(2) f On Bohr's spectrum of a function By HANS WALLIN ARKIV FOR MATEMATIK Band 4 nr 11

(1)

A R K I V FOR MATEMATIK Band 4 nr 11

C o m m u n i c a t e d 28 O c t o b e r 1959 b y L E ~ A ~ T CA~LESO~r

O n B o h r ' s s p e c t r u m o f a f u n c t i o n By HANS WALLIN

Let q~ be a measurable and bounded function. We will examine the complement of the set of real values t for which

T

- T

Eggleston [2] has shown t h a t the exceptional set is not necessarily enumerable, but t h a t it has Hausdorff measure zero with respect to ~logr~ , e > 0 . The fol-

k - / lowing more general results are true.

T h e o r e m 1. The set where (A) is false has Hausdor// measure zero with respect to every increasing/unction h(r), h(O)= O, such that

1

f

^h(r)dr^r ^{~ .}

0

T h e o r e m 2. There exists a closed set o/positive logarithmic capacity where (A ) /ails to hold/or a suitably chosen 9).

Suppose [ T ( x ) ] < 1 and let us put

T

1 f eUXd g~ (t) = 2 T qJ (x) x

- T

and note two preliminary relations

c ~

b b

If IgT(to)l>b then Igr(t)l> 2- if I t - t 0 1 < ~ .

2 Tg T (t) = fq)l (x) eaXdx,

- o o

(1)

(2)

159

(2)

H. W,S-LHN,

On Bohr's spectrum of a function

where ~ (x) = q~ (x) for Ix I ~< T a n d ~o (x) = 0 for Ix I > T.

According to the P a r s e v a l relation we h a v e

oo oo

j

~-~ 2Tgr(t)l 2tit= Icf~(z)l~dx<<-2T,

- o o - o o

which gives (1). (2) follows from the fact t h a t

T

IgT(t)--gT(to)l=]2 ~ f~o(x)(e'tX--eU~ <~

- T

T T

< ~ le,tZ_e,t.~[dx 2 isinlt-toIx]dx<T _tol<b.

- r 0

if It-tol<p. b

P r o o f of t h e o r e m I . L e t h (r) be an a r b i t r a r y function satisfying the conditions in t h e o r e m 1. I t is enough to show t h a t t h e set where lim I g r ( t ) l > a has zero H a u s d o r f f measure with r~spe.~t t e h (r) for arbitrarily small values of a. Suppose a is chosen, 0 < a < 1. I f

m r (a)

is the Lebesgue measure of the set where l gr (t) l > 2 ' t h e n (1) implies

4 ~

m r (a) < a2 T" (3)

L e t NT (a) be the m a x i m a l n u m b e r of points t for which I gr (i) I > a ~ a n d which

g

are f u r t h e r m o r e situated a t a distance of a t least ~ f r o m each other. According to (1) and (2) we h a v e

N~(a/" "~<~,

N r (a) < 16~ aa (4)

Hence,

NT(a)<N (a)

which is independent of T.

4 ~ a a

I t follows ~ - ~ ... (?) and {4), ~ince -~:~_>~ .: ~. ~ , t h a t f~r e v e r y T I g r ( t ) l > _ in 4 ~

a t m o s t N (a) different, intervals of length ~ each.

160

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A R K I V F O R M A T E M A T I K . B d 4 n r 1 1

N o w let us consider the set of values t where

lim J gr~ (t) I > 2 ' (5)

?L--~ oo

a n d where (T~}~ ~ is chosen so t h a t

Tn+l

= k = k ( a ) > l . Tn

(5) being true for a certain t means that, for this value of

t, I gT,(t)] " a

/ ~ for infinitely m a n y n. This obviously means t h a t (5) can be satisfied only in such sets E which can be covered infinitely m a n y times b y intervals {In}, where I n has length

4 ~

a ~ T~-~ a n d each I~ m a y be used only N (a) times at the covering. I t is thus possible, for n o arbitrarily large, to cover E with intervals

In, n>n o.

^{B u t}

c/T~o N(a)h(cT;1)<N(a) f h

n=~o+l

lo-gk r(r) dr'

0

4 g

where c = ~ f f 9 The right h a n d side, however, is arbitrarily small, a n d so (5) can only be satisfied on sets E of h-measure zero. N o w suppose t o is such t h a t (5) is n o t satisfied. Choose n = n ( T ) such t h a t

T~+I>T>~Tn.

F o r T large enough we have

- - T n T

-- 2" Tn

_<a

T~ 2

if k = 1 + 2" This yields li--m

[gT (to) I ~ a.

W i t h this it is shown t h a t the set where lira I gr (t) I > a, has vanishing h-measure.

l a r o o f of t h e o r e m 2. L e t us consider g2~ (t), where n is a positive integer. We can always find a bounded function ~ such t h a t ] g2~ (to) [ >/89 where n and t o are arbitrary. Namely, if ~ is already chosen i n ( - 2 ~ , 2 n) such t h a t g2n (to)l~ > 89 and we w a n t to have the inequality Ig2n+l (tl) l~> 89 satisfied, we choose ~0 (x) = e *~ e -'t-x for 2 4 < x ~ 2 n+l w r 9 I I , he e 01 = arg{g2n (tl)}. B y (2) it then follows t h a t such sets which can be covered infinitely m a n y times b y intervals {In}F, where In has length 2 -n, are exceptional sets. There is a closed set with this quality which has positive capacity (Carleson [1]), a n d so the theorem is shown.

Eggleston's result a b o u t the set where (A) is false is shown as a consequence of theorems, due to Erdhs a n d Taylor [3], concerning the set of values of t for

t ~

which the nonintegral parts of t h e sequence {nk } k = l a r e n o t equidistributed, 161

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H. WALLIN, On Bohr's spectrum of a function

where nk is an increasing sequence of integers for which nk+l--nk< C, some con- s t a n t C.

T h e t h e o r e m s of ErdTs a n d T a y l o r say t h a t the set where the non-integer p a r t s of {nk t} are n o t equidistributed m a y be nonenumerable, b u t t h a t it has H a u s d o r f f

/ 1\-1-~

measure zero with respect to ( l o g r ~ , e > 0 , if {nk} satisfies the conditions

\ - - /

above. The theorems (1) a n d (2) a b o v e say, according to Eggleston's result t h a t the exceptional set, where the non-integer p a r t s are n o t equidistributed, m a y h a v e positive c~pacity, b u t t h a t it has vanishing H a u s d o r f f m e a s u r e with respect to h(r), where h(r) satisfies the conditions in t h e o r e m 1.

I t m a y be noted t h a t if, instead of boundedness, we assume t h a t ~ (x) = O( ] x I~), /~ < 89 when Ix I-->~ a n d t h a t ~ is b o u n d e d in e v e r y finite interval, we get a n analogous result. T h u s it can be shown b y a similar m e t h o d as a b o v e t h a t in this case the exceptional set where (A) is false, has vanishing a - c a p a c i t y if a > 2 f t . T h e fact t h a t (A) holds a l m o s t everywhere, if f l < 89 is a result of W i n t n e r [4], who p r o v e d t h a t if ~0 belongs to L 2 in e v e r y finite interval a n d

T

f I ( )l dx=O(T

--T

for T - - ~ a n d some s > 0 , t h e n the exceptional set has Lebesgue m e a s u r e zero.

R E F E R E N C E S

1. L. CARLESO~, On t h e connection b e t w e e n H a u s d o r f f m e e s u r e s a n d capacity, A r k i v ]Sr mate.

matik, B . 3, n r 36 (1957).

2. H. G. EGGL~STO~r, T h e B o h r s p e c t r u m of a b o u n d e d f u n c t i o n , Proc. A m e r . M a t h . Soc., vol. 9

(1958).

3. P. Ea~DSS a n d S. J . TAYLOR, O n the set of convergence of a lacunaxy t r i g o n o m e t r i c series a n d the e q u i d i s t r i b u t i o n p r o p e r t i e s of related sequences, Proc. L o n d o n M a t h . Soc., 7 (1957).

4. A. WI~rTNER, O n F o u r i e r averages, A m e r . J . Math., 63 (1941).

1.62

Tryckt den 30 januari 1960

Uppsala 1960. Almqvist & Wiksens Boktryckeri AB