Proc.
766
169.
On
Japan
Lacunary
Acad.,
Trigonometric
By Shigeru
of Mathematics,
Department
(Comm.
44 (1968)
[Vol.
Series.
TAKAHASHI
Kanazawa University,
44,
II
Kanazawa
by Zyoiti SUETUNA,M. J. A., Oct. 12, 1968)
§ 1. Introduction.
In [3] we have proved
Theorem A. Let {nk} be a sequence of positive integers and {ak}
a sequence of non-negative real numbers for which the conditions
(1.1)
nk+>nk(1 +ck-a),
(1.2)
N
AN = (2-1
k=1, 2, ...
a)112->+ oo,
as N--> + oo,
k=1
and
(1.3)
aN = o(ANN-a),
as N-~ + oo,
are satisfied, where c and a are any given constants
(1.4)
c>0
and
0<a<1/2.
Then we have, for all x,
N
(1.5)
lira {t; t e E,
N-+~
such that
a,, cos 22rnk(t + ak) < xAN} / IE
k=1
=(27r)_h/2
x
exp(_u2/2)du,*)
where E c [0, 1] is any given set of positive measure and {ak} any given
sequence of real numbers.
This theorem was first proved by R. Salem and A. Zygmund in
case of a -0, where {flk}satisfies the so-called Hadamard's gap condition (cf. [4], (5.5), pp. 264-268). In that case they also remarked that
under the hypothesis (1.2) the condition (1.3) is necessary for the validity of (1.5) (cf. [4], (5.27), pp. 268-269).
Further,
in [2] P. Erdos has pointed out that for every positive
constant c there exists a sequence of positive integers {nk} such that
nk+l > nk(1 + ck-1/2), k>1, and (1.5) is not true for ak =1, k>1, and
E = [0, 1]. But I could not follow his argument on the example.
The purpose of the present note is to prove the following
Theorem B. For any given constants c>0 and 0<a<1/2,
there
exist sequences of positive integers {nk} and non-negative real numbers {ak}for which the conditions (1.1), (1.2) and
(1.6)
aN = O(ANN-a),
as N-> + oo,
are satisfied, but (1.5) is not true for E=[0, 1] and ak = 0, k>1.
The above theorem shows that in Theorem A the condition (1.3) is
*) E~ denotes
the
Lebesgue
measure
of a set
E.
m
No. 81
Lacunary Trigonometric Series. II
indispensable for the validity
for 0<a<_1/2.
of (1.5).
767
In §~ 3-5 we prove Theorem B
§ 2. Some lemmas.
i. In this section let {Xk(w)} be a sequence
of independent random variables on some probability space (Q, 3, P)
with vanishing mean values and finite variances.
Putting E(X) = Qk
and sm= E ak, the theorem
m
of Lindeberg
reads as follows :
k=1
Theorem.
(Cf. [1] pp. 56-57.) Under the hypotheses
(2.1)
sm -~ + oo and o = o(sm), as m-* + oo,
a necessary and sufficient condition for the validity of the relation
m
(2.2)
lim P{sm1
x
Xk(w)<x}=(27r)-112
m--->~k-1
for all x is that, for any given
J-~exp(-u2/2)du
>0,
lim sm2
Xk(w)dP(w)
m-~k=1
JXk>,sm
= 0.
From the above theorem the following lemma is easily seen.
Lemma 1. Under the hypotheses (2.1), the relation (2.2) implies
that, for any given >0,
m
(2.3)
lim
m-~
P{X k(w) > s}=
0.
k=1
ii. Lemma 2. Let m and l be any given positive integers. Then
there exists a positive constant c0, not depending on m and 1, such that
{t; t e [0, 1],
E cos 2~cm(j + 1)t >(l+ 1)/3} >2c01'.
j=0
Proof. This can be easily seen from the relation
~
2)t
cos 22rm(j+1)t= sin 2~cm(l+3 /
-1/2,
j=o
2 sin ?rmt
provided if sin Trmt 0.
§ 3. Construction of sequences.
In the following let c > 0 and
0<a1/2
<_ be given constants in Theorem B. First let us put
p(i)==[j 1/a],
(3.1)
1(j)= Min {[pa(j)/ c], p(j +1) - p(j) -1 },
jo=Min{j; l(j)>1}.*)
Since p(j+1)-p(j)~a-1j(1-'
)la and p~(j)~ j, as j-->+ oo,**) we have
(3.2)
1(j)~ j3(aQj,
as j-~+ co
where
(3.3)
(a)=
1/c,
Mi
n(2, 1/c),
if 0<a<1/2,
if a=1/2.
Next we put
n1=1
and nk+1= [nk(1 + c1c ) + ii,
If np(j), j > jo, is defined, then we put
*) [x] denotes the integral part of x.
*) f (j)~g(j),
as j-~ + co, means that f (j)/g(j)-4,
for k + 1 < p(jo)
as j-> + ao.
n
N
768
S. TAKAHASHI
[ ol. 44,
- J np(j)(1 + l),
if 1 <l <l(j),
pc~)+l [n
)+l-1{l + cp-a(j)} + 11,
if l(j) < l <p(j + 1) -p(j).
Further we put, for j > jo,
(3.4)
np(j)=2gcj),
where
Min[m ; 2m/ np(J)-1>1 + c{p(j0) - 1}-a, if j = j0f
(3.5)
q(j) = Min[m; 2ml np(j)-i>1 +c{p(j)-1}-a
and 2m> np(j-i)j3],
if j > jo.
Then it is clear that the sequence {flk}satisfies (1.1).
On the other hand we define {ak} as follows :
-
(3.6)
if p(j) <k<p(j)+l(j),
, if otherwise.
Then we have, by (3.6) and (3.2),
m
a k=
1,
1/k2
for some j>j0,
(3.7) A(m)+I(m)=2
2p-1
E {l(j)+ 1}+ 0(1) ti ~(a)m2/4,
=20
as m~ + oo.
Since pa(j) j, as j-> + oo, this sequence {ak} satisfies both of the conditions (1.2) and (1.6).
Further,
if we put
SN(t)=
ak cos 2lrnkt,
then
we have,
by the
k-i
definitions
(3.8)
of {nk} and {ak}, uniformly
m
l(j)
3=jo
l=0
Sp(m)+l(m)(t)-
in t,
cos {2r2q(j'(1 + l)t} + 0(1),
as m-+
oo .
§ 4. Independent
functions.
Let xk(t) be the k-th digit of the
infinite dyadic expansion of t, 0 < t < 1, that is,
(4.1)
t=
xk(t)2-k,
(xk(t)=0
or 1),
k=1
then {xk(t)} is a sequence
[0, 1]. Putting
of independent
functions
on the interval
q(j+1)-1
(4.2)
~~(t)=
k=q(j)
xk(t)2-k'
for j>_j0,
we define
1 l(j)
,I
~j
(4.3)
cos 272(3)(l + 1)i j(t)dt,
0 l=0
l(j)
Y3(t) _
cos {2'r2(l
q(j'+
1)~ j(t)} -
l=o
1
zj=
m
Y;(t)dt
and
2
tm-
2j.2
=2o
0
Then
we have,
by (4.2) and
(3.5),
l(j)
sup
t
l(j)
cos {2rc2q(j'(l+ 1)t} t=o
cos 2jr{2q(~'(l+ 1)~ j(t)}
t=0
0(j)
O(sup
1
xk(t)2-k
2q(j'
k-q (j+i)
=O(2c2)j2 k=q(j+1) 2-k)
= O(2q(i)-q(i+1)j2) = 0(j-1),
(1+1))
l=0
as j-~ + oo.
xm
m
No. 8]
Lacunary
Therefore,
Trigonometric
Series.
II
769
we have
0(j)
(4.4)
sup
Y (t) -
cos 27c2~(j)(l + 1)t
t
as
=
J--~+oo,
a=0
and, by (3.2) and (3.7),
m
(4.5)
tm=2-1
(l(j) + 1)2+ 0(log m)
j=jo
Ap(m)+l(m) j3(a)m2/4,
Thus we obtain
(4.6)
tm-~ + oo and Zm= o(tm),
Further, we have, by (3.8) and (4.4),
m
(4.7)
Sp(m)+b(m)(t)-
Y (t) = 0(log
as
m->
+ oo .
as
m-*+oo.
m)
j=jo
= o(Ap(m)+l( m))
uniformly in t, as
,
Since (4.5) and (3.2) imply that 4j3(a) tm / 5 < {l([m / 2]) + 1}/4,
we have, by (4.4)
{t; 0<t<1,
j=m/2
m
[t;0<t<1,
m>
m0,
tm/5}
~ cos 2TC2q(j)(l+1)t>{l(j)+1}/3]
j=m/2
by Lemma
2, we have
(4.8)
for
oo .
0(j)
>
and
Yj(t)(>~~(a)
m
m->+
,
{t ; 0 <_t < 1, Yj(t) > N~(a) tm/5}
lim
m--~
l=0
j=m/2
m
>lim 2c0
r
§ 5.
(5.1)
m-~
l-1(j)
j = m/2 =
m-~
j=m/2
Proof of Theorem B. Suppose that (1.5) holds, that is,
lim {t ; 0<t<_1, Ap m)+l (m)
S p(m)+l(m)(t) ~ ~}
=(2Tc)-1/2
exp(-u2/2)du.
Then by (4.7) and (4.5), we have
m
(5.2)
lim I{t ; 0 < t < 1, tm'
Y j(t) < x}
m-'
x
= (2?r')-1/2
exp (- u2/ 2)du.
On the other hand (4.1), (4.2), and (4.3) imply that {Y3(t)} is a sequence
of inde endent functions with vanishing mean values and finite van-p
antes. By (5.2) and (4.6) we can apply Lemma 1 to {Y2(t)} and obtain
lim
~
m--'= j=m/2
This contradicts
~Y (t) ~>
~ t ;0<t<1,
tm/5} =o.
with (4.8).
References
[1]
H. Cramer:
Random
Tract (1962).
Variable
and its Probability
Distribution.
Cambridge
770
[2]
[3]
[4]
S. TAKAHASHI
P. Erdos:
On trigonometric
Int. Kozl., 7, 37-42 (1962).
S. Takahashi:
On lacunary
503-506 (1965).
A. Zygmund : Trigonometric
(1959).
[Vol. 44,
sums with gaps.
trigonometric
Series.
Magyar
series.
Vol. II.
Tud.
Proc.
Cambridge
Akad.
Japan
Kutatos
Acad.,
University
41,
Press