) m i ~
ix(x
~ b(¢ 1 . ¢n;m) exp ¢ . = 0 , -+I .... J
n it(Xm+~¢kXk)dt 1
to
be
107
where
the
(Sl,..., nate,
sum on the En)
right-hand
side
is not taken
~ (0,...,0,-i,0,...,0),
and where,
"-i"
over
in the m-th
coordi-
generally C - -
(5.27)
b(el,
this, Because
,en;m)
...
(e
from
(5.25).
m
of course,
of (5.17)
a
= 2 r m
follows
and since
we've
already
,...,e
1
dealt
n
)
with
the
constant
term
n
of
we have that n,m p e r f o r m i n g the i n t e g r a t i o n
Thus,
R
no f r e q u e n c y
x +[e.x. m I ~ J
of
R
n,m
is O.
in (5.26), n
iN(x +[~.x
)
m 1 J J
1 iN
(5.28)
R
0 n,m
(t)dt
i [b
= ~
e
(El,
-
.... en;m)
1
n
i(x +[~.x ) m 1 J Therefore,
because
(5.29)
of
J
(5.27),
IMN(Rn,m) i < 4 [la( --
. s I,
n . • ,e
and
4 (5.3o)
IMN(Rn,m ) I -< Nd n
where From
the
(el,...,en)
estimate
(5.20)
we
(5.31)
for all
,.. el
(el,...,e n (0,...,0,-I,0,...,0)
Thus,
. ..
I , '£n
compute
Is(
2 n-
el'
# (0,...,0,-i,0,...,0
)#
(~)2n-2(1)+
%la(
)I = 2n( "'en
... + ( k ) 2 n - k ( I ) 2 -
(
-i ) +
+.. .+1- 2i = 2n- i-2
m) 1 < 2 n+2 - 2 n,
n2 n-I
2
N ,
IM~(R
2nl ) +
108
and
(5.32)
IMN(R n
,m
2n+2_2 )i < - -Nd n
by (5.29) For
R
and (5.30),
a calculation
n,O
respectively.
similar
(5.33)
to that for
IM~(Rn,O)L
2
R
gives
n,m
2n+l-2
and 2n+I_2
(5.34)
IM~(R~,o)I i - - Nd n
From
(5.32),
(5.34)
and
(5.24) we have 2n+i_2
(5.35)
IMN(~
)l < i+
Nd
n
n
and
(5.36)
IMN(C.e j
ix.x i J R (x)) I > ~ r (i n j
2 n+2 2 ~d n
N
for
~=i,...,
n ,
since
i [ i N j0 2 r.dt J
i = 2 rJ
Therefore
(5.3~)
(I+
Now because
2 n+l N -2) sup fP(x) t h sup IP(x) I IMN(Rn)I n xe[O,N] xs[O,N]
of (5.24),
(5.32)
and
(5.36)
2n+2_2 ) Nd 2 IMN(Rn, j n
(5.38)
thus there
is
~.c~ J
such that
I <2n+2_2 ~ Nd n
h
IMN(PRn) I
109
2
(5.~9)
n+2
and,
from
-2
- M~(R
Nd
0
.) n,8
n
(5.38) ,
l~jl
=
2 IMN(Rn,j)I/(~)
n+2
-2 < 1
.
n
Consequently,
by
(5.24) ix
(5.40)
J R
j
(5.37)
(5.39),
.x
MN(C.e
Combining
and
and
n
i = 2 r (i-~.
(x))
(5.40)
j
(i+ 2
Nd
j
n
we h a v e
n+2 (5 • 41)
2n+2-2 ) ,
n I ~r
-2) sup Nd n xE[O,N]
2n+2_2) (i ..........
IP(x)t >_7 i J
~n
Now Nd
-(2n+2-2) n
Nd + ( 2 n + 2 - 2 ) n
and Nd - ( 2 n + 2 - 2 ) n
lim N ÷ ~
=
Nd
i
.
+(2n+2-2) n
Thus.
from
(5.41),
if
0<s<
i . 2
there
is
N
such S
and,hence, n 1 ~r.
IP(x) 1 >_ 7
sup xc[O,~)
For
s
i -- T4
we
choose
N
such 1
--
2
i J
that
<
Nd - 2 n
n+2
+2
Nd + 2 n + 2 - 2 n
YN>N S
sup IP(x)l ~ s [ r x~[0,N] J
(5.42)
that
110
and
so we get
(5.42)
for 2n+l+2n+2_3
N > 2
therefore,
take
N > 3
d 2
n
n+2 d
n q.e.d.
Before is strong Proof•
giving
the proof
independent
if and only
(of T h e o r e m
FN(X)
~
2
(i- ~jT)
5.1)
that
if
N=I
the Fej6r
that is
kernel
{yl,••.,yn} ~
IN
for
+
N )(eiNX+e -iNx)
=
(ei2X
+e
-i2x)
+...+(i- N+l
x+2(l-
)cos
F
As
for
(where
x. a
~) and
for
2x+.. • + 2 ( ~
cos NX
= i+(I- ~)(e
ix
+e
-ix)
is a g e n e r a l i z a t i o n
N
given is
y. 0
given
¥j,j=l,..., but
of the
.
J
x
typical
n , with
considered i¢.
c.=r.e J J
= l+cos
corresponding
as an element
we define
factor
the
of the
generalized
product n
N
R (x) s n
to Lemma
n
N>I.
then
so that
Comparable
all
F
(in E)
N l_l__)(eiX -ix) Z(l- ni~)einxN+l = l+(1- N+I +e -N
FI(X)
such,
5.1 note
{YI' • "''¥n }
if
Consider
1+2(1- N T I ) C O S
Note
of T h e o r e m
n s (x.x+~.)
5.5.1 we w r i t e
I~jI~N
N
j=l R
N
n
(~i ..... ~n
O
j
as
)exp
i x ( c l X l + ' ' ' + E n nX )
in R
n
.
xj=yj group Riesz
111
Since
{x. : j = l , . . . , n} J esis
of
is
Theorem
strong
5.1]
e.=O in ( 5 . 4 3 ) . J Consequently if some E.#O J
we
independent have
that
[from n [c.x.=0 i J J
Thus
1 2T
IT e x p J-T
and
this
F
: E +
for
we
M ( R N) n
where of Again,
for
R
ix(E
converges T
the
-
x +...+e x )dx 1 1 n n
to
0
define
as
M(F)
=
and
the
only
hypothif
each
T([eox.a )
T([c.x.) J
equality
'
T ÷ 1 lim --~ T + ~
~
FT I F J-T
T 12T [ RN = a ( n n ~ I-T n v,...,v,
lim T + ~
second
if
and
, sin
(5.44)
(5.9)
follows
and
see
that
= 1
trivially
from
the
definition
N
n
P(x)
n ix.x ~o.e a i J
=
,
n
M(PR ~) = (1---~i ) jr.
(5.~5)
n
1 J
i(xx +¢ ) To
prove
(5.45)
note
that
if we
consider
m
re
m
then
m n
iX(Xm+[S .x. ) 1 J J
i(XXm+¢ m ) (5.46)
RN(x)r e n m
and
by
=r
~j
the
b
1~41<j (~m
m
independence
x +[£.x.=0 m
j#m
Whenever
i lim --~ T ÷ ~
(5.47)
as When
some
e.=0 J
in for
(5.44). all
j#m
a.#O, J
j
j#m
FT
t J-T
;m) e .....
exp
ix(x
en
if
and
only
j
,
n +Ze.x.)dx m 1 3 J
= 0
,
if
all
~.=0, O
112 rm
(5.48)
where Because
i¢
fT
lim ~ T + ~
j
b -T
the
of the
dx
(O,...,-l,O,...;m)
"-i"
is the
way we've
m-th
defined
R
=
r
m
m
e
a(
0,.
..
,-i,0,...)
slot. N
,
n
-i¢ (5.49)
a(o, . . .,-1,0, . . .
Combining
(5.46)-(5.49)
) = e
and
summing
(as we
noted
over
m ( l - l_l_)N+l
m
we
get
(5.45). N
But
now,
since
FN>O_
quently,
¥~>0,
M(PR N) < s u p n -that
such
o
I P ( x ) I M ( R N) n
M ( R N) n
= i
(5.50)
is t r u e
that
and
we h a v e
for
all
Rn_>O
, and,
conse-
YT>T -- o
so, b e c a u s e
of
IT N j-TRn
(5.45)
and the
fact
,
x£ESup IP(x) I ~
(5.50)
§ 1.5)
i ~T N i -< I--~ J-T1P R n I _< xe[-T,T]sup IP(x) I - ~
M(pRN)-en
Thus
~ T
in
N
n i (i- ~-~)j[=ir.j
and h e n c e
(5.10)
follows. q.e.d.
Remark
The
closed
sets
F
are
Helson.
the
new
In fact,
Fk
is
countable
closed
(there
a small
5.3
is
Infinite
II
in
k
Kronecker 4.2
n
:n=l,...},
translate
r
so t h a t
support
extra
i
- {0,
if we
sets
In E x a m p l e
k
only
problem
for
Fk by
by
Prop.
k>_2 ,
an i r r a t i o n a l 5.5b
discontinuous the
k=2
and the
we fact
measures,
Fk
see
that
that is H e l s o n
case).
Sets and Prop.
4.5 we
indicated
a natural
method
to
113
construct we'll
infinite
construct
extension proof
can be r e f i n e d
and
a perfect
of K r o n e c k e r ' s
of Prop.
Theorem
strong
5.5
Vs>0,
5.5c
independent Kronecker theorem.
and by u s i n g
set.
E . By using
First
In fact, general
this
we must make
in a m a n n e r
Riesz
procedure
one f u r t h e r
similar
products,
to the
Theorem
5.1
to state: Let
{YI'" "''Yn'n} ~ E _
3 N>O
such
that
sup mE[O,N]
and,
sets
as in T h e o r e m
4.1,
be s t r o n g
independent.
V~>O
Then
VCl,... ,e e~ n n imy. n l~c.e e I ~ (l-s)~Icj I ; i 3 I
Ve>O,
such
3 N>O
that
Val,...,anSE,
3
me[O,N]
for w h i c h
le
The basic
construction
L. C a r l e s o n
although
perfect
sets,
Wik
imy . is . J-e J I<e,
at the
and the
of the
time
j=l .....
following
he was
argument
only
n
.
example
interested
to c o n s t r u c t
is due
to
in c o n s t r u c t i n g
Kronecker
sets
is due
to Rudin. Example
5.1
As
in E x a m p l e
4.2 we define
E
- A
Ek
,
i 2
k
and
k
12j-l,
k C
12j
-
i~-l j
j=l ....
2k-1
,
where
{I~:j=l,..., 2 k } is a c o l l e c t i o n of disjoint intervals. We now J i n d i c a t e the i n d u c t i v e step we need to form Ek given E k-I so that
E
will be K r o n e c k e r . Given
I k-l,j j=l, ... , 2 k-l,
and let
Jk2j_l, jk-12j~-- I k-IJ.
be two
114
disjoint
open
intervals.
By Prop.
4.5
{w}~{y~:j=l,..., J is
strong
k
there
independent is
N k >0
in
such
. We
now
for
each
interval
k
with
and
for k y. J
center
use
Theorem
a k¢~ 2
1 < ~
and
with
~ ns[O'Nk]
2k
the
sup
2
take
following
I~'-'~
k
1 ~
I <
for
for
each
which
k
I ~
J~
a closed
J
properties:
k iny. J I,, < 1 k
[e i n Y - e
k n~[O,Nk],~sl. J
sup
that
hence,
J --
(5-52)
(5.53)
such
5.5;
, ¥ j = l .....
j=l . . . . .
each
k
7.~J. 3 3
2k}
V~I,...,
that
k in7. i~. le J-e Jl
(5.51)
Now,
E
k
choose
(mod 2"n-)
J By
this
procedure To
let
prove
¢¢(0,i)
for
any
form
this
there !
s
E is
By
E
perfect
. Since
~>0
I¢(Y)-@(k)I
we
E
is
Kronecker
is
compact,
¢
6>0
so t h a t
if
(5.53)
there
is
j=l,..., in
a proper
Hence, closed
homeomorphic
y,
image
all
of
rem
and
keI.°, J the
of
I
circle.
each
each
j=l,...,
the
unit
C.C {z:Iz1=l} 2 --
i¢(¥)_¢(~)
unit
for
for
subset
k
any
j
is
at
Therefore
we
extend
of
most we
¢
can
from
tension
I. J
o
+ C. J ~e
of
Clearly,
Doing
¢
this
from
¥k>k --
E
for
to
and
[O,1]
and
~¢E
, k m(I .o) k J o) is O
use
the k E~l.° J
< 6
to
for
contained
follows k
in
a
since
¢(E ~I.°) J
Tietze
Thus
then
consequently
so
, and
continuous.
; this
¢<1
]@Izl
is
extension
a continuous
for
not theo-
func-
each j we f i n a l l y get a c o n t i n u o u s k 2°k U Ij O ' w h e r e l@e I~I j=l
, Vj=i,..., o
y,
such t h a t k 2 o ¢(E~I
circle
k tion
Iy-kl<6,
o
2 o
¢¢C(E),
is u n i f o r m l y
k
k
let
2 k,
¢ (y~) e
j
exists.
We
now
choose
ex-
115
k>k
with
o
the
(5.54) this
properties
k>B/e
sup {lCe(V)-¢e(y~)l:j=l
is t r i v i a l
this
k
such
that
to
we h a v e
do
since
from
¢
(5.51)
and
,. . . . . 2 k
is f i x e d e and b e c a u s e
and
sup j=l,...,2 k
this
n
(5.56)
and
any
yeE
l•e(Y)-einyI
k
l@e(Y~)-e
k y. J (5.54),
we u s e
follow tension
by of
Proposition pendent
<
continuous.
that
A I < k
3ne[O,N
(5.56)
Let
if
and
E
be
a finite
closed
+ i/k
< e ,
y ~ l ~ ; the t h r e e i n e q u a l i t i e s O (5.52), respectively. Since 0
an i n f i n i t e
number
Kronecker
first
step
of l i m i t
set
take
where
I. is the c e n t e r of D each I~ form disjoint J k.e int J (5.52),
k]
"
of
(5.56)
is an ex-
e
F~
E
closed
countable points with
closed Ii'''''
limit
disjoint
points
intervals
strong Xk
12 and j,l and (5.53)
I~ J closed
intervals
I~J,l'
2 lj,l
inde-
There ll,...,
II
.
i''"
In
For
are done.
5.6
At the
uniformly
;
k iny . l*e(Y~)-e OI +
I*e(Y)-*e(Y~) I +
(5.55),
we
set w i t h
an i n f i n i t e
Proof.
¢
in
< e/3
,
k iny . iny le 8-e I < s/3 + I/k
where
y E ! .k} J
lCelZl k inyj
(5.55) For
that
is I k.
II '
k
where
E ~int are
I~ 2#A , and w h e r e c o n d i t i o n s ( 5 . 5 1 ) , J, s a t i s f i e d ( m o d u l o the o b v i o u s n o t a t i o n a l
modifications). We n o w p r o c e e d each
precisely step
to
as in the
choose
the
proof "I's"
of E x a m p l e so t h a t
5.1 m a k i n g
each
"l"
sure
in
is s t i l l
included.
q.e .d .
116
As
a special
Proposition tally
5.7
is
arbitrary By
5.1
it
is
and
Theorem
4.3
we
is
Kroneeker
then
that
Kronecker
sets
abelian
group.
57,
as
Theorem
locally
compact
a judicious
use
now
result
ratios
such of
have
m(E)=O
and
hence
E
is
to-
he
C
X C
CE(E)
the
compact
If
which
subsets
existence
began
Let
such
that
be
the
is
of
certain
is
disconnected
theorem
~
(5.57)
his
to
0
in
[ la E x a m p l e
on such
sup
Kronecker.
Kaufman's sets
theorem
for
symmetric
totally
non-construc-
in
the
any
sets
disconnected
elements
E
of
are
group
cP(F), in w h i c h
O~p<~, the
.
of
category
is K r o n e c k e r
xs~
sets
Kronecker
perfect
a first
I@IE1
no
proved
a perfect
X
Kronecker
modification
real-valued
f(E) C
%cC(F),
there
with
E
be
totally
extension
of p e r f e c t
converged
(E)
set
Tietze
Katznelson's
dissection
let
the
are
show
Closed give
of
184-5 ] . Kaufman
5.6
Proof.
to
on the
pP.
and
for
E
true
5.8
We
[59;
not
routine
Proposition
tive
If
of
disconnected. It
an
case
in
, let
C(E)
set
set
. Then
and
for
in
[ O,
there
all
2~)
is
fsC~(E)-X,
E
X(¢,s)
be
the
set
of
fsC~(E)
that
l@(¥)-eiXf(Y)
I <
ysE
Note
that
X(@,s)
is
closed
in
C
(E)
. In
fact,
if
f
÷ f
in
n
f ~X(~,~)
, then
Yn,
Yx~E
n
ixf (v) (5.58)
sup ysE
thus~
supI¢(y)-e ycE
taking
any
xe~
ixf(~)I h Isupf¢(Y)-e ysE
n
I@(y)-e
and
any
I ~ s
E>r>O
ixf (y) n
I-supfe ycE
;
,
ixf (¥) n
-e
ixf(v)
fl
C(E),
117
for
all
large
n
an i n e q u a l i t y the Assuming
fact
have
which
that
f C (E)
r
maps
that
Hence, is
given
x,
independent
is a r b i t r a r y , E
YE>O
s u p lI@ ( y ') - e i'X f ( Y ) ~ c-r , yeE of n ; this, coupled with
gives
homeomorphically and
f~X(¢,~)
onto
a Kronecker
¥ @ s C ( f ( E ) ) , I@I~i,
le
sup
ixt
-
(t)
I
~ xgN
set we
for w h i c h
<
taf(E)
that
is
(5.59)
sup
le
ixf(y)
-~of(Y)l
<
TeE
Note
that
$of~C(E), @
defined
I@ofl~l
; and,
conversely,
is an e l e m e n t
with
f
I¢I~i, ~ For
the
converse
of
above, xE~
let
~,ygE
; then
the
feC
for
that
(E) of
I@I~l
unimodular
we h a v e
such
a homeomorphism Let
C(f(E)),
between
as
I¢I~i
from
E
; that
is,
elements
f
of
that
induces C(E)
VE>0
and
and
(5.57);
a Kronecker
we
show
f
set.
xmN
l*(~)-*(Y)l
~#T
take
¢ with
- I(eiXf(X)-*(x))+(*(Y)-eiXf(Y))l
@(X)~¢(T)
I ¢ ( X ) - @ ( T ) I > 2E;
by
and
E
(5.57)
a C(f(E)) .
~@~C(E),
holds.
assume
onto
(t)
(5.59)
(5.57)
and
-i
le i x f ( ~ ) -e i x f ( T ) I = le i x f ( ~ ) -¢(~)+~(~)-~(~)+¢(Y)-eixf(Y)l
If
, then
÷
tt~-P @of
Hence,
CsC(E),
by
: f(E)
bijection
if
so t h a t
let
x~
satisfy
: E ÷ R
is
118
leiXf(
Therefore
le
ixf(k)
injective Since
f
is
¢
ixf(y)
-e on
E
continuous
I>£
ixf(¥)
and
hence
!
f(k)#f(y)
.
; that
is,
f
is
. and
E
is c o m p a c t ,
the
bijection
f
: E + f(E)
is b i c o n t i n u o u s . Next
we
note
that
from
f(E)
the
fsC
(E) ly
that
there
in
C(E)
and
We
now
maps
if
prove
g¢CE(E)
is
this
and
C(f(E))
¢eC(E),
is
such
that
, given
q>O
dense (5.57) , and
is
immediate
a bijection
onto
I¢I~i,
in
C
f
the is
for
(note
set
which
that
(5.57)
and
unimodular
a homeomorphism.
a Kronecker
3 xs~
(E)
from
between
because
homeomorphically
X(¢,e)
x~E Let
E
¥¢>0 ~
Kronecker;
fact
functions Thus
is
if
and
(5.57)
on-
holds.
fcCX(@,e)
if
holds) set
x=lO/2 N
Because
E
is t o t a l l y
is
closed
E. J
(5.60)
so
sup
disconnected
and
that
(Ej}
for
is
each
[,(y)-,(x)l
¢(y)
= e take
can
write
a disjoint
E = ~,~E. j=l 3
collection;
where
further,
E. J
take
< ~/2,
sup
le
ixg(~)
-e
ixg(y)
I < ~/2
l,ycE. J
ia. J
for
some
l~jl, 16jl!~
ycE. and J for e a c h
let
e
ixg(v)
= e
i8. j
for
some
ycE.; J
j
Define
x
Clearly
f¢CE(E )
since
gEC
suplf(y)-g(~)l yeE
To
check
the
j
k,ysE, J Let
we
thus
f
is
that
f
satisfies
some
J
5
in
an
(E)
j
, and
<
sup Ix-iT-~l IAjiN
O
neighborhood
--
(5.57)
for
x
<
2__m~ <
--
X
of we
i0 X
-
n
;
g note
that
for
TeE. J
,
119
ia.
I¢(Y)-eiXf(Y)l
Thus
each
Now,
let
-i8. J)
l_elXg(Y)e
C
X(¢,E)
is o p e n
and
~. + 0 and c h o o s e J set of e l e m e n t s of
if
spaces
fiX,
fcCE(E)
such
that
= ~ + (e
are
dense,
C(E)
given
our
so t h a t
n
ix
Therefore,
,f(y) n,j
by
our
X(¢,E)
for
each
n,J
n
remarks
and
each
, and
e>0 Then
ix +
argument,
X
is
a first
j
there
is
cate-
x
.EE n,J
I < ej
supl¢ (T)-~(y)l<~. n j ycE
I ~ I ¢ ( Y ) - ¢ n (Y)I
a Kronecker
dense.
.f(y)
n
previous
is n o w h e r e
I@ I ~I) to be d e n s e in the ' n m o d u l u s I ( s u b s p a c e s of s e p a r a b l e
previous
~cC(E),l¢1~l
any
so
separable).
, then
TeE
Consequently,
and
with
ix
I¢(y)-e
iS. ixg(¥) J-e )
{ @ n : ¢ n ~C(E)
supl * (T)-e
¢
i~. i(a.-8.) J J J-e i x g ( v ) e I <
le
+
2
X ~ ~ _ ~ X ( @ n , e.) ; by J n,j g o r y set in C (E).
Define
Jl
(5.60).
metric
Now,
I~(Y) -e
•
L + 2
by
~
I¢ n ( y ) - e
f
maps
let for
e.<e/2 J ycE
.f(v) n,j
E
and t a k e
] < ~,+~. < J J
homeomorphically
onto
set.
q.e.d.
We n o w
Theorem
~.~
Proof.
Take
position
of
Then
V~>O
T
prove
Varopoulos' i m p o r t a n t
Kronecker
sets
TEA'(E)
and
(e.g.
there
is
are
let
sets
of s t r o n g
J and
[104,
spectral
{T.j~T@F ° : j = l , . . . , k }
§ 2). n.sl J
theorem
@.~ J
such
that
be
pp.3831-3834]
-
resolution.
a finite
decom-
120 k c - - + [llTjll A, 2
Since
E
is K r o n e c k e r
1
we have
k ie.^ O [ [ e T.(n
s
j=l
that
V6>O
l~(y)-e
sup
J
iny
3n
J
)l
for w h i c h
1 < 6 ,
~cE
k i@.
¢(~)
where
=
Xe ae
in.y J
XF
1
Consequently,
by T h e o r e m
2.7,
there
is
n
such
that
i~.^
le
S T . ( n . ) - T . ( n ) I ~ £/2k, J J J
j=l,...,
k
Therefore k
k i0.^
~tlTjlIA,-~ <_ lie 1
1
and thus Now,
by Prop.
J
J
1
I ITI Iv<~
(e.g.
m(E)=O
so that
5.7,
k^
£
~T (n)I- ~ <_ l~Tj(n)I : l~(n)l <_ llT1t A, ,
§ 2). Ta~(E)
because
of T h e o r e m
2.5. q.e.d.
Remark
In T h e o r e m
~I ITjl IA,=I ITI IA , cial
aspect
5.7 V a r o p o u l o s for any finite
of s h o w i n g
that
showed
a pseudo-measure
5.4
Wik
we've
Proposition Proof.
of
T
TeA'(E)
sets . The
is a m e a s u r e
The T a u b e r i a n
cru-
is to get
nature
of this
Sets
For the fact,
in K r o n e c k e r
decomposition
[IITjl type
that
Wik
sake
tacitly 5.9
of c o m p l e t e n e s s assumed
Every
implies
on several
Kronecker
Helson
we state
the
following
in
occasions.
set is Wik and every Wik
is obvious.
which,
set
is Helson.
121
For
E'
From
Kronecker,
§ 1,4
take
let
e>O
¢eC(E),
and
l¢I~l
psi(E)
, satisfy
111~Ill-I<~,~>l Since
E
is K r o n e c k e r
there
:is
n
such that
-in7
sup l~(¥)-e
I < ~/2
i <
Thus ^
^
la~u(n)-I 1~1 I x
-~+l 2
<_!. 1 2 ' ~ u ( n ) - ~ ( ¢ ) l
~II:L sup
+
I~(~)-II~llll
¢(y)-e-lny I < c ;
yeE
^
2~l~(n)l
but
:
II~IIA,-I Consequently,
I~tI 1
~1111
and so
< ~ .
I lUl IA,=I IUl II q.e.d.
Proposition E
is Wik;
Proof. As
>.i0 in fact,
From
always,
Let
E~
F
EU{0}
§ 1.4 we know
be countable
and strong
independent.
Then
is Wik. that
every
pe~(E)
IIWI[A,
is discontinuous.
E-{y I .... }
n
Let
~-~c.~ 1 J Yj
and
~ -~c.~ n 1 J Yj
; since
I IW-Wnl Ii = ~ Icjl n+l Since
{yl,...,yn } strong
is strong
independent
in
~
is d i s c o n t i n u o u s
+ 0
as
n +
independent
in
r
E . Thus,
we have
by K r o n e c k e r ' s
{w, yl,...Vn } theorem,
122
Ilull 1 !
Given
e>O
IIU-~nll i +
we choose
Thus ' because
N
IIWnll 1 =
such that
llW-Unll 1 +
Vn>N,
IIU-UnlII -> I I P - P n I IA,
n
Ii<~/2
-> I I l~I IA '-
I.nllA,I
IlUnlIA , ~ I[~IIA, when Combining
~/2
+
,
,
n>N
these
ly
I IP-~
llUnll A,
inequalities
11~11l! I[~I A' +e
gives
, and c o n s e q u e n t -
ll~ll I = II~IIA, q.e.d.
Remark
Wik I l l S ]
not K r o n e c k e r countable
given
(see E x a m p l e
strong
§ 4.4, we r e f e r Also,
has
Helson
sets
5.7,
latter
are U-sets.
is again proved.
sets
Kronecker sets
Further,
). Prop.
are H e l s o n , there sets
Wik set w h i c h
~.lO
tells
sets
(e.g.
non-Helson
of s t r o n g
§ 8);
spectral
it is not
U-sets
are t r i v i a l l y
of m e a s u r e
(1.7)
it is easy to
see that
from
is
us that
and as we m e n t i o n e d
are i n d e p e n d e n t
are
are U-sets
Since
of a p e r f e c t
and [ 5 8 ]
§ 6 to see that
from Theorem and t h e s e
>.2
independent to
lution
an e x a m p l e
known
in sets. resoif
O, P r o p . 5 . 7
if
E
is
^
Kronecker
and
Note
~E~c(E) that
then
in T h e o r e m E
lim(m n n
every
it is n a t u r a l
countable that
such
lim(m
is dense
{zE~:IzlS]1~II l}
in
4.1 we proved:
is s t r o n g
~l~l~l
As such
{p(n):neZ}
independent
on F, 3
{m } ~ Z n -,y) = ¢(y)
to say that
E
÷4
YF ~ E ,
such that
is u n i f o r m l y
finite, YyaF
independent
if for
closed n
FC E and V C a C ( F ) , I¢I~1, there is {m } ~ Z -n -,y) = ~(y) u n i f o r m l y on F . C l e a r l y , u n i f o r m l y inde-
n
pendent
sets
independent if
E
are s t r o n g
by the T i e t z e
is c o u n t a b l e
independent.
Thus,
it is n a t u r a l (or Helson)
independent;
then since
extension E
- noting
Rudin's
theorem.
is K r o n e c k e r
countable
to ask w h e t h e r
and K r o n e c k e r
sets
Further,
if and only
are u n i f o r m l y
it is o b v i o u s
if it is u n i f o r m l y
uniformly
independent
sets
every uniformly
independent
set is Wik
example
of a s t r o n g
that
independent
are Wik,
set w h i c h
123
is
not
then
Helson E LJ{O}
Example and
~.2
thus
shows
(e.g. is W i k
÷+
We
an
a Wik
that
§ 6).
give
set,
there
Finally, E
of
is n o t
positive
n
when
a countable
Kronecker
> 6
n
that
, s
E
is
countable
independent.
sequences
e
and
strong
example
which
are
is
observe
[84, {6
n
+ 0
n
strong
},
p. {~
n
independent
348] }
• To
such
set,
do t h i s
one
that
,
whenever
6
<
le
-e
I
n
<
e
n
,
then
1
inn <
Next, of is
choose
[0,
y
2w)).
strong
o
~F
with
Finally
independent
le
y /n o
we
iBn -e
1 I <
irrational
choose
y
(y
inductively
n
iY o
<
le
§ 8),
EUF
is
so t h a t
as {y
an o
element
,...,
y
n
}
iY n
-e
I<~
n
(e.g.
considered
and
6
Before
o
proving
that
Varopoulos
n
the
showed
union
that
if
of two E
is
Helson Wik
sets
and
F
is H e l s o n is H e l s o n
then
Helson.
Notes The
Bohr
ly u s e d
technique and
sis
by
Wik
sets
developed
Kahane is
to
and
due
to
prove for
Salem
Kronecker's
the
[43;
Varopoulos.
study 45]
of
• The
theorem
has
arithmetic deepest
been
sets
study
of
most
effective-
in F o u r i e r Kronecker
analyand
6.
Independent
Sets
6.1
Introduction A Rudin
ty.
In 1 9 6 0
§ 7 every in
1954
E
every
1954,
Related
We Salem's sets
set
of
serve
give
of
the
ing
that
surely
the
onto
proof
[105]
group
has
Kaufman the
Wiener Rudin
Rudin
exist.
As
we
sense
in the
from
On
the
an
§ 5.4 other
a large
in
class idea
a strong
of
not
In
set
abso-
of
and
so,
sets
are
of
alare
strong known
in-
wheth-
strict
§ 6.2
to
we
multi-
prove
symmetric
set
and
sets
such
transform
independent
in
result.
perfect
was
we
inde-
is not
this
§ 6.3.
(as
Pyateckii-Sapiro,
are of
this
E
Rudin
sets
in
strong
sense;
it
hand,
which
Rudin's
closed
Rudin that
exposition
proof
countable
wide
see
proved
sets
convergence, and
E
- Helson
every
multiplici-
shall
non-Helson
Also,
Kronecker
M-sets
into
strict
every
absolute
recall
that
of
still
a pre-
multiplicity. [60]
same
theorem,
that
both
multiplicity.
from
Rudin's
of
existence
[93]
though
a U-set
given
multiplicity
although
proved
has
result
strict
, is
of
wide
5.10).
a U-set.
Rudin's
Recently, rem,
is
[70]
strict
strict
this,
existence
Gehee
famous
are
set
the
Me
to
even
sets
that
the
sets
set
independent
(Prop.
are
course,
Helson
plicity;
§ 5.4);
such
in
strong
e.g. [ 4 3 ]
independent
that
a U-set are
sets
of
showed
is
is H e l s o n
Kroneeker
dependent.
a strong
showed
and
convergence,
not-noting,
is
there
§ 4.4
set
though
er
set
so
in
E
[89]
Helson
mentioned
lute
Rudin
- and
pendent
set
of M u l t i p l i c i t y
sets. every
sets
point
along
process
has
with
(A.30)
Finally
we
given
of view.
the
Also,
a series takes
of
of
of
Kahane
locally
Salem's
[47;
48;
results,
subsets
Varopoulos'
metrizable
definition
proof
related
certain
mention
non-discrete
(where
another
of
compact
independence
5 0 ] has by
F
rather
theo-
show-
almost difficult abelian
is m o d i f i e d
a bit). In worse,
6.2
§ 6.2
supply
Salem's
by
§ 6.3
a good
deal
we more
follow
[43]
but,
whether
for
better
or
for
detail.
Theorem
Recall mined
and
{n k}
that
if
, then
E there
is is
any
perfect
a natural
symmetric singular
set
in [ - w , w )
probability
deter-
measure
125
~c~(E) al
; in
fact,
derivative
of the
construction are
except
of
on [ - w , w )
f
for
a
classical we
, not
6
27))
the and
f ( ) =
is
continuous
(uniform) iation
limit
and
on of
~
is
f
f
all
@sC(F)
. ~y
that
convenience
~ we
is
-
the
271
~ cos j=l
-
271
for
each
the
§ A.1).
and
product by
~
take
that
E
k
; f
is the
of b o u n d e d
var-
[ 43,
pp.
14-15]
wnqlq2...nj_l(l-q.)j
singular
measure
for
E
, and
for
and
i bk < ~
we
w n q l q 2.. .nj_ l ( l - q j )
= w(-n,m)
~ [0,i] N
with
ordinary ak,
bk
i < bk < ~
~ ~ (~I,~2,...)sG
gk
a k >0
such
we
¢(~)df(y)
w(n,m)
considering
Next,
(6.1)
Since
now
,
is l i n e a r o f f of k a continuous function
R cos j=l
O < ak
and,
÷ [O,1]
f
Cantor-Lebesffue
= ~(-n)
induced
(e.g.
(although
the
m
(n)
Consider
[O,I]
. For
write
that
p
f
co
w(n,m)
measure
§ 2.1
[-~'~)
y~E~
calculation
^
Observe
of
distribution-
j
-= I J
a standard
u(n)
say
first
by
^
We
the
function
fk:
for
is
<~ ¢>
for
define
• and
k
defined
is
2k
[-w,~
the
~
notation
J
k
fk
7,
Cantor-Lebesgue
preserve
[0,
at
natural measure
probability on
each
that
, bk ÷ 0
, form
O
Lebesgue
such
,
~ ~ (~i,~2,...)
~ ak(l-~k)+bk~k
have
the
i ~
"
and
we
construct
the
perfect
126
symmetric tween the
set
~
and
space
each
E
determined
such
perfect
of t h e s e
of w h o s e
E
's.
elements
We n o w
X
define
n
: ~ + ~
{gk } . T h u s
symmetric
sets,
Consequently,
~ the
by
has
the
random
and
X
and
( ~,
form
we h a v e
p)
so we
a bijection induce
p
beonto
is a p r o b a b i l i t y
space
E
variables
: ~ ÷ ~, n e Z ,
n,m
m=l,
2,...
^
as
Xn ( ~ )
Lebesgue
~ ~(n)
and
singular
~ u~(n,m)
Xn,m(~)
measure
for
E
(and
is t h e
, where ~
and
Cantor-
are r e l a t e d
as
^
Using
(6.1)
and t h e
definition
of
u(n,m)
we
have
m
2WXn,m(~)
~ Y n ( m l ..... ~m)
= j=lHYn,j (ml' .... ~')j
where
Yn,j'(~l'
....
m')j ~ cos n~ [ a l ( 1 - ~ l ) + b l ~ l ]
...
[aj_l(l-~j_1)+hj_z~j_1] [Z-aj(Z-~j)-bj~j] .
Therefore,
from
the
definition
of
[ IXn,m(~)Imap~ (2w) m J~
Lemma
a.
-
For fl JO
Proof.
--
b.
e =l o
c.
C
a.
and
e
1 Cm
m
m
For
u=St
the
=
0 ( ,---- ),
o~]~, q+l
and
q
m
, a positive
I
2~
to
leos
ulmdu
~ c m
JO
0
as
m ~
of
(6.3)
-*
right
integer,
F2w
i
decreases
m
fl I IYn(~ 1 ..... ~ m ) I m d ~ l . - . d ~ m JO
1 ... JO
8>21T, q = [ B / 2 ~ ] ,
I Icos(~+13t) Imdt <
(6.3)
,
rl
I
(6.2)
p
hand
side
is
q+l q
above).
127
t6
i
(6.4)
2~(l+q) t
i
loos(~+u)Im~u 8 Jo
< --8
2w(l+q) r j
1 2~q
Icos(~+u)l m d u !
Icos(~+u)Imdu
0 where
the
Further,
inequalities since
q
follow
is
since
an i n t e g e r ,
q~6/2w~q+l
the
right
.
hand
side
of
(6.4)
equals
2~(l+q) ............ ,1
r 2 ~v
2~q
J0
icos(~+u)lmdu
+
1
[
2~q
j 2w
r4W
r2w
l I 2 ~ J0 b
is
c.
Ic°s
+
q 2~q
1 J2w
',,'lCOSt~+u)lmdu
=
q+l
=
m
q
clear.
First
note
that
tI c o s k t j
If
ul m d u
Icos(~+u)Imdu
m
is odd
dt = ~i cos k-i t sin
rw/2 ul~du
-- 2
I
J0
if
cos
m
.
r3w/2 u du
-
I
co~
m
u du
,
J~/2
is even,
t2~
r2~ Icos
ulmdu-
JO loss
m
J0
I
Without
I cos k-2 t dt j
then
[ 2w I c e s and
t + k-i k
I
cos
m
u du
.
JO
of g e n e r a l i t y
we
f2w i I cos2Pu 2w JO
du -
do the
calculation
for
m=2p,
p=l, . . . .
Thus
m-i raft=3 m-5 m m-2 m-4
"'"
re-(m-l) re-(m-2)
i m-i m-3 -2w m m-2
|" 2 W c o s 2 p - 4 J0
i) l) = (i- m (i- m-2
u
...(i-
du
=
i re-(m-2))
128
Now,
(_~)2<~--
(and
I- --~I
, and
so,
from
the
theory
of i n f i n i t e
p=l products ,
lim S2p p ÷ =
exp{
P [ 3=1
i__} 2j
exists. On the
other
e
2p
hand,
exp{
P [
P
j=1
~7 }
-~2p
so t h a t
since
we h a v e
the
i i 2 10g p } e x p { 2 l o g p}
[
i
j =i
2j
exp{
i . 1J _ l o g p) lim ~( p ÷ ~ J=l
existence
of
is 2 of E u l e r ' s
lim S2p/7 p ÷ ~
and h e n c e
of
,
constant
lim S2pJ2p. p ~ ~ q.e.d.
Using
the
Proposition
6.1
large)
m
let
notation
For have
given
each the
n>0
before
the
Lemma,
(and s t a r t i n g
property
that
for
Then
--
n al...ak_l(bk-ak)
there
is
with
C
, independent
of
n
n
-
sufficiently
l
n
(6.5)
we p r o v e
~ 2
(and
m
--
.
), s u c h
that
for
all
n
n
m
m ( iXn I n) 2 /m
n !
n (~(X)
is the
Proof.
Note
expectation
of
X
e.g.
§ A). ^
that
from
(6.6)
the
product
IXn(~)I < Ix --
for Also
formulas
all
n,m
and
~
.
for
(~)l n , m
'
~(n)
and
~(n,m)
,
129
X n,m (m) = X n,m-I (m)cos ~n~ I. "'~m-i (I-~ m )
(6.7)
where
~
a random
corresponds to
E
. Thus
cos wn$1...~m_l(l-~ m)
is
variable.
From (6.2), (6.7), and integrating first with respect to
(6.8)
~m ,
$ ( I X n , m Im) =
i .m( I fl[ m-i md~0m) .d~ (7) ]0" " "]0 1j=l H Yn ,j (~i ..... ~j )Im(fl ]oIYn,m(~l ..... ~m )I ] d~l'' m-l<--
sup I IYn,m(t°l,''-,~° m) Imdt°m~ ( IXn,m_ll m) 0<~.
Y n,m (~i'''" ,~m )
for
8---nw(b
m
=
cos(e+Sm m )
-a m ) [ al(1-~l)+b~l]
... [am_l(l-~m_l)+bm_l~m_l]
we have rl (I 1 IYn,m(~l,...,~m)Imd~ = 1 Icos(~+8~ )Imd~ 20 m JO m m
(6.9)
Because of (6.5) and for a fixed
n, B>2w, for --
~i=...=~0 m-1 = 0 , where
m=m n
Thus, by (6.9) and the Lemma, fl 2M O<~.
(6.10)
where
M
is a constant independent
Observe that from the definition
of
Xn,k,
of everything. IXn,kIk+l
,
130
(6.11) Also,
~(IXn,k Ik+l) i ~(IXn,k Ik)
we
perform
calculations
of (6.8)
example,
this
Consequently,
last
sup t e r m
by using
is less
(6.10)
~ ( I X n , k Ik )
for
we get the
than
or equal
to
2M//k,
l<_k<_m
(6.11), (2M) m
~"~(Ix n,m t TM) Therefore,
and
r1 sup I IYn,k(~l ..... ~ k ) I k d ~ k , l<_k<_m , r e s p e c t i v e l y . 0<w .
and
For
the
result
by
<
/m-~-!
- -
(6.6)
and setting
Cz2M q.e.d.
Theorem
6.1
Given
fixed
EE(O, ~)
, and
let
O<
ak
1 --, k=l,... ,
satisfy 1 1 . ½-2~ ! ak< 2
(6.12)
1 and
ak+
1 (k!) ~ _< b<
1
(k!) Then
almost
strict
Proof. k
every
, where
Note
that
for any
is d e f i n e d
by
(6.1),
is a set
ak
since
ak+(k!)-a
k
k2~ -I -c 1 i ~+(kI) < ~ + ~ .
al'..ak_l(bk-a k)
To see this,
for
k ,
(6.13)
first
observe
.2e-~ a.>__O , ~=i,..., J Thus
~k
of
multiplicity.
2~-~< ~ -s 2 -(k!)
Also,
E
from k-i
(6.12)
(k!)~-C that
bk- ak >__(k! )-e
and
large,
131
al'''ak_l(bk-a
k)
k-1 n
(k!) ~
.~-2e 0
j=l and
(k!)~_2e
.~-2E O
> k-I
j=l because k-i H j=l
Hence,
we
Next,
for
have
(6.13).
any
n
, let
(6.14)
m
j
be
n
[ (k-i)!
=
defined
[ (m + l ) ! ]
~-~
, a > 0
.
by
< -2 <
n
n
!) E-~
(m
--
n
i that est
is,
for
integer
0
, consider
which
From
this
we
Combining
(6.14) also
(6.13)
and
!
thus
(n/2)i/a>m --
and
(n/2) a -
take
m
get and
lim n
m =~ n
(6.14)
we
see
that
for
n
any
] a-~
< 2 --n<
a
1
...a
m
_
l(bm
n
We
shall
now
in
particular,
show
that
for m
First
note
that
if
n
(6.5) all
n
A7 /~ n
(6.16)
A>O
there
is
m
is
m
-a
n
'
)
n
holds. and
for
all
6¢(0,~)
,
[ - o(
(m n + i ) '
)' n ÷ ~
large m
A n
[(m ÷l),] 1-25( n
I/a
follows.
[(mn+l)X
and,
larg-
(m + l ) ! > ( n / 2 ) n
n
that
(6.15)
the
n
"
~
)2 _ A 2 126
n
2 (m "
m2 ~ n
n
+i)
1-25
<
n
such
132
A 1
2 •
where the
2 A 226
k
A2 k . ~•
,~°.•
satisfies
smallest
""
2 A (m _ j ) 2 ~ n
•
A2/k26>l,
integer
for
,.
2 A -~Tm (mn+l) n
" oe
A2/(k+l)28
which
i/j~28
,
and w h e r e
; thus,
(6.16)
j
follows
since
A i
is
independent
of
2 J . ,
2 A 28
---
k
n ,
A2J(m +1) n
lim mn(mn_l
n
is f i x e d ,
= A 2j
. . . . . ( m n - J + l ) ] 2~
and
2 A
lim n
(k+l)
2 A
.....
25
(m -j)
26
- 0
n
Next
observe
that 1-28 =
[ ( m n + l ) ! ] 6-~
(6.17)
in fact,
(6.15)
0
(n
i-2~),
n
÷
~
;
implies 1
([(mn+l),
] e-~)½-s
< (2)~-s --
Combining
(6.16)
and
(6.17)
we h a v e
m (6.18)
/~-7., -
n
that
for
all
A>0
1-26 l-2e
n
A
= K(I_)I-2~
n
o
(n
),
n ÷
n
Taking
any
r>O
we
use
Prop.6.!
, so for
A~rC
,
and
for
is
all
1 33 m m
[ (IrX n In)<--
n=l
where
the
convergence
~
n
<
Jm
n =I
follows
by
n
co
'
(6.18)
and
the
fact
that
i<(I-2~)/(1-2~) Thus
by
the
"triangle"
every
r>O
inequality,
; and,
= (~I rX
1
therefore,
by
m (m)l
the
n) d~
converges
definition
of the
for
Lebesgue
m
Hence
integral,
for
a given
r>O
for
p(S
r
)=I
each
r
there
, [IrXn(~) I n converges a.e. 1 is S C ~ w i t h the p r o p e r t i e s r --
that
and m n
¥~S
, lim
< i
IrXn(~) 1
;
r n
consequently,
(6.19)
¥~eSr,
lim n
.
A
Now,
p( ~ ~ S k) k=l
= 1
; and
from
(6.19),
lim
Ix (~)l
for gives
all lim
~e~S k=l
k
IX (~)I
and
= 0
< 1
n
n
This
IrXn(~) I _< i
for
for
-- k
all
k
almost
, and
every
~0(E~)
so
n n
q.e.d.
Remark
i.
In
Lemma
c we
have
actually
shown
that
lima
/m
exists,
limit.
See
m n
and,
modulo
also
[ 93,
2. tially
the
an
p.
explicit
537]
Salem
formula
or
two,
have
computed
the
•
has
conditions
given on
the
refinements ak
and
of bk
Theorem are
sharp
6.1
so
[ 43,
that p.
essen-
102 ] .
134
6.3
The
Existence
Let
E
of
be
Rudin
Sets
a perfect
symmetric
set
determined
by
{~k } , and, k
with
the
m(E~)0
notation
= 2~
of
"'~k
§ 2.!,
A
we
have
E = ~Ek k=l
T - translation
of
, Ek =
E
is
0 E~k O j=l
,
a sequence
and
{TkE k}
i'
where
kEk
k k k T.E. - { y a F : y = x j + k , 3 J Cantor-Lebesgue singular where
2 ~
k k
k k k J J
j=l
and
on
2 k}
kt-~ keEL| |E} • A T - translation of W , ~ the J m e a s u r e for E , is a s e q u e n c e of m e a s u r e s
and
Tk~ k }
- { T .k E .k: j = I , . . . , J J
k ~-'---~@j k ; h e r e @ keA(F) E. E. J 3 Ek k k p, p~j , and T.~. is the
is
equal
to
i
for
each
¢~C(F)
translation
k ~.
of
since
With pothesis
will
Proposition that
for
(6.21)
k k T ~
the be
.6...~2
any
~0 above
+>
and
I I~I Ii=i
notation we
converges
in the
< ck
J
J
k k T ~ S0
have
using
Theorem
. Then E
II
so
that
6.1,
Ill =l
the
•
hy-
there
is
{ek:ek>O},
Zek<~
, such
satisfying
j=l,...
2k+l
and
p =
topology
, 2
"
~
Tk k
and
prove
of
"
weak
J
we
and
~e~o(E)
T - translation
I~.k+i--Tpkl J
J
i +(~++k-)d~k.(~)
:
JF
satisfied,
Let
0
t
k k
that
to
,
J J
Note
equal
k k T.E.
to
J
(6.20)
and
J
J J
Thus,
k E.
on
to
TWe~
(TE)
, for
some
wE~F.
135
Proof.
We
~a e imY
first
then
note
k k ~ ~J a ~ o ( E ~ E - ) J
that
~k(n ) _
12w <~'
Za em
; in fact,
i(m-n)¥>
if
^
= 2amP(n-m)
k ( Y ) --E. O , so that since
m
m
Zlaml<~
and
~e~o(E)
lim
^k u.(n)
m
,
= [a
lim
~(n-m)
= 0
m In) ÷
Let
k k k ~ -T ~
Clearly,
from
k k k 9.-T.~. J J J
and
(6.20)
, k -in~ . ^
^k v.(n)
=
e
O :_(n )k-
J
J
and 2k ^k
(n)
=
[ j=l
Consequently,
k p.e~ 3
because
0
~k .(n) D
,
(Ek~E) J ^
(6.22)
Note
that
lim
for
each
j=l,...,
2k
~k(n)
and
for
^k ^k+l . u.(n)j = ~ 2 j _ l (n)
this
is i m m e d i a t e
Ek+ICEk ej
Thus
for
each
-
k
from
the
= 0
each
n
^k+l, + ~2j ~n)
construction
;
of
E
j
and
n
^k+l (n) v
- ~k(n)
=
2 k+l _inT k+l Z e J ~k+l(n ) j=l J
since
Ek+l 2j-l'
136
2k _in~ k
(6.23)
2k + l
m
[ e
^k
U (n)
=
_ i n T k+l.
~
J
e
^k+l,
~.
m
[ e
the
last
sum
in
(6.23)
2 k _inT k m:^k+l , [ e [~2m-1 [n) m=l
Combining
this
with
~k+l(n)
(6.23)
terms
2 k+l e
=
j=l
=
w j k + l e ~ o ( E k + l ~.E .) j N
for w h i c h
X
k+l -inT. (e J
, (6.24)
--inT
e
implies
l~k+l(n)-~k(n)l<
that
Nk
really
since
we
take
e
depends
choose
e
only
in t e r m s
k
for
InI
that if
k,
E,
and
of
N
k
for
each
p--
2 a"
'
k
there
~ ; this
; in f a c t
is i m p o r t a n t
for
a given
and u s i n g
k
2TM
I ][
(e
i
< k
Nk22k+2
the
(real)
_ i n T k+l. J
mean
value
theorem,
-inT k ^k+l, P)I~. (n)l
e
j=l
a
< --
2k+l k+l
211j
II 1
[
k+l J
k P
iNk2k+l --
is
InI~N k
so t h a t
k
g
Hence,
on
k
P)~k+l(n) j
1 2k
k Note
for
_inx k P ~k+l(n) 8
j=l
Because
and t h a t
gives
~k(n)
-
^k+l.
2k+l
has
^k+l, + ~2m in)]
2k+l
(6.2~)
, ~
~2m-1 ~nJ + ~2m
m=l that
-
j =i
2 k --inT k m~^k+l
Notice
[n)
J
m=l
I 2
k
137
Therefore,
for
each
k
and
(6.25)
n
,
i~k+l(n)
_ ;k(n)l
<
1 2k ^
From
(6.25)
n
and
the
; and,
since ble
fact
that
<=,
Ii=l
exponentials
limit
argument
lim
^
vk(n)-v(n)
exists
for
each
2
l lvkl
because
the
~
, T~=VC~(TE)
are
- they
total are
in
for
C(P)
either
some
and by
easy
or
an
TE~
P ,
easy
dou-
impossible.
^
Finally,
we
show
lim
v(n)=0
IOk+m(n)
- ~k(n)l
. In
fact,
any
m-i
<
k,
n,
and
m
k+m l
j =0
and
for
:
Z
2 k+j
1
j =k
2j
so
Igk+m(n)
lim
~k(n)l ~
m ÷ ~
[
1
j=k
2j
Thus
1O(n)t
_<
Z
Consequently,
given
+ I v^ k ( n ) l
1
j=k
2j
s>O
we
choose
k
such
that k
and
for
this
l~k(n)l<
k
we
C
F
from
take
(6.22).
Mk
so t h a t
Hence,
for
for
all
In}>M k,
1
S
2j
2 '
Ini>M k
l~(n)l<E q.e.d.
Theorem
6.2
Proof.
Let
show
that
There
E ~E
be (see
exist
a perfect Pro~,
T - translation We
use
the
technique
Rudin
of
in
symmetric
6.2) of
sets
E
Example
is
F
set
.
of
strict
strong
independent
satisfying
(6.21).
4.2
to
construct
multiplicity. for
This TE
We
a suitable will
do
it.'
138
k Thus,
for
each
k
we
let
Hk~E
2
be
the
union
of
all
hyperplanes
k (in 2
) of
the
form
2
k
Z
0
n.y.
j=l
j
,
j
where
2
k
Z Injl>o,
Injl
j=l
From
Example
4.2
there
it
are
j
ana
is
sufficient
2k
closed
j=l
2
.....
k
d
to
sets
construct ik
TE
j=l
J
so t h a t 2k
for
such
each
that
J max{m[I_):j:l J
k I , .~Hk=A
and
~.~Z,
--
2k } < --k'
.....
, where k
ik
(6.26)
Note
that
TE k
is
ksTkE k
p
--
j
so
and
~
that
.
x
Ik 2
for
that
(see
proof we
there
the
is
k
k k ~ k ~ ' E ' 'JJ
if
definition
of P r o p .
have
k k ~eT.E. J J
some
by
2
(~ k --
which
follows
(6.21),
for
..
property
; this
from
x
yeF
the
TE = s u p p
in m i n d is
j
ik 2
x
of
and with
cause this
set
Ek+I~E~
then
With
the
_ ik 1
that
6.2
for
for ¥cTE
k
+ y
where
k+IEk+l %k+ISTp P
of
~k and
notation). and
k
there
,
co
(6.27)
I~-~L
< --
In
fact,
taking
~k
÷ Y
Z ~ m m=k
' as
- r k
above,
and
letting
be-
kk ~kS~.E.jJ
,
139
IZk-Xk+21<
Ck' ... ' IZ k+j - Z k + j + l
k+j'
" ;
".
N
and
if
~>0
we
IXk-V]
have
~m'
N
large
enough,
so t h a t
m=k ly-Xkl<S+~e m k
Because
of 2
the k
for
approximation closed
k j
for
s>O
, which
(6.27)
{z~r:lY-zl_
k
2
as
in
(6.26)
(6.28)
do
this
we
From
the
the
k k yet.E.} J 3
each
k
there
form
$
shall
3
show
= A,
k=l,...
;
J kf,~ Hk = A,
k=l,...
,
, for
suitable
since
chosen
small
ek
(6.28),
definition
will
be
and,
hence,
of
Hk
(6.29),
, (6.29) k 2 • ~ n jYj
TE enough
proves
means
= 0
; (6.29)
implies
(6.28)
in the
definition
of
the
theorem.
that
,
2k with
are
k
J k = T IkE IkX . . .xTk Ek k 2k 2
that
, of
for
prove
(6.29)
Recall
we
Ik~Hk
to
k
, some
j=l
Ik
that
(6.27).
which 2
Defining
gives
we g e t
Ik ., j = l , . . . , 3
sets
I.
and
all
~ Injl>O, j=l
{yj:yj-~eE~E~}__ J
InjI~k
. Hence,
, has
no
(6.29)
solutions
of the
will
if we
hold
form
can
take
TE.
140
k
k
(~i'''''
2
k so
~ k )~E
that
2
k 2 k [ n.T.
(6.30)
~ <E>
,
j=l JJ 2
k
[In;l>o,
where
Injl<2~
i
In
§ 6.2
we
took
ak,
plicity
set
rem
4.4
k
let
bk + 0 E
and,
,
which
we
consequently,
find
(Theorem
for
any
6.1),
strict
m(<E>)=O
multiby
k For
each
the
S
C~ k --
Theok
2
be
the
union
of
all
hyperplanes
(in
E2
)
of
form 2
k k n.T. J J
,
k 2
where
;~e<E>
and
[
Inj l>0,
Inj
f~k
•
1
Clearly
m k(Sk)=O 2 m(<E>)=0
(where
k
m k 2
is
Lebesgue
measure
in 2
) because
. k
Thus,
there
is
can
be
( T Ik , . . . , chosen
k )e 2 such t h a t (6.30) h o l d s , and t h i s k 2 so t h a t (6.28) is v a l i d s i n c e m k(Sk)=O 2
point
q.e.d.
Notes We
refer
to
continuous Littlewood, More
Salem's
collected
works
for
singular
measures.
Also,
classically,
Hil!e-Tamarkin,
recently,
we
refer
to
and the
his
research
Wiener-Wintner work
of
Kaufman
in t h e
there (e.g. and
structure
of
is
the
work
[93]
for
references).
Hewitt-Zuckerman.
of
7.
Helson
Sets
7.1
E~uivalent From
§ 1.2 we have
is e q u i v a l e n t son
Definitions
to
if and only
Smulyan
if
~(E)
equivalent [1142
in
§ 7.3
7.2
Wik's
, Kahane
by n o t i n g
of Prop.
A.18
rem
in s e v e r a l
W
sets;
is Hel-
closed
(and K a h a n e [ 5 4 ] Helson
E
which
by the K r e i n -
and C a r l e s o n
in A'(F).
sets
) give
are several
we m e n t i o n
one
for this
(Salem-Zygmund-Kahane)
where
(m I ..... m n )eZ n
for w h i c h
~e~
proved
right
yj~F,
hand
Given
~Im~ I<2 k of
d e t e r m i n e d by the
w h i c h we
the
sup
analogue
of Prop.
and then p r o v e d the
in
the theoA.!8 corre-
case:
the t r i g o n o m e t r i c
(7.1).
set
, the
, an d Let
of all
IS k(Y1 . . . . . 2
gave
the m e a n - v a l u e
sum is over
determines
polynomial
be the (e.g.
yn)(O~)t
canonical
Example
<_ K /knR
all
a sequence
such that
( ~ i ' " ' " 'Yn )
in
~ [±rmCOS(mlTl+...+mnYn+@m )
m = m ( m I ..... m n)
side
mentioned
of T h e o r e m A . I I was
concerning
in n - v a r i a b l e s
inequality
of n - v a r i a b l e s ,
space
argument
result
inequality
in the p r o o f
Kahane [ 3 9 ]
S2k(Yl,...,yn)(~)
for the
Sets
the K a h a n e - S a l e m
step
polynomials
Salem-Zygmund
(7.1)
is
Helson
the S a l e m - Z y g m u n d
By a c l e v e r
variables
for t r i g o n o m e t r i c
7.1
of H e l s o n
s e c t i o n we prove
use
there
, which,
being weak
to c h a r a c t e r i z e
. An i m p o r t a n t
bility
~(E)
A(E)=C(E)
easy to see that
A'(F)
and S a l e m [ 4 3 ]
Theorem A.II
±l's
to
in
t h e o r e m that
Properties
We b e g i n
Theorem
It is thus
if
•
In this
sponding
if and only
Helson
is c l o s e d
inequalities
Arithmetic
§ 4.3
E
is e q u i v a l e n t
In r e p r o v i n g
norm
Sets
A~(E)=~(E)
theorem,
refined
of H e l s o n
,
A.I).
of
probaThen
142
where
K
is The
of
Theorem
proof A.II
Prop.
A.18).
L emma
7.1.1
and is
let an
a constant of
and
the
. Then
interval
of
Theorem so we
Given
ms~
(independent
only
there
is
everything)
(e.g. [ 54,
prove
radius
Proof.
of n - v a r i a b l e s ,
x...×l , where i n i/2 k+l) s u c h t h a t
loss
of
generality
2
By
the
mean
theorem
all
of
As s u m e
Iyjl
that
drop
several
partials
on
the
line
(F i d e n t i f i e d
i ej
also
,
the
"~"
variables
we
S(O .... ,0)
the
8 z ( e l , . . . , e n)
we
in
s(~ I .... ,~n )
where
I. C F j --
2
convenience
value
(7.1),
i
without
for
=
are
Vl
~S ~X 1
with
and
"2 k"
have
+ . . . + y
evaluated
between
at
n
some
(0, .... O)
[ - w , w ) ) , and
~ . A_s T I' if ~__. / ]- ~~s J J
point and
(yl,...,yn).
let
~x. ~s
#o
~S ~.
- 0
J
~S ~X n
5 0
'
if
O Since 8
~S IS(y I .....
to
> ~ lls k ( - ) ( ~ ) l l ~
I Is k Iloo -- Is k(O,...,O)(o~)[
and
(corresponding
that
2
(y1,...,yn)eH
Assume
lemma
¢o
i
is k(~'z . . . . . Yn)(~)i
all
) is
R=~r 2 m similar
HHI
2
for
and
33]
polynomial
a hypercube
i/2 k (i.e.
p.
Kahane's
trigonometric
length
(7.2)
7.1
of
yn ) -
S(O
....
,0) I <
1
2 k+l 3
l
~s
2k+l ~j -j71 J
143
Now
define
the
trigonometric
R(y)
Notice Also,
that note
R'(O) that
because equal
to
BS Bk. J definition
the
e.=0, J 2k-i
of one
variable
~ S ( 8 1 + e i ¥ .... , 8 n + e n y ) .
= ~ej
from
polynomial
±i
of the
, we have
the
polynomials
degree
of
R
R(y)
and less
S , and than
or
since n
I [ m.~.L < 2 < 1 j-i Consequently,
from
Bernstein's
j J
--
inequality
(Prop.
A.17),
k (7.3)
IS(y I ..... yn )
IYj[
for From
S(O
0)[
.....
£
- l l s ( o .... 2k+l ' 2
O) I
< i/2k+l
(7.3),
~Is(o
. . . . .
o)t ! Is(Yl
. . . . .
1
~n)i
k+l
Is(o
. . . . .
o)l <
IS(Y 1 . . . . .
Yn)I,
2 and this
is
(7.2). q.e .d.
Now
Theorem that most
for
for
7.2
Let
every
KEnk
the
Kahane-Salem
E~
[0,
2w)
y l , . . . , y n e [0,
points
ycE
of the
theorem.
be 2~)
Helson.
There
and e v e r y
is a c o n s t a n t
integer
k
, there
K E such are
at
form n
(7.4)
where
Proof.
Imj I <__ 2k-I
Consider
•
(m I ..... mn)
such
that
[Imjl
< 2k
, and,
in
(7.1),
144
set
r =i
if
(and was
~m.y.aE
m
J
not
counted
by
a previous
m)
and
0
3
otherwise. Then,
let
~
satisfy
points Take
~a~(F)
Theorem
of the such
form
7.1,
(7.4)
noting which
R
that
are
is the
E
in
of
number
.
that -in([m~yj)
(7.5)
[(n)
where By the
way
Recall
from
the
±l's
we h a v e § 3.4
side Consequently,
of
r
we
(7.5)
m
correspond
defined that
= [±r e
¢
~ .
, UE~(E)
can w r i t e
for
for t h e s e
m
to
and
(7.1)
certain
I I~I ll=2WR
in the
form
of the
right
hand
Cm
,
m
I IWI IA, ~ K /knR
Since
E
is H e l s o n
we
thus
have
2~CER
4 ~ 2 C 2ER 2
and t h e r e f o r e
R < nk(K2/(4~
Hence,
~ CEI IWI II ~ K /knR
< K2knR
,
"
2 2
CE)). q.e .d.
Remark Theorem which
By an
7.2
noting
the
ball, for
I.
that
result
the
Ll-lattice-ball,
since
We
to F o u r i e r
refer series
2. for
we
E
pick
observe
using
the
for
an
in i n f i n i t e l y
We k n o w
that
to be
Helson
the
above ~
Kahane-Salem factor
in
L2-1attice 7.2
directly
technique,
in the
of the
proved
statement
b y the
Theorem
has
proof
seems
of L e m m a
Salem-Zygmund
result
variables.
conclusion all
Benke
to the
to p r o v e
application many
since
G.
, is r e p l a c e d
that
up an e x t r a
to [ 41]
technique,
is e q u i v a l e n t
~Imji<2 k
We
L2-1attice-ball,
difficult
cient
different
(~Imj12)1/2<2 k
the
7.1.1
entirely
of T h e o r e m
independent
7.2
sets
is n o t
satisfy
suffithe
con-
145
clusion as
we
er
the
sets.
whereas showed
in
condition
now for
• We
F×F
the
7.2
this
be
is
7.2
to
~ix~2
by
due
combinatoric
an
open
sufficient
present 7.3)
Uj~c(F),
is
given
C.
non-Helson
sets,
problem
to
for
Herz
countable
[29]
an
interesting
to
Salinger
card
Proof. that Since
X
the
= card
Because
of
~.=m, j=l, J Iml(F)>O x~(l
2
product Y
j=l,
2
and
Theorem . Let
)~F,
, and
Then,
n K. J,k
A.> m
X xy C F n n -we
~ m×m
Varopoulos
assume
that
Yn
X
closed
3
, Y
xeF
and
where
n
xJ
n
assume
F
such
--
.
without
intervals
I ~(
)~F)
C n
loss
of
generality
.
n
m((I
the
and
n
(7.6)
Consider
necessary
> 0
measure).
= n
n
choose
xJ n
Helson
property
is
n
wheth-
• new
n
that
as
preliminaries.
lUlXU21(F)
(where
independent it
might
(Theorem
some
Let
hand
matter
sets
need
strong
other
Theorem
Helson
the
perfect
Theorem of
7.1
has
On
apply
first
Proposition F ~
of
discussion We
are
§ 6.
condition A
[953
there
b an
>
'
) n
(i-
I J
"
n
, J
~(c n
such
n
, d n
that
) , and n
- - ~ )2( I n × J n ) n
squares
--- [ a n + J ( b n - a n ) / n '
an+(J+l)(bn-an)/n]
C +(k+l)(b n
j,k=O,...,
n-i
. Clearly,
n
-a
n
)/n ]
x [Cn+k(bn-an)/n
,
n m(K. ) = j,k
(b - a n n
)2/n2
Then n-1
"m((In×J
)N;) n
=
Irr J; I xJ n
xF d~ m o
X ~ ,k=O
n
Ir xF d m~ o JJ Kn
J ,k
22 Kn
oo
,
146
where
n-1
~(x,y)
If
@
on
=
[ XF(X+j(bn-an)/n, j ,k=0
y+k(bn-an)/n)
then
Kn
oo
m((In×Jn)~F
) <
( n 2 - 1 ) m ( K noo ) =
(i-
1 2 ) ( b n - a n )2
,
n
which Thus
contradicts
~(Xo,Yo)=n 2
Consequently,
by
for the
X
Y
(7.6).
some
(x
o
definition
z
n
of
{Xo+J(a
e {Yo+k(a
n
,yo)~K n oo
n
n
-b
-b
@
n
n
, we
set
)/n:j=0,...,
n-l}
)/n:k=O,...,
n-l}
,
2 and
Xn ×Yn ~ - F
since
@(Xo,Yo)=n q.e.d.
Pro p osition
7"2
~je~c__ (F),
Let
j=l,
2
I~I.~21(E)
Then
~n
3 C
C +D C E n n -Proof. Since From
n
, D
, and
Let
~ F n --
C +D n n
F
7.i
consists
we
take
X
have , Y
n
X xY Given
n
> o
card
of
that
n
C
2
= card
n
distinct
D
n
= n
, C ~D = A, n n
points.
I~I×Z21 ( F ) > O ~F
n
such
that
. card
X
--
= card n
Y
= n
and
n
CF
n
n
--
we
find
p(n)=n4+l,
(7.8) n
that
assume
~ {(X,y)srxr:X+ycE}
I~I×~21 ( E ) > O Prop.
such
, and
Cn
and
n=card
Dn
for
C =card n
X+y=X'+y'
which D
Cn ~
Xp(n)'
, C N D =A, n n n
implies
X=X',
y=y'
and
Dn~
Yp(n)
Vl,X'cC
n
"
, y,y'cD
n
147
This
gives
the
result;
eard(C and The
proof
If
n=l
We
can
n
card
take
) = n
by
Cn+Dn~E_
(7.8)
D = n n
X~X2,
and
the
X~..n(n)×YP(n) CF -
facts
that
card
, and
C ~ D =A n n
.
and
let
X2
and
assume
been
CI={X},
DI={Y}
Y2
are
two
element
m
the
sets
sets;
also
(7.8)
trivially. that
for
c m -~ {~i ..... ~m } C-Xp(
(7.8)
, and
n
yEY 2
since
satisfied
have
since
induction.
X#y
n>l
fact,
2
C = card n
any
choose
let
+D
is by
is Now
n
in
chosen
such
m)'
that
Dm
{¥i ..... 7 m } ~ Y p ( m )
card
Cm=card
Dm=m,
'
Cm." ~ D m = A
' and
holds.
m
Then
(7.9)
j +y k,i,j
X+Yi=X
is t r u e
for
m2(m-l)
(7.10)
is
true
for
obvious. possible ble
the
other C
m+l
m
For
values (7.9)
of
l
hand by
note
which
of
Xp(m)
has
ticular,
that
more
,m
and
keF
; and
m
X#X J
,
,
leF
satisfy
for
each
(of
pair
¥i,~_J
have
X=X. J
m)
y. there I t h e r e are m-1
;thus
there
are
m
possi-
are
mm(m-l)
(7.9).
than
3 2 m - m +i
points
and
so we
form
choosing
km+ISXp(n)
so t h a t
of
X. , and f o r e a c h J - for if yk=Yi we
¥k
values On
values
k=yi , i=l,...,
is (7.10)
. .k=l, . .
X
m+l X
m+l
does is
not
- {XI ' ' ' ' ' k m }
satisfy
different
either
from
all
(7.9) the
or
(7.10).
elements
of
In p a r C
m
and
D
m
•
148
We
determine
Ym+l
in
a similar
way
adjusting
(7.9),
for
example,
to
read
y+li=Yj+lk,
obviously all
the
i,
we
k=l,...,
have
elements
no
m+l,
problem
of
C
j=l,...,
in and
m+l
m,
taking D
and
Ym+l
since
m
y#yj
;
different
we
adjust
from
(7.10)
to
read
~=X., 1
and Clearly,
we
take
i=l,...,
-
Ym+iEXp(n)
(7.8)m+ I follows
{Tl"
.
immediately
m+l
}
'''Ym from
°
this
procedure.
q.e.d.
Theorem
7.3
Let
Proof.
Let
KE
Now,
any
k>2
for
E C ~_
be
F
be
the
Helson.
constant
let
n>2KEk
If
of
; and
~3. S ~ c ( F ) ,
Theorem
7.2
choose
C
--
n
2
Thus
all
This
contradicts
points
lar,
and
, D n
conditions
of P r o p .
7.2
of the
form
Theorem
that
at m o s t
2
, then
assume
~ n
F
I~IM~21(E)>O.
satisfying
the
--
•
~ . + y . , X.EC l J l n 2 (since n >KE(2n)k)
7.2
j=l,
KE(2n)k
points
and
y_eD j n
which
of the
,
says,
form
are
in
E.
in p a r t i c u -
X.+y. are l j
in
E,
q.e.d.
Let
Theorem
Proof.
7.4
also
If
E
Clearly
theorem Thus
us
with
m~m=m fact
the
observe,
is
Helson
mC~e(F)
that
then
it
of
Prop.
5.7
and
Prop.
5.9,
that
m(E)=O
is
identified
by
the
Radon-Nikodym
IsLI(F)
so the m >--O
light
, and
function
and
in
result .
is
immediate
from
Theorem
7.3
and
the q.e.d.
149
7.3
Uniqueness Recall
son set sets
(from
of He lson
Sets
§ 5.4 and ,§ 6"!)
that
is U . On the
are
Kahane
Properties
U
in the wide
and Salem
ing H e l s o n
other
give
sets
E
hand we shall
sense;
a proof
by the
which [90,
Lemma
we give
we now pp.
state
7.5
measurable,
§ 1.4)
Lemma Let
E
there
is
Since
E
If 7.6.1
KE>O
HelHelson
result [26]. characteriz-
such that
I [~[ I1
in the
convenient
following
measures
form
(e.g.
be Helson. @aA(E)
where
Then
such
7.6 for all
that
=
@ : E ÷ ~, @
for each
bounded
and
~c~(E) o
@W
the
map
(E)
÷
Co(Z)
is continuous.
then
has
c
o
(z)
the
sup-norm
and
~o(E)
has
the
such that
for
norm.
by d u a l i t y ,
all
book
that
of c o n t i n u o u s
in T h e o r e m
is Helson,
is a h o m e o m o r p h i s m ,
Thus,
is
~
7.6)
fundamental
in their
there
[;(n)[
~e~o(F)
o
It II 1
(Theorem
if every
characterization
~
Proof.
that
known
).
(Wiener)
We use
theorem
lim [n I +
Wiener's
(e.g.
118-119]
7.6.1
Theorem
uses
of this
KE
prove
is H e l s o n ' s
condition
YBc~I~(E),
The proof
this
it is not
for each
$S~o(E)'
there
is
fsLl(z)
~P/t(E)
(7.11)
Again,
by duality,
the right
hand
side
of
(7.11)
is
] ,(y)d.(y) dF
ing the F o u r i e r
image
of f),
and so
(,
be-
150
f (7.12) JE Now,
, the
for
functional
~o(E)
+
¢ r
is
continuous;
and
consequently
it
is r e p r e s e n t e d
by
~
as
in
(7.12). Hence,
for
all
such
~
there
is
CeA(F)
so that
¥ ~
(E),
Cp=~p
~ O
q.e.d.
Theorem wide
By
To
show
~e~o(E), Lemma
Now
E
be Helson.
note
~o(E)
is
a set
of u n i q u e n e s s
in the
= {0}
7.6.1 that
~
is
if
y
continuous is
are
without
shall
disjoint
show
this
that
is,
loss
joint,
for
hence any
and
any we
CeC(E)
open
show
: l>y}
V - {IEF
: X
in
is
that
UNV#A
¢eC(E)
of g e n e r a l i t y
sets
F
supp
point
-= { I E F
phenomenon we
and h e n c e
an i n a c c e s s i b l e
U
Take
E
~0
assuming
We
Then
sense.
Proof. Let
Let
7.6
such
impossible,
if n o n - e m p t y
p ~ F~E_
of
F~_
that
that
[0,
O~F
2w)
then,
,
U N V#A
giving U,
is p e r f e c t .
a contradiction;
V~_ F
are
open,
dis-
then
(note
can
apply
, and
let
that
Theorem 8>0
and
E
Helson
implies
F
Helson
have
the
and
7.~). a neighborhood
N
prop-
151
erty
that
]Xu-¢ ]
(7.13)
on
N
; this
is
permissible
>
since
we
are
dealing
with
×U
and
ug~ v#A Let
[@[~i
be
defined
by
JII~-xuIXNaIVI= JlI~-xul~x.d. , f
(7.1~)
which
we
can
do b y
the
Radon-Nikodym
theorem.
Define
f l~-xuI ¢-Xu
~
,
on
N
,
elsewhere
g = 0
g
is m e a s u r a b l e
with
respect
to
W
and
if
CW = ×U P
we
have
r
] (¢-XU)gd.
= 0 ;
1N
whereas
by
and
I (~-Xu)gd. IN
(T.15)
Since
(7.13)
N
contains giving
an
the
open
desired
(7.14)
= I I~-XuIdl. JN set
the
right
I > 61 dl.I IN hand
side
of
(7.15)
is
positive,
contradiction.
cl.e .d.
Theorem ed
that
vergent fact son
if
sets
A(E)
Taylor
that
7.6
her are
is
series theorem
Carleson
has
been
the
space
then is
E
generalized of is
by
restrictions U
in
the
a generalization
sets
( [ ll4]
M.
, Notes
Chauve to
wide
uses
the
§ 2).
E
[14] of
sense. Wik
; she
prov-
absolutely
con-
Of
the
course,
theorem
that
Hel-
152
Example
7.1
Mal!iavin
are
sets;
and
if
U and
only
these set
if
he
basic
a set
and
pp.
example
and
also
Theorem
B.22).
a proper
property
y
7.6
is
of
]
the K
subset,
a set
7.4
of
spectral
F u!ther
In t h i s
and
criteria
for
Edwards tain
sums
[83]
of
and
jections; convergence to
the
Proposition ¥¢eC(E),
convolutions.
begin
with
7.3
I¢I~i,
~
if
is
S
with every
resolution. the
is not paper
case;
that
Helson.
We
for
set,
details
take
y:n=l,...}~
The
P~K
K - P have
the as
Helson giving
in
functional
7.Z)
7.8)
terms
of
is the
Grothendieck-Dieudonn@ such
Sets
of u n i f o r m
(Theorem
conditions
pseudo-measures.
(Theorem
in t e r m s
second
true
for
first
and
naturally
suggests
is
analysis
due
limits due
to
to of
R.E. cer-
Rosenthal
existence theorem problems
of p r o on w e a k related
.
the
yeA(F)
know
Kronecker
supports
The
uses
Given
set
Thus,
is not
which
z {7n'
theorems
sufficient
§ 2
C
two
conditions
of
a perfect
prove
gives
approach We
we
of m e a s u r e s
such
Varopoulos'
Criteria
and
theorem
that
to
to
spectral
resolution
which
Helson.
this
a Helson
resolution.
strong
let
that
resolution
~ C + P
to be
gives
know
spectral
. Then
a set
[ 21]
of
interesting
showed
Analysis
section
of
be
resolution
Functional
we
is
reader
E
is
it
a set
Let
÷ y
n
sets
spectral
spectral
refer
perfect
that
strong
4668-4670
a set
that
definitions
of
resolution
[ 104,
[68]
Theorem
the
to be the
is
constructed
state (see
it
spectral
Varopoulos is,
from
observations
of
proved
following
E C
F
, for
easy
Assume
result.
~>0
which
and
sup
yeE
l*(Y)-~(~)]
< 1-6
and
O
such
that
1 53
Then
E
is Helson.
Proof.
Let
pe~(E)
lI
Then
=
2~l[~(n)~(-n)l
supl
~
-iny>
I t t~t t A.
n
Thus
(7.16)
(i-6)I I~] I1 + K s u p I < D , e - i n y > I . n
The
right
hand
side
of
(7.16)
hence
from
§ 1.4
is independent
II~II l ~ ~up{l<~,~>I:~C(E),l~l~l}
<-
of
@EC(E),
I¢I~i
, and
(1-6)tl~ll I + KII~II A,
Therefore K
ll~lll <-7 ll~IrA' ' and
K/6
is independent
of
~
-
that is,
E
is Helson. q.e.d.
R.E. analysis
with
to Marcel
Edwards'
theorem
a ~udicious
Riesz
[90,
We say the
use
Chapter
E C
F
is really of the
fact
that
in "pure" L2(F)~
functional
L2(F)
= A(F)
¥$~C(E)
condition
3 K~K(@)>O
limit
(on E)
RE
such
2
if
that
of functions
¢
is the unihaving
the form
n
Xak(fk~gkl
(7.17)
,
i n
where
(due
i; 2 ] ).
satisfies
form
a result
llfkll 2, ilgkil2<-¢~ and
Ziakl<-i I
154
Theorem
7.7
Proof. son,
If
E
is Helson
By the M. Riesz
there
is
CE>0
2
is satisfied•
mentioned
that
above
¥~c~(E)
and because
E
is Hel-
,
sup{I]r f . g d ~ l : l i,,f i i 2 ,,1 1 g,,] 1 2,,! l } , q
E
~
m
B
~
~
d
I
B
.
we have
¢ l]FCd~l
(7.18)
RE
¢
I1.11 l_C ¥¢eC(E)
result
such
<
Thus,
then
¢ sup{l[JFf* gd~l: [ Ifl 12'
~
I Igl 12 -< /CEI I@l I~ }
'
r since For
a given
Then
X °°
I]E¢~I
¢cC(E)
let
that
is,
form
(7.17)
the dual
of
still Thus,
X¢
is the b a l a n c e @ each
by the
CeX~ °
form
C(E) ~(E)
p. 198]
q(C(E)
of the
[34,
187]
theorem since
~(E))
and
has
¢
hull
of functions
fk~gk~X¢
. This
theorem
the
~(C(E),~
X °°
is norm
of X
having
follows
(e.g. [34, (E))
-
the
from
p.192]
topology
). is
•
again,
X °°
- closed
limit
Hahn-Banach
C(E) p.
Iigii 2 ~/CEII~IL~}
is the weak
[lakl~l
when
Hahn-Banach
[34,
{f~g:llf112
convex
where
the b i p o l a r Now,
II¢II.II~II 1
_<
¢
is convex
and
closed
o(C(E)
in ~(E))
C(E) -
closed. Finally,
observe with
that
CaX °°
¢
¢~X; o
means
l] Sdu I<_i
for
all
the p r o p e r t y ¢
II f gd~1<_l , l lfll 2 , llgll2~/cEll¢ll~
(7.19)
in p a r t i c u l a r , (7.18),
and,
if hence,
satisfies
(7.19)
CeX oo • Note that ¢
;
¢ I lCdul<__l J we can take any then
from K(¢)<
CEII*II~ q.e.d.
155
The can b e
converse
adjusted
Example
7"2
We
for
which
supp
are
Helson
and
discrete
do
now
we
first
that
is if
7.7
only
P
if
closed
Helson
T
is
Rosenthal's
From if
fact,
show,
and,
and
further,
subset
support
closed
of
Helson
- ~(E)
of H e l s o n
then
2
C(E)
TeA'(E)
subsets
RE
T
sets
is
a
the
ant
commutes
(or
is
have
a more
sets.
In o r d e r
on p r o j e c t i o n s .
surjection theorem
subspace
special
invariant
invariant
the
with
(7.2o) (7.20),
V. 8
a linear
a linear
P for
of
induced
Q
is
an
Rosenthal mean
Assume
subspace
If
: X + Y linear
X
then
to
X
is
is
a
operators a projec-
on
that
and
Let
take
X C Y ~_ X
A continuous
L=(Z) to be
be
a
a weak
projection
: X + Y
sup
translation)
If
situation.
subspace.
=
(T f ) ( m ) ~ f ( m + n ) . W i t h n [ 82, T h e o r e m 1 . 1 ] :
invariant
Helson
remarks
Hahn-Banach
Pf
To p r o v e
on
exists.
translation
Y
theorem
background
Y
translation
X,
Theorem
don't
countable
consider
(where
an
sets
to
category
supp
P
and
a second
Y~----X , t h e n
yet),
we
closed
showed
easy
In
some
p2=p
: X + Y Now
give
and
topologies
where
is
countable.
present
space
projection
norm
T
We
a linear
tion
consider
note
so
Theorem
measure.
this
(no
to
to
norm if
(T
-n
this
: X + Y invariant
explicitly
topology VneZ
PT
n
)f
setup
is
from
and
L=(Z))
is
invari-
VfeX
,
and
nomenclature,
a continuous
continuous
projection
projection
constructed
Rosenthal
P
P
in t e r m s
then
there
: X + Y
of
Q
L~(Z)
for
every
closed
projection
QF
: A~(E)
+ A~(F)
F~
E
there
is
a continuous
156
Then
E
is Helson.
Proof.
From
(7.20)
let ^
-*
QF : A~(E)
A'(F)s ^
be an invariant
continuous
projection
(where
QF is induced
in the obvi-
ous way from QF ). For
TeA~(E),
Thus
VncZ
PF(T)¢A'(F)
define
and
¥TaA'(E)
by
PF(T) z QF(T)
we have
PF(elnYTY) QF(TnT7)^ TnQF(T) TnPF(T) =
by the t r a n s l a t i o n
P
because
invariance;
and consequently
is continuous,
F
(7.22)
PF(~T) = CPF(T), V@¢A(F) from in
If
=
• iny PF(elnTT~ ) = e PF (Ty)
(7.21)
Hence,
=
ue~(E)
(7.21)
and since the t r i g o n o m e t r i c
and
C ~
E
is Borel,
(7.23)
~c~7~'~(c)
¢eA(F)
~ <~,Xc¢>,
and t h e r e f o r e
each
such that
¢=i
on
PF ~ = $PF p = PF (¢~) = PF(PF
by
are dense
we define
(7.22)
and since
PF
¥~eC(r)
Ua~(E)
= ~F + ~E-F' F ~ _ E
Now take
polynomials
A(£)
Clearly
,
can be written
as
close@.
F . Then,
for
U e~(E)
,
+ CUE-F ) = WF + PF(@WE-F )
is a p r o j e c t i o n
with
WF CA'(F)S
157
~ef¥1(E)
Because
such
is regular
there
is a sequence
of closed
sets
E
%-- E-F n --
that
(T.24)
tim
I~I(E
) : I~I(E-F) n
Next
take
CneA(F) @n=l
It is easy
with
on
on
@n =0
En
, and
F
to see that
(7.25)
lim
in fact, f I j E-F
0<@
the p r o p e r t i e s
if
@cC(F)
r j (E-F)~J(E-E)
n
I ]%n~E_FI Ii = 0 ;
and
I l~I I ~ l
¢¢n d~l
then
-< l~I ((E-F)-En ] =
I~I (E-F)-I~I(En)'
n
so that In p a r t i c u l a r ,
we have lim
(7.25)
from
I I%n~E_FI IA,=O
(7.24). and hence
n
(7.26)
PF ~ = ~ F ' ~ 7 ~ ( E ) by
To prove
(7.23)
E
and the
is Helson
we
continuity show
~(E)
of
,
P
F closed
in
A'(F)
Assume
lim
I [T-~nl IA , = O
n
for By
(7.26)
TeA'(F)
(and the
lim
and
~ne~(E)
continuity
of
PF )
] I~nF-PFTI IA, = lim
n
I IPF(T-~n) I IA, = 0
n
^
In particular,
lim n
BnF(1)
- lim n
~n(F)
exists
for each
closed
F
158
Now,
by
the
Dieudonn@-Grothendieck
measures
~ ¢~(E)
theorem
converges
[16;
in t h e
24 ]
, a sequence
~(2~(E)
~(E)')
of
topology
n
if
only
and
if
~
(G)
converges
for
every
open
G~E
n
Consequently
~
converges
in the
weak ~
n
This
does
it
topology
^
since
for
.
~
each
m
lim
to
a measure
~
.
^
~
(m)=~(m)
and
so
T=~
n n
q.e.d.
Remark
i.
given is
The
locally
and
complicated
topological in
compact
interesting
too
this
7.8
for
closed
all
for w h i c h A(E/F) is
that
we
a norm
if F~
give
= C(E/F)
For
such
. It is
by
the
takes
Pelczynski
CeC(F) [ 76,
into
prop.
sets
E
is
an
(e,g.
also
easy
to
§ 2).
is
Helson
classical
Borsuk
: C(F)
+
and best
To p r o v e
R
¢
is
the
result
the
this
proof
let of
: A(E) ÷ A(E/F) F f o r m of T h e o r e m 7.8
A(E)
extension
= C(E)
and
theorem
there
6 ]
then
noted
P
I-UR
:
extension that
U
of
¢
induces
on
E
.
a continuous
jection
where
a
C(E)
a continuous
[
are
P
this
then
The
from
projection it
spaces
hold
]
true. see
a
theorems
algebra
[lll
is
for
map
u
which
7.8
Helson;
If
is b o t h
of W e l l s
then
is
of
disconnected
hold
work
is
E
linear
to
a continuous
a converse.
decreasing
which
Theorem
then
. Also,
totally
sets
the
classes
Dieudonn@-Grothendieck
a theorem
on F}
E
of
to
there
minimal
example,
to be
converse
~ {¢£A(E):¢=O
Theorem
so t h a t
a class
seems
The
determine
space
for
direction
to
hard.
with
basis
2. A(E/F)
problem
restriction
of
C(E)
¢
to
+
C(E/F)
F
,
. This
is
the
converse.
pro-
8.
Concludin~ As
we
not
known
The
following
which
if
are
Remark
have every
remarks
closed
related
sets
of
spectral
§ 7.3,
and
classified
certain
in p a r t i c u l a r ,
They
also
gave
ter
results
and
Malliavin
ize
such
given
every
by
of
in
are
the
read
the
be
(very
the
mutually
set,
Dirichlet
sets
are
another
that
Kronecker
sets
are
Dirichlet,
are
strong-U
( [52]
Dirichlet
sets
relation
between
example,
there
aren't
we
3.
We
on
the
refer 4.
Helson
sets
, Notes
given
are
countable
- note also
to
[71]
be
we
Helson
§ 2). [ 53]
note and
closed
strong 5.2
an
the
of
the
synthesis
it
is
of
has
open
as to
to
see
that
of
Kronecker
sets
Besides are
the
Dirichlet
Also,
Dirichlet
sets
basic
properties
of
sets
study
is
in
Dirichlet
and and
of t h e
[ 63]
sets
two
significant
arithmetic
treatment
of
sets
E
(for which
regard).
Varopoulos union
characterif
comprehensive
Kronecker
in t h i s
Katznelson
exhaustive
that
lat-
set
independent
elegant
to
trivial
finite
a more
and
and
subsets.
These
of
determine
is C
interesting
that
technique
Helson.
A survey
[ 44]
resolution,
papers
generalization.
all
aren't
Helson,
the
a
locally
analyticity.
to
show-
mentioned
non-S
a generalization
such
and
between for
U.
text
[ 68]
Katznelson
has of
it
is
course,
which
Example
mention
relation
Finally to
in
Dirichlet,
Kronecker
Meyer
sets;
is
sets
is
is
the
arbitrary
spectral
• Note
whereas
and
Dirichlet
in
as
of
exclusive
resolution of
the
difficult)
[ 95]
are,
are
set
problem
Varopoulos
sets
there
sets
F)
and
a probabilistic
Helson
and
to
a set
sets
C
up
it
set
Malliavin
(for
Kahane
not
on
a
notes,
Helson
taken
1962
this
approach
is
in
Cantor
a set
are
the
every
not
U sets
1963
that
and
if
topics
are
spectral
sets;
Y.
set
even
Congress
which
give
throughout
problem.
Then
that
which
Helson
S
several
strong
2.
fact
to
Salinger
every
set
showed
Another
or
generalized
sets
useful
and
S
this
case.
S
[ 56 J
sets.
is
times
resolution
conditions
are
analyticity
to
[23]
group
and,
set
International
Filippi
abelian
several
indicate
the
in
whether
Helson
At
that
been
mentioned
1.
ed
compact
Remarks
has
of h i s shown
strong
work
of
properties basic
the
results.
union
spectral
of
of t w o
resolution
160
sets
to
be
strong
spectral
is b a s e d
on
Drury's
union
two
Sidon
edly
of
important
for
resolution
technique sets
is
further
[ 20]
Sidon. work
in
C 109, when
the
Varopoulos' the
llO]
area
. Part
of his
method
latter
proved
that
paper
[ llO ]
is u n d o u b t -
of H e l s o n
sets.
the
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125
Compactness
of M e a s u r e s "
iii.
B. Wells
"Weak
112.
N. W i e n e r
"Generalized M!T
de Helson"
CRAS,
251-253.
sis"
Wiener
Proc.
51-112.
"Groups
ii0.
Synthesis
Harmonic , 1964.
(1970)
in Harmonic
Analy-
109-152.
Analysis"
PAMS
20
Selected
(1969) Papers
124-130. of N.
269
113.
N. W i e n e r
ll4.
I. Wik
Non-Linear
Problems
in R a n d o m
"On L i n e a r D e p e n d e n c e
in C l o s e d
Theory
Sets"
MIT
Ark.
, 1958.
Mat.
(196o) 209-218. 115 •
116.
A.
Zygmund
"Some
Examples
of Sets
Mat.
5 (1964)
207-214.
Trigonometric
Series
with
Linear
Cambridge
Independence"
U. Press,
1959.
Ark.
INDEX (by section
number)
Sets Absolute
convergence
Analyti city
7.3,
Carleson
7.1
Helson
S
1.2,
Sidon
8, B.3
4.3,
5.4,
1.2,
4.1,
Strict
multiplicity
Strong
independent
4.4
4.4, Strong
M
5.3, 4.1,
6.1, 7.3,
5.1,
5.3,
8, B.5
1.2
p S
5.1,
Strong
1.2,
Symmetric
2.1
1.2,
6.3
3.1,
3.4,
U in the wide Wik
Proper
Bary
names
"name"
4.3
Benke
7.2
Bochner
S.N.
2.3,
7.2, A.IO
B.3 5.1,
Borel-Cantelli
kernels,
Doob Drury
Chauve
6.1
5.4, 6.1,
8 1.2,
3.4,
5.1, 7.3
7.4 5.1
8 A.II
A.2
3.1
Eymard 3.4, 5.3
R.E.
7.4
B.3
6.2 Fej6r Fekete
1.5, 5 .i
6.1, 7.3
etc.)
A.9
Edwards,
4.2,
1.2,
5.4
1.5,
Dvoretsky
5.2
7.4
Cantor-Lebesgue Carleson
5.4,
4.3,
Einstein-Smoluchowski Cantor
4.1,
sense
5.3,
Dirichlet
4.1,
Borsuk
theorems,
1.2,
Dieudonn6-Grothendieck
Bernstein,
Bohr
(including
1.2,
6.1
resolution
7.3,
6.1,
1.2,
8
B.3
Rudin
8,
5.4, 7.3, 8
U
U
5.2,
spectral
K
5.4,
1.2, 7.3,
B.5
4.1,
1.2,
8, A.9,
B.5
resolution
8
5.2
B.5 q Kronecker
1.5, 7.3,
8
Spectral
1.2,
Independent N
1.3,
B.2~ B.3,
6.1, 7, 8, B.5
I
1.2,
8, B.5
Calderon
Dirichlet
6.1
5.2,
5.3
A.9
261
Filippi
Reiter
8
B.3
Richards Grothendieck
Hadamard
B.4
5.1
Hausdorff-Young Helson Herz
Riemann
3.2
Hardy-Wright
7.3, 7.2,
Kahane
B.I, B.3
6.1,
B.I,
Kahane-Salem Kaplansky
B.5
7.1, 7.2, 7.3,
A.II,
3.3,
B.2,
8,
5.1, B.2
Riesz,
F.
1.4
Riesz,
M.
5.2, 7.4
Rosenthal
2.3
B.3,
B.I
7.4
Rudin
4.4,
Saeki
B. 3
Salem
6.1,
i
5.3,
5.4, 6.I,B.I,
6.2, 7.1,
B.3
7.3, A.II,
B.I
B.5 Salem-Zygmund
4.3, 7.2
Salinger
B.3
7.2, A.IO
7.2
v
Katznelson Kaufman
5.3, 8, B.I,
5.3,
Kolmogoroff KSthe
6.1,
B.2,
Lipschitz
7.1
4.1,
4.2,
5.2,
5.3
A.II,
B.2
4.4 1.3,
Schwarz
B.4
Kronecker
Sarat
Schwartz
B.I
5.1, A.2
Krein-Smulyan
B.5
4.3,
5.1,
McGehee
Stechkin
A.10
Stegeman
B.3
Steinhaus
4.2,
de la Vall@e Varopoulos
4.4
Poussin 2.4,
5.3,
7.3, 8, B.3,
Wells
6.1
Wiener
A. ii
Wik
.
Pelczynskl
1.2,
1.4,
B.4
Poisson
3.2
Pollard
B. 2
Young Zygmund
V
Pyate ckii-Sapiro
Rademacher
5.4, 7.1,
de Wilde
7.4
Pietsch
Raj chman
6.1,
B.4, B.5
5.1 6.1, 7.3,
A.9, A.II,
p
5.4,
7.4
Weyl
3.1
8
Paley
1.5
7.3, 8, B
Menshoff Meyer
B.5
3 .4
7.2, Malliavin
B.2, B.3,
A. i 3 •I
6.1
B.4
3.1 A.II
7.3
B.2
A.7,
