Mathematics. - On the uniform distribution modulo 1 of lacunary sequences. By P. ERDÖS and J. F. KOKSMA. (Communicated by Prof. J. G. VAN DER CORPUT.)

(Communicated at the meeting of February 26, 1949.)

§ 1. As is weIl known one caBs the sequence oE real numbers UI' u2' ...

uniformly distributed modulo 1, if the number N' of those among the numbers

u,-[u,], U2 - [U2]' ••• ' uN-[UN],

which fall into an arbitrarily given part a :s; u < fJ of the unit interval o :;;; u < 1 satisfies the condition

N' N ~8-a, iE N~ 00.

The dif ference I ~ - (fJ-a) I is always :;;; 1 and for fixed N ;;;;: 1 one calls

its upper bound, (if (a, fJ) is supposed .to run through all couples 'with 0;:;;; a < fJ:;;; 1), the discrepancy D(N) of the sequence. IE

ND (N) = 0 (N), • (1)

it is trivial that the sequence is uniformly distributed modulo 1 and as was proved by WEYL 1), inversively (1) is a consequence oE the distribution modulo 1, defined above.

One gets an interesting special case when putting

Un = () ln (n = 1, 2 •... ) . (2) wh ere

(3)

denotes an increasing sequence of integers. FATOU 1) already proved that such a sequence is everywhere dense modulo 1 in the unit interval Eor almost all values of 0, provided that the sequence (3) is lacunary, i.e. th at for some positive constant !5

(n = 1. 2 .... ). • (4)

HARDY-LITTLEWOOD 1) and WEYL 1) proved that for each sequence of integers (3) the sequence of numbers (2) is uniformly distributed modulo 1 Eor almost all O. Hence Eor such sequences (1) holds. FOWL.ER 1), KOKSMA 1) and DREWES 2) deduced improvelI!ents of (1). In the special case

ln=2n,

1) References in "Diophantische Approximationen", Erg. d. Math. IV, 4 (1936) by J. F. KOKSMA (Kap. VIII and IX).

2) A. DREWES, Diophantische Benaderingsprob1emen, Thesis Free University, Amster­dam (1945).

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the problem is equivalent to the question how the digits O. 1 are distributed in the dyadic expansion of (J. Here KHINTCHINE 1) proved very sharp results.

Generally speaking. the problem is somewhat easier to handle for lacunary sequences (4) than in the general case (3). In this paper we consider the case of lacunary sequences of numbers

un =f(n. 8).

which form a generalisation of the sequences defined by (2). The method used in this paper leads to great difficulties. if one tri es to apply it in the g~neral case. In a following paper we treat the general case with an other ruethod. which in the specialised cases which are considered in the present paper would give a slightly less sharp result than we deduce here.

§ 2. In this paper we prove a general theorem in which as a special case is contained the following

Theorem 1. Let d denote an arbitrary positive constant and w(n) a positive increasing function of n = 1. 2. ... with w (n) ~ 00, if n ~ 00.

Then [or any sequence of positive numbers À,1' À,2' .... whïch satisfy (4), the discrepancy D(N) of the sequence (2) satisfies the inequality

ND (N) = 0 (Ni logl N (log log N)i w (N)) . • (5)

for almost all (J.

This estimate is sharper than all known results. The exponent ! in the factor Ni cannot be improved. as KHINTCHINE proved th at in the special case À,n::'= 2n • we have

ND (N) = Q (Ni -Vlog log N).

Another application of our theorem is the foIIowing

Theorem 2. For almost all values af (J;;;;: 1 the discrepancy af the sequence

satisfies the inequality (5). if w (n) denates a pasitive increasing functian such that w(n) ~ 00 as n ~ 00.

That the sequence (J. (J2. ••• for almast all (J is uniformly distributed (modulo 1) had already been proved by KOKSMA 1). whereas the sharpest estimate for the discrepancy of this sequence known till now was given by DREWES 2).

§ 3. The theorems quoted above are contained in the following Theorem 3. which itself is a special case of the main Theorem 5.

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Theorem 3. Let e<b, b>O be given rea 1 numbers. Let [(1,0). [( 2. 0) . ... , denote a sequence of real functions which are defined on a ;;;;; 0 ;;;;; b, such that

{i; (n + 1.8) =- (1 + b) fó (n. 8) > 0: {i;' (n + 1. 8) =- (1 + b) {i;' (n. 8) =- 0 (n = 1. 2 •... )

for all values of 0 on a ;;;;; 0 ;;;;; b. Let w (n) denote an increasing function of n = 1.2 ..... such that w(n) ~ co as n ~ co.

Then for almost all 0 on a;;;;; 0 ;;;;; b the discrepancy in the uniform distribution of the sequence

{(l, 8). {(2. 8) •... satisfies the rdation (5).

Remark. It is clear that the sequences of the theorems 1 and 2 satisfy the conditions of Theorem 3. In the first case we put without loss of generality a = O. b = 1 and in the second case we put a = 1 + b. b> a and after application of Theorem 3. we let

b .... O. b .... oo.

The reader will find the deduction of Theorem 3 from Theorem 4 in § 8.

§ 4. Forthe proof of our main theorem we deduce a lemma (Lemma 2). which has some interest in itself. For the special case. considered in Theorem 3. it runs as follows:

Theorem 4. Suppose that thé conditions of ,Theorem 3 are satisfied. Let K denote a positive constant. Then for almost all 0 the following statement is true: If N and k are integer such that 1 ;;;;; k ;;;;; NK, then

1 nfl e2ni kf(n,Ol 1-== C (8) Nt logt N (log log N)t w (N).

where C(O) does not depend on N or k. The reader finds its deduction in § 9.

§ 5. Before we state the main theorem. we make some

Preliminary Remarks. Let N and r denote positive integers. Out of the N integers n = 1. 2 •...• N. we can form N' different r~tuples; such an

r~tuple we shall denote by (nv .... n,). There are C:' different r~tuples N

among them for which n1 ;;;;; n2 ;;;;; ... ;;;;; n,. Such a special r~tuple we shall denote also by {nl ..... n,}. The elements n10 .... n, of the r~tuple

{nl • .... n,} have a number of different permutations. which we shall denote by A {nl' .... n,}. Then we obviously have

A I nl ..... nr I -== rI . (6) and

I A Inl •...• nrl =Nr. (7) ~nl'.' . ,nrt

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For later purposes we put

A1= ~ A2Inl •...• nrl . ...... (8) InIJ ' .. ,nrl

Definition. If {nl' .... n,} and {mlo .... m,} are two different r~tuples of the above kind. we say that the first is greater than the second:

Inl.·.·. nrl > I mI •• ··• mrl.

if and only if for some 7: (1 :;;; 7: :;;; r)

n ... > m... , n1' = mI! (e = 1. + 1. 1: + 2 •...• r).

Condition A. Let g(x. 0) for x = 1. 2 ..... N denote a function of 0 on the segment a:;;; 0 :;;; b such that for each coup Ie of r~tuples {nl • .... n, } > > {mlo ...• m,} the function

r r cp(8)=cp(nl' ...• nr:ml •..•• mr:8)= ~ g(nl!'O)- ~ g(me.8) • (9)

e=1 e=1

has a derivative for a :;;; 0 :;;; b. which is continuous. #- O. and either non~ decreasing or non~increasing in the segment a :;;; 0 :;;; b. We th en put

1JI=IJI(nl •...• nr:ml •...• mr)= ~ (10) Min I cpó(nl ..... nr: mI.·.·. mr; a). cp'o(nl' ..•• nr: mI' ...• mr: b) I. ~ .

BN=N-r 2 ~ A Inl ..... nrl AImI ..... mrlj in ...... nrl> Im" ... ,m r: (11)

IJl-I (nl •.•.• n r : ml •...• mr).

Now we state our main theorem:

Theorem 5. I. Let a and b denote rea i constants with a < b. Let f (n. 0) for n = 1. 2 ... denote a real function of 0 on the segment a:;;; 0:;;; b. Let No be a positive integer. Let r = r(N) and s = s(N) be positive in~ tegers whïch are defined for each integer N ;:;:: No, stich that

S (N) -=:: N.

Let for each integer N;:;:: No and each integer a = 1. ...• s(N) the Na func~ tions

ga (X. fi) = f(a + (x- 1) s. 8) (x= 1. 2 ..... Na = [Ns a] + 1) be considered and let the condition A be satisfied with ga instead of g and with Na instead of N.

11. Putting

Bi.; = Max BN •. . . . . . . . (12) l ~ '!J'$.s '7

we assume that a non~decreasing sequence 111 ( 1 ). 111 (2), .• , of positive numbers exists such that the series

co ~ ([N-sJ ) (-2r n~No S • I (b-a) r! Nt + Bi.; log NI? 111 -s- + 1 ~ . (13)

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converges. Then almost all numbers (J of a :s; (J :s; b, have the property that the discrepancy D(N) of the sequence [(1. (J). [(2. (J) .... satisfies the inequaHty .

ND(N)-=K1Si.Ni.tp([Ns lJ+l)logN forN=:=-Nö •• (1-t)

where Kl denotes a numerical constant, whereas No denotes an index depending on (J.

§ 6. We now prove

Lemma 1. Let N and r denote positive integers and let g(x, (J) satisfy the condition A. Then for each fixed integer h # 0 we have

b

11 1: e 2:>ihg(x.O) 12r

d8 = (b-a) AN+ 2~ BN Nr (I ~I-= 1). . (15) x=1 :nh

a

Proof.

1

1: e.21fihg(X.O)12r = Î ; e2nihg(x.8)t ~ ; e-2ni hg(x. 0) V x=1 (X=I ~ ~X=I ~

I e2nih(g(n .. O)+ ... +g(nr.0))· I e-2nih(g(n .. 8)+ .•. +g(nr.8)).

where both sums are to be expanded over all N' r~tuples of integers

n(! = 1. 2 •...• N.

Applying the preliminary remark of § 5. we write

A In n I e -2nih(g(n .. O)+ ... +g(nr.8)) l' ••• , r

I A2Inl ..... n,I+2 I I In, ..... nrl In ..... n r : > :m ...... mrl

by (9). Hence by (8)

b . I 1 x~ e2ni /zg(x.O) IU d8 = (b-a) AN + a b. (16)

+2 I I Alnl •...• nrIAlml •...• mrIIcos2:nhC/)(8)d8. Jn" ... ,nrl> Im ...... mrl

a

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Now choosing the new variabIe of integration u by the substitution

u = cp (8),

we find

du = CP' (0) dO and

b 4> (bJ I cos 2nh cP (0) . dO = I cos 2n hu cp~rO) a 4>(~

and therefore using BONNET'S theorem and (10) b IJ cos 2n h cP (0) . dO 1-== n hl IJ' •

a

Hence by (16) and (11)

b

I1 xgl e2nlhg(x,OJ rr de = (b-a) A1 + 2:r;:r B,v' ([ ~ [-== 1). a

Q.e.d.

Lemma 2. Let the conditions 1 of the main theorem 5 be satisfied. 11. Defining BN by (12), we suppose that a non~deereasing sequenee 'IJ' ( 1 ), 'IJ' (2), ... and a non~deereasing sequenee of integers A ( 1 ), A ( (2), .. . (A(N) ~ 3, if N ~ No) exist, su eh that the series

n!NoS I (b-a) r!.ti (N) + 2BN log.ti (N)j ~ 'IJ' ( [N S SJ + 1) r2r

• (13a)

converges. Then almost all numbers () of a::;;; () ::;;; b have the property that for all integers h = 1. 2, ... , A(N)

Inie2nih/(n,BJI-=2SiNl'lJ' ([Ns

1J+1)' if N~No(O) . (17)

Proof. Let N be a fixed integer ~ No. Let (h, a) denote a couple of integers which satisfy the inequalities

1 -== h -= .ti (N), 1 -== a -== s.

Then the Lemma 1 with

g (x, 0) = g~ (x, 0) = f(a + (x-I) s,O), N= Na

learns b

I1 x~: e2nlhga(x,0) rr dO -== b-a) r! N~ + ~ BNaN~ • . (18) a

because of (6) and (7).

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Now let S(N. h. a) denote the set of all numbers () on a:;;; () ~ b for which

111 e2nlhga(X.O) 1 ~ N~ VJ (Na) . . . . . • (19)

Then by (18) we obviously find for its measure mS (N. h. a) the inequality

hence

m S (N. h. a) -== ~ (b -a) rl + ! BNa ~ !VJ (Na)l-2r

and therefore by

[N-sJ Na ~ -s - + 1 and (12) :

mS(N.h.a)-==~(b-a)d+! BivHVJ([Ns sJ+l)f2r.

Now let S(N) denote the set

S(N) = :E S (N. h. a). (h.,)

wh ere the summation is to be extended over all couples (h. a) which satisfy 1 ;:;;; h;:;;; A(N). 1;:;;; a::S; s. Then we have

mS(N)-==~(b-a)rlsA(N)+SBiv:~7! OVJ([N., SJ+lr2r

< s !(b-a)rl A(N) + 2Biv log A (N)} ~VJ ([N s sJ + 1) r2r

•

(20)

Each () of a::S; () -;;, b. which does not belong to S(N) (N"2:, No) has the property that the inequality

1 X~le2nlhga(X,O) 1-== N: VJ (Na) < (N s 1 + 1 r VJ ([N s IJ + 1) is valid for all couples (h. a) which satisfy the inequalities

1 -== h -== A (N) • 1 -== a -== s. Therefore we have for such a ()

I; e2nlhf(n,oJI-==Ii: i e2nlhga(x,O)I-==2siNiVJ([N-l]+1).

n=1 a=lx=1 S

for all integers h = 1. 2 • .... A(N). Now as mS(N) satisfies (20) and as the series (13a) converges. almast

all numbers () of a::S; () ::s; b belang to at most a finite number of the sets

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S(N) (N = No, No + I, ... ). Therefare far almast all fJ af aS; fJ S; b an index No can be faund, such that

1 n~l e2nih !(n,O) 1-== 2si Ni V' ([N s IJ + 1)'

whatever the value af the integer h = 1. 2, ... A(N) may be. Q.e.d.

§ 7. In arder ta prave the main thearem, we quate the fallawing thearem, which is an improvement praved by EROÖS-TURÁN 3) af the ane dimensianal case af a thearem af VAN DER CORPUT -KOKSMA 1).

Lemma 3. If Ul' U2' ... is a real sequence and if D (N) denotes its dis­crepancy, then far each integer m 2:: L we have

ND(N)-==K~m~ 1 +h~l ~ li/2nihunl~· ... (21)

where K denotes a numerical constant.

Proof of the Theorem 5. Put

Un=f(n.8), m=[VN], A (N) = [VN].

Then (using Lemma 2) fOl; almast all fJ we have by (17) and (21), if

N 2:: No (fJ)

ND (N) -== K VN + K[~l ~ st. Ni V' ([N-l] + 1) h=l h s

-== Kl si Ni V' ([N si] + 1) log N. Q.e.d.

§ 8. Proof of Theorem 3. Be w(N) the functian af Thearem 3 and Iet No be a sufficiently large integer. We shall prave that the functians [(n, fJ) af Thearem 3 satisfy the canditians af Thearem 5, if we put far N 2:: No

s (N) = eOg (~ + (5) log log N ] ; • (22)

r (N) = [ ~g N ] + 1 ; V' (N) = Vlag N2 . V w (N) lag w ([VN]

where c5 denotes the canstant af Theorem 3. Now for N 2:: No we consider the s = s(N) sequences

g~(x, 8)=f(0+(x-l) s, 8) ( 1 -==0-== s; x= 1. 2, ... , N a = [Ns 0] + 1). a) P. EROÖS and P. TURÁN, On a problem in the theory of uniform distribution.

Proe Kon. Ned. Akad. v. Wetenseh., Amsterdam, SI, 1146---1154, 1262-1269 (1948); Ind. Math. 10, 370-378, 406---413 (1948).

272

By hypothesis we have (the prime meaning diHerentiation by 0):

f' (n + 1. 8) =:: 1 + 15 f' (n. 8) - •

hence for N ;;;: N ()

, ( + 1 8) • 100100N ga ~ ( 8) . ==- (1 + d)S > (1 + 15) 111001l+J) = (log N)"/. > r + 1.

ga x.

Therefore if we take two r.tuples

we have. using the notation of the Preliminary Remarks:

CP' (8) = g~ (nl' 0) + ... + g~ (nr. 8) -g~ (mI' 8)- ... -g~ (mr. 0)

==- g~ (nT. 0)- rg~ (nT-I. 0) > 9~ (nT- 1. 0) ==- f' (1. a) = Co (23)

and we conclude that if we range the r.tuples {nv .... n,} in order of in· creasing magnitude. the corresponding sums

with each step will increase by at least the amount Co. whatever the value of 0 (a ::;;:; 0 $ b) may beo Hence. if the r.tuple {nv .... n,} is fixed. we have by (10) and (9)

as there are at most N; ~ N' r.tuples {mlo .... m,}. Therefore we find by (11) and (6)

B :::::: N-r 11 + dog N ~ A I I rI (1 + 1 N) Na - a r ~ nl ..... nr = - r og Co In" .. .• nrl CD

by (7). Hence we find by (12) a fortiori

BN oooc:::: 2 r r Ilog N < rr log N for N ==- No. Co

Thus we find that the general term oE our series (13) is at most

t (b-a)srr Ni +srrlog2 NI I1/' ([iN] + 1)1-2r oooc::::

oooc:::: Cl S (log NV Ni . ilog N- 2r I i w ([VNlW2r

by (22) and therefore by (22)

2 looN

oooc:::: Cl s Ni I i w ul" N])i 100 V",([VN]) < ~ L iE N ==- No.

Hence. our series (13) converges.

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By hypothesis we further have f(~+l,O) ;;:;: (1 + c5) f;~,O) ;;:;: O. Repeating the proof of (23) with f" instead of f' and g~ instead of g~,

we find CP" (e) ;;:;: O. Hence eb' (e) is non-decreasing. From our result we conclude that for almast all e the inequality (14)

holds. i.e. because of (22):

ND (N) -== K 2 Ni (log NP (log log N)l y' w (N), if N =- No (8).

Hence (5) follows immediately.

§ 9. Pro of of Theorem 4. We shall use Lemma 2 and we put

A(N) = [NK], s (N) = [ log (~ + d) log log N J. r(N) =

= [-iK + 2) I~~ N ] + 1, tp (N) = -V log N2 log w ([} N])

I (24)

w(N), ~. where K. c5 and w(N) are defined in Theorem 4. Then in exactly the same way as in § 8 it follows th at

BN -== rr log N for N =- No

and thus the general term of the series (13a) is at most

I (b-a) srr NK + srr. 2Klog2 NI I tp( [-VNJ + 1) 1-2r-==

-== C2 s (log NV NK (-Vlog N)-2r Iw ([ll N]) 1-2r

by (24) and therefore by (24)

-(HK)loIlN

<c2s·NKlw([-VN])!log",([YI\]) <N-t, if N=-No.

Hence, the series (13a) converges. From our result we conclude that (17) holds for almast all e on a :s; e :s; b, i.e.

1 ni e2"1 h/(n,O) 1-== K3 Nllogl N (log log N)l w (N), if N=- No· (8).

Pram this Theorem 4 follows immediately.

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