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
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) implies4 ~
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
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 in- finitely 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 length4 ~
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 tc/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
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