Harmonic Analysis on Totally Disconnected Sets

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann,...

Author: John Benedetto

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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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109 •

Paris,

on Spheres"

Camb.

Phil.

379-387.

Algebras

and H a r m o n i c

la r 6 u n i o n

Analysis"

271

(1970) of

de deux e n s e m b l e s

Acta Math.

Continuous

Functions

A e t a Math.

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,

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