Harmonic Analysis on Totally Disconnected Sets (Lecture Notes in Mathematics 202)

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

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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

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

O,BneZ

sub-

if

and

E

is

a Dirichlet

set

if

Ve>O ~neZ

such

that

sup ll-elnTl 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

for

I.i

if

~ -

,

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 ~ --

,

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

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

Y -- i , . . . } C r

that

n

co

i

(1.3)

~

Now,

assume

=

¢

[ b n 1

is

~

y

on

C(F),~Ib

a Borel

measurable

I0,

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;

(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~I 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

* @,~>

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)

£


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)

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

ty 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

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 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:Isll}

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:Isll}

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

,

Isli

,

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. IslN,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~l0

ee = e

[c

n

s

n

analytic

satisfies for

Isl0

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)

+

~>0

as a + 0

se-

52

Hence,

by

Lebesgue since ~Ic

dominated ~la

n

I< ~

convergence and

c

a (1-e-a[nl)l e ; hence, taking ao for which IIjK~ ~a]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 1k -- 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

jO

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 =

0

if we

Y =

I CkYk ' k=l

Yk

in

(4.24)

elements

n k0, 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

: OO

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 JIl

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 [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 _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
_ 7

sup xc[O,~)

For

s

i -- T4

we

choose

N

such 1

--

2

i J

that


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
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,-~ O

¢eC(E),

and

l¢I~l

psi(E)

, satisfy

111~Ill-Il Since

E

is K r o n e c k e r

there

:is

n

such that

-in7

sup l~(¥)-e

I < ~/2

i
{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 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


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

O0

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

l2w, for --

~i=...=~0 m-1 = 0 , where

m=m n

Thus, by (6.9) and the Lemma, fl 2M O__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
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 ~ 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
y}

V - {IEF

: X0

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