by gm G. sup (cf. 1 ch. 5, w 1). A problem on the union of Helson sets

A p r o b l e m o n t h e u n i o n o f H e l s o n sets

E. GALA~S

Institut Mi~tag-Leffler, Djttrsholm, Sweden

Let G be a locally compact abelian group and let G be its dual group.

Definition 1. A compact subset E c G is called Kroneeker if for every con- tinuous function f on E of modulus identically one (If(x)l =~ 1, V x E E ) and for every ~ > 0 there exists g E G such t h a t

sup

(cf. [1] ch. 5, w 1).

x E E

We shall denote b y M(G) the set of all bounded complex valued R a d o n measures on G and b y M(E) the elements of M(G) with support in a compact subset E of G.

W e shall denote b y C(E) the set of all continuous complex valued functions on E.

Definition 2. A compact subset E of G is called a Helson o~-set (Ha-set) ff there exists a constant ~ > 0 such t h a t

tl~lt~ = sup l,~(x)i ~ ~ tl~lt

for every ~u E M(E), (observe t h a t then 0 < ~ ~ 1).

I f K is a compact subset of G we shall write Gp (K) for the group generated by g m G.

I n this paper we shall prove the following theorem.

T~OREM. Let K be a totally disconnected Kronecker subset of G and D a countable comloact Ha-subset of G such that

Gp (K) N Gp (D) ---- {0}.

Then K U D is an Ha-set.

194 ARKIV :FOR NATEMATIK. YO1. 9 :No. 2

_~emarks. Varopoulos [3] has p r o v e d t h a t if K is a n y t o t a l l y disconnected c o m p a c t Hi-subset of G a n d D is a n y H~-subset of G t h e n K 0 D is Ha(~) w i t h fi(a) > O.

The interest of our t h e o r e m lies in t h e f a c t t h a t in this special case we h a v e

=

l~owever we m u s t p o i n t o u t t h a t the conclusion o f o u r t h e o r e m fails if we replace t h e condition ))Kronecker~) b y ~)HI~ for t h e set K . To see this observe t h a t t h e set { ( x , 0 ) , ( - - x , 0 ) , ( 0 , ~ y ) , ( 0 , y ) } c R ~ ( x , y C R ) is n o t a n Hi-set of R 2 despite t h e f a c t tha~ it is t h e u n i o n o f t w o Hi-sets

E 1 ~--- {(x, 0 ) , (-- x , 0)}, E 2 = {(0, - - y ) , (0, y)}

t h a t satisfy Gp (E~) f] Gp (E2) = {0}.

W e shall prove first:

L ] ~ M A . Let K and D be as in the theorem with D = { x 1 , x 2 . . . . , x , } a f i n i t e set. T h e n for every Z E G there exists

Z, "-> 1 uniformly on K

r~---> o o

for all j = 1 , 2 . . . r.

{X.},,~, Z,, e G such that and X,(xi) --> Z(xi)

n---> o o

Proof. L e t L be t h e set of points t E T" for which there exists a sequence {Z, e }~=1 such t h a t Z,[~ - ~ 1 u n i f o r m l y , a n d (Z~(xl), Z,(x2), . . . . g~(x~)) --~ t

in T . " - ~ ~

L e t us also d e n o t e b y H ~ {Z(Xl), g(x2) . . . Z(x,)}. W e observe t h a t b o t h L a n d H are closed subgroups of T" a n d t h a t L c H . W e shall prove t h a t L = H .

T o w a r d s t h a t let us suppose b y contradiction t h a t L ~= H . There exists t h e n a character O C T~ o f T ~ such t h a t

O ( L ) = I a n d O(H) % 1. (1)

t h a t

O(lim ( g n ( x l ) , . . . , gn(x~))) = lim O(z,(Xl) . . . g,(x,)) -~ lim X-(Y) = 1.

n n n

c O

B u t this implies t h a t i f {~vn E G},=I is a sequence o f characters such t h a t Since O E T " a n d O(H) 4= 1, there exist n 1 , n 2 . . . n, E Z such O(•(Xl) , Z(x~) . . . Z(x,)) = z ( n l x l + n~x 2 + . . . + n~x~) = Z(Y) V g e G where y = nix 1 + n~x~ t - . . . . + n,x, :/: O.

L e t n o w {xn}~=l be a sequence of characters such t h a t t h e sequence of points {(gn(xl) , . . . , g,(xr)) E T'}n~=l converges t o w a r d s a p o i n t o f L, (1) implies t h e n t h a t we have:

A P R O B L E M O N T H E U N I O N O F H E L S O N S E T S 195

~n [K ---> 1 uniformly on K t h e n ~n(y)--> 1

It--> ~3

and from t h a t using Corollary i of [2] we deduce the required result t h a t y E Gp (K).

Proof of the theorem.

Case 1. D

Let

/ x E M ( K U D ) ;

I~ E M(D).

ll~xll~

ll~l[

I]/~=ll~ => ~l[~I[.

We first observe t h a t there exists ~ E R a n d Z l , Z2 E G such t h a t

Re e'~,(zl) >_ 11/611 - ~- (3)

Indeed choose q E R and Ze E G such t h a t (2) holds. L e t us show t h a t we can find h E ~ satisfying

(3).

We can choose ~ E

C(K), etep

[~1 ~ 1 such t h a t

f ~ d ~ >_ I I ~ l l - 8/2

a n d then since K is Kronecker we can approximate ~ uniformly b y h E G as close as we like. This shows t h a t Zl can be chosen to satisfy (3).

Now from the lemma we can find a sequence ~n E G such t h a t ~v~ --> 1 uniformly on K a n d

We have t h e n

and Therefore

YJ" [D --+ Z ~-1z2 [D"

iq~ ^

R e e ~2(XxY-) - + R e e ~ ~ ( Z 2 ) 9

I1~11oo >~ sut) I~(x~w.)l ~ sup Re {e'~p(X~.)} ~ II/~I1 -4- ~1I/~.11 -- 2~

n r t

>_

~(11~11 + I1/~II) - 2~ ___ ~1>I1 - 2e.

S i n c e e is arbitrary the conclusion of the theorem follows.

Case 2. D

L e t # E M ( K U D ) we can write

P = / 6 +/~2 +/~3, /11 q

M(K), /as,/~3

M(D)

196 A R E / V F O R I~TEMATI]~. r e 1 . 9 ^{N o .} 2

with the support of /~2 Tying in a finite subset of D and 11/~311 ~ s (s ~ 0).

Now b y case 1

Therefore

1[Pll

And since e is arbitrary this completes the proof of the theorem.

I t is m y pleasure to express m y gratitude to Dr. N. Th. Varopoulos for his advice and critisisms and the Mittag-Leffler institute for its stimulating hospitality.

References

1. I=~UDIN, W., Fourier analysis on groups. Interseience, No 12, 1962.

2. V ~ o P o u L o s , N. TH., Sur les ensembles de Kroneeker. C. R. Acad. Sci. Paris 268 (1969), 954-- 957.

3. --))-- Groups of continuous functions in harmonic analysis. Acta Math. 125 (1970), 109-- 154.

Received May, 1970 E. Galanis

D e p a r t m e n t of P u r e M a t h e m a t i c s 16, Mill L a n e

Cambridge E n g l a n d