TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 362, Number 11, November 2010, Pages 5967–5982 S 0002-9947(2010)05026-3 Article electronically published on June 16, 2010

ON THE LAW OF THE ITERATED LOGARITHM FOR THE DISCREPANCY OF LACUNARY SEQUENCES

CHRISTOPH AISTLEITNER

Abstract. A classical result of Philipp (1975) states that for any sequence (nk)k≥1 of integers satisfying the Hadamard gap condition nk+1/nk ≥ q> 1(k =1, 2,...), the discrepancy DN of the sequence (nkx)k≥1 mod 1 satisﬁes the law of the iterated logarithm (LIL), i.e. −1/2 1/4 ≤ lim sup NDN (nkx)(N log log N) ≤ Cq a.e. N→∞ The value of the lim sup is a long-standing open problem. Recently Fukuyama k explicitly calculated the value of the lim sup for nk = θ , θ>1, not necessarily integer. We extend Fukuyama’s result to a large class of integer sequences (nk) characterized in terms of the number of solutions of a certain class of Diophantine equations and show that the value of the lim sup is the same as in the Chung-Smirnov LIL for i.i.d. random variables.

1. Introduction

Given a sequence (x1,...,xN ) of real numbers, the value A(N,a,b) DN = DN (x1,...,xN )= sup − (b − a) 0≤a

is called the discrepancy of the sequence mod 1. Here A(N,a,b) denotes the number of indices k ≤ N with a ≤xk **measure. There is an extensive literature dealing with the asymptotic properties of the sequence nkx (see Gaposhkin [10] for a survey until 1966), but the precise order of magnitude of its discrepancy has been found only for a few special sequences (nk). R.C. Baker [2] proved, improving earlier results of Erd˝os and Koksma [8] and Cassels [7], that for any increasing sequence (nk)of integers the discrepancy of nkx satisﬁes**

− 1 3 +ε DN (nkx)=O(N 2 (log N) 2 ) a.e.

Received by the editors June 2, 2008 and, in revised form, February 19, 2009. 2000 Mathematics Subject Classiﬁcation. Primary 11K38, 42A55, 60F15. Key words and phrases. Discrepancy, lacunary series, law of the iterated logarithm. This research was supported by the Austrian Research Foundation (FWF), Project S9603-N13.

c 2010 American Mathematical Society Reverts to public domain 28 years from publication 5967

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5968 CHRISTOPH AISTLEITNER

for every ε>0. (For simplicity, here and in the sequel we write DN (nkx)instead of DN (n1x,...,nN x).) In the case nk = k, Kesten [14] proved 2 log N log log N D (n x) ∼ in measure. N k π2 N The proof depends on the classical connection of the discrepancy of kx with the continued fraction expansion of x and uses deep probabilistic ideas. Philipp [16] proved that if the sequence (nk) satisﬁes the Hadamard gap condition

(1.1) nk+1/nk >q>1,k=1, 2,...,

then the discrepancy of nkx obeys the law of the iterated logarithm, i.e.

1 NDN (nkx) (1.2) √ ≤ lim sup √ ≤ Cq a.e., 4 2 N→∞ 2N log log N

where Cq is a constant depending on q. This result also has a probabilistic character: comparing with the Chung-Smirnov law of the iterated logarithm

ND (ξ ,...,ξ ) 1 (1.3) lim sup √ N 1 N = a.s. N→∞ 2N log log N 2

(see e.g. [17], p. 504), valid for uniformly distributed i.i.d. sequences (ξk)k≥1, Philipp’s result shows that the sequence nkx behaves like a sequence of indepen- dent random variables. The probabilistic analogy is, however, not complete: the asymptotic properties of the sequence nkx depend also on the number-theoretic properties of the sequence (nk) in an essential way. For example, Kac [12] showed k that in the case nk =2 the sequence f(nkx) satisﬁes the central limit theorem for all “nice” periodic functions f, and Gaposhkin [10] showed that this remains valid r if the fractions nk+1/nk are integers or if nk+1/nk → α,whereα is irrational for r =1, 2,.... On the other hand, Erd˝os and Fortet (see [13], p. 646) observed that k the central limit theorem for f(nkx) fails if nk =2 − 1. As a consequence of the arithmetic connection, the lim sup in (1.2) can be diﬀerent from 1/2, and it is still an open problem if the lim sup is always a constant almost everywhere. Very recently, Fukuyama [9] determined the lim sup in (1.2) for the sequence k nk = θ , θ>1 (not necessarily integer). He showed that the lim sup Σθ equals 1/2 if θr is irrational for r =1, 2,...; i.e., in this case we get the same value as for an i.i.d. sequence. For other values of θ, the dependence of the lim sup on θ is very delicate. For example, Fukuyama showed that √ Σ = 42/9, if θ =2, θ (θ +1)θ(θ − 2) Σθ = if θ ≥ 4isaneveninteger, 2 (θ − 1)3 √ θ +1 Σθ = √ if θ ≥ 3 is an odd integer. 2 θ − 1

k It is worth observing that even though in the case nk =2 the sequence nkx obeys the central limit theorem by Kac’s theorem, the lim sup in (1.2) diﬀers from 1/2, revealing a further delicacy of the problem.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE LAW OF THE ITERATED LOGARITHM 5969

The purpose of our paper is to extend Fukuyama’s results and to compute the lim sup in (1.2) for a large class of integer sequences (nk). Noting that N NDN (nkx)=sup f(nkx) , f∈A k=1

where A denotes the class of functions f = 1[a,b) − (b − a), 0 ≤ a

(1.4) ank − bn = ν, 1 ≤ k, ≤ N.

In particular, Gaposhkin [11] showed that f(nkx) satisﬁes the CLT N t −1/2 −u2/2 (1.5) lim P x ∈ (0, 1) : f(nkx) ≤ tσN =(2π) e du N→∞ −∞ k=1

with a suitable norming sequence σN provided the number of solutions (k, )of (1.4) for nonzero integers a, b, ν is bounded by a constant C(a, b) independent of ν. Here, and in the sequel, P denotes the Lebesgue measure. This criterion covers r the examples nk+1/nk →∞and nk+1/nk → α,whereα ∈ Q,r≥ 1, but it is unnecessarily restrictive. Deﬁne, for (nk)k≥1 and ν ∈ Z,

L(N,d,ν)=#{1 ≤ a, b ≤ d, 1 ≤ k, l ≤ N : ank − bnl = ν} and L(N,d)=supL(N,d,ν),L∗(N,d)=supL(N,d,ν). ν>0 ν≥0 For ν = 0 we do not count the trivial solutions a = b, k = l in L(N,d,ν). In our recent paper [1] we showed that f(nkx) satisﬁes the CLT for all “nice” periodic functions f provided for any d ≥ 1, (1.6) L(N,d)=o(N)asN →∞, and this condition is optimal. Moreover, if we also assume that for any d ≥ 1, (1.7) L∗(N,d)=o(N)asN →∞, √ then the CLT (1.5) holds with the norming sequence σN = f 2 N. In view of this precise characterization result, it is natural to expect that the value of the lim sup in (1.2) is also connected with the behavior of the Diophantine functions L(N,d), L∗(N,d). Our main result below shows that this is indeed the case and provides a near optimal criterion for the validity of the exact LIL for the discrepancy DN (nkx) ∗ and the similar quantity D (nkx) (“star discrepancy”), where N ∗ ∗ A(N,0,a) − DN = DN (x1,...,xN )= sup a . 0

Theorem 1.1. Let (nk)k≥1 be a sequence of positive integers satisfying the Hadamard gap condition (1.1), and assume that for any d ≥ 1 we have N (1.8) L∗(N,d)=O for some ε>0. (log N)1+ε

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5970 CHRISTOPH AISTLEITNER

Then ND (n x) 1 lim sup √ N k = a.e. N→∞ 2N log log N 2 and ND∗ (n x) 1 lim sup √ N k = a.e. N→∞ 2N log log N 2 Clearly L(N,d,ν) ≤ d2N for all N,d,ν, and therefore condition (1.8) is not far from optimal. We note that ε in (1.8) is allowed to depend on d. It is possible that the conclusion of Theorem 1.1 remains valid if instead of (1.8) we assume only L∗(N,d)=o(N), but on the basis of the diﬀerence between the conditions of the CLT and LIL for partial sums of independent random variables, it is more natural to expect that for the exact LIL for DN (nkx) one needs a condition with at least a log log N factor stronger than L∗(N,d)=o(N). Note also that replacing L∗(N,d) by L(N,d) in (1.8), Theorem 1.1 becomes false: the modiﬁed condition is√ easily k seen to hold for nk =2 , but by Fukuyama’s result, the lim sup in (1.2) is 42/9 in this case. It is easy to see that condition (1.8) and thus the exact LIL for DN (nkx)is r satisﬁed if nk+1/nk → α,whereα is irrational for r =1, 2,... or if (nk) satisﬁes the “large gap condition”

(1.9) nk+1/nk →∞ as k →∞. As the following theorem shows, the exact LIL breaks down continuously if we weaken the growth condition (1.9). For (nk) satisfying the growth condition (1.1) for a “large” value of q, the lim sup in the LIL for the discrepancy of (nkx) will be “almost” 1/2. This falls in place with a result of Berkes [3], who found a similar phenomenon for sums f(nkx) for “nice” periodic functions f.Writing 1 1/2 2 f 2 = f(x) dx , 0

he proved that for (nk) satisfying (1.1) for a “large” value of q the lim sup of N f(n x) k=1 k 2N log log N

will be “almost” f 2 a.e.

Theorem 1.2. Let (nk)k≥1 be a sequence of positive integers satisfying

nk+1/nk >q≥ 2,k=1, 2,.... Then NDN (nkx) 1 −1/4 lim sup √ − ≤ 8q a.e. N→∞ 2N log log N 2 and ∗ NDN (nkx) 1 −1/4 lim sup √ − ≤ 8q a.e. N→∞ 2N log log N 2 Our next theorem, which is a by-product of the proof of Theorem 1.1, provides N the law of the iterated logarithm for sums k=1 f(nkx) under Diophantine condi- tions. We write Var[0,1](f) for the total variation of f on the interval [0, 1].

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE LAW OF THE ITERATED LOGARITHM 5971

Theorem 1.3. Let f be a function satisfying 1 (1.10) f(x +1)=f(x), f(x) dx =0, Var[0,1](f) < ∞, 0

and let (nk) be a sequence of integers satisfying the Hadamard gap condition (1.1) and the Diophantine condition (1.8) for any d ≥ 1.Then N k=1 f(nkx) (1.11) lim sup √ = f 2 a.e. N→∞ 2N log log N Similarly to Theorem 1.1, the Diophantine condition in Theorem 1.3 is nearly optimal. Note that by Koksma’s inequality (see e.g. [15], p. 143) the ﬁniteness of the lim sup in (1.11) follows from Philipp’s LIL (1.2) and thus the essential new information provided by Theorem 1.3 is the exact value of the lim sup. It is worth pointing out that replacing L∗ by L in (1.8) the value of the lim sup can be diﬀerent k from f 2, as the example nk =2 shows; see Fukuyama [9].

2. Main lemma The crucial step of the proof of Theorems 1.1–1.3 is Lemma 2.4 below, giving an exact LIL for lacunary trigonometric polynomials. With this lemma established, our theorems will be proved in Section 3. We start with some preliminary results. Here and in the sequel log x will stand for max{1, log x}. F ≥ Lemma 2.1 (Strassen [18]). Let (Yi, i,i 1) be a martingale diﬀerence sequence M E 2|F E 2 with ﬁnite fourth moments, let VM = i=1 (Yi i−1) and assume V1 = Y1 > 0 and VM →∞. Assume additionally V lim M =1 a.s. M→∞ sM

with some positive sequence sM ,and ∞ (log s )10 (2.1) M EY 4 < +∞. s2 M M=1 M Then M Y (2.2) lim sup √ i=1 i =1 a.s. M→∞ 2sM log log sM Condition (2.1) and the Beppo Levi theorem imply the a.s. convergence of the

series ∞ 10 (log sM ) E 4 |F 2 (YM M−1) sM M=1 ∼ ∞ −2 10E 4 |F and hence by VM sM a.s. the series M=1 VM (log VM ) (YM M−1)isalso a.s. convergent. Thus ∞ 5 (log VM ) 2 x dP (YM

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5972 CHRISTOPH AISTLEITNER

Therefore by Corollary 4.5 of [18], M Y lim sup √ i=1 i =1 a.s., M→∞ 2VM log log VM

and since VM ∼ sM a.s., this implies 2.2). Lemma 2.2. For any function f satisfying (1.10) we have b 2 1 2 ≤ | | ≤ f(λx) dx f(x) dx f ∞ a λ 0 λ for any real numbers a

for all integers N1,N2 ≥ 0 and a number C depending only on p, d and q.

Lemma 2.4. Let (nk)k≥1 be a sequence of positive integers satisfying the Hadamard ≥ gap condition, let d 1 and assume that (1.8) holds for some ε>0.Letp(x)= d j=1(aj cos 2πjx + bj sin 2πjx) be a trigonometric polynomial of order d.Then N k=1 p(nkx) lim sup √ = p 2 a.e. N→∞ 2N log log N Proof. We shall assume for the simplicity of writing that p(x) is an even function; the proof in the general case is essentially the same. Thus let d p(x)= aj cos 2πjx. j=1

We will assume that p 2 > 0, since otherwise the lemma is trivial. We will also assume without loss of generality that p ∞ ≤ 1andthat|aj |≤1, 1 ≤ j ≤ d. Here, and in the rest of this section, C will denote positive constants, not always the same, depending (at most) on p, d and q. We divide the set of positive integers into consecutive blocks Δ1, Δ1, Δ2, Δ2,...,Δi, Δi,... − + of lengths 4logq i and i, respectively. Letting i and i denote the smallest, resp. largest, integer in Δi,wehave n(i−1)+ − − (2.3) ≤ q 4logq i = i 4,i≥ 2. ni−

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE LAW OF THE ITERATED LOGARITHM 5973 ∈ ∈ For every k i≥1 Δi,leti = i(k) be deﬁned by k Δi, put m(k)= log2 nk + 2log2 i and approximate p(nkx) by a discrete function ϕk(x) such that the follow- ing properties are satisﬁed: v ≤ v+1 m(k) − (P1) ϕk(x)isconstantfor 2m(k) x< 2m(k) ,v=0, 1,...,2 1; −2 (P2) ϕk(x) − p(nkx) ∞ ≤ Ci ; (P3) E(ϕk(x)|Fi−1)=0. Here Yi = ϕk(x)

k∈Δi

and Fi denotes the σ-ﬁeld generated by the intervals v v +1 + , , 0 ≤ v<2m(i ). 2m(i+) 2m(i+)

Since p(x) is a trigonometric polynomial, it is Lipschitz continuous, and so

C v v +1 |p(n x)−p(n x)|≤Cn 2−m(k) ≤ for ≤ x, x ≤ , 0 ≤ v<2m(k). k k k i2 2m(k) 2m(k)

Thus it is possible to approximate p(nkx) by discrete functionsϕ ˆk(x)thatsatisfy (P1) and (P2) only. Then for k ∈ Δ and any interval I of the form v , i 2m((i−1)+) v+1 for some v we get, letting |I| denote the lenght of I, 2m((i−1)+) −1 −1 −2 |I| ϕˆk(x) dx ≤|I| p(nkx) dx + Ci dx I I I m((i−1)+) 2 p ∞2 ≤ + Ci−2 ni− 2 4i n − + ≤ (i 1) + Ci−2 ni− ≤ Ci−2,

by (2.3), Lemma 2.2 and since p ∞ ≤ 1. For every x ∈ [0, 1) we can ﬁnd an interval ∈ −| |−1 of type I for some v such that x I, and we put ϕk(x)=ϕ ˆk(x) I I ϕˆk(t) dt. Then these functions ϕk(x) satisfy (P1), (P2) and (P3). We put

M E 2|F Ti = p(nkx) Ti = p(nkx),VM = (Yi i−1). ∈ ∈ k Δi k Δi i=1

We put

M 2 | | sM = p 2 Δi , i=1

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5974 CHRISTOPH AISTLEITNER

where |Δi| denotes the number of integers in Δi. For given M we want to estimate VM − sM 2.Wehave T (x)2 − p 2|Δ | i 2 i 2 − 2| | = p(nkx) dx p 2 Δi k∈Δ ⎛ i ⎞ 2 d 1 d = ⎝ a cos 2πjn x⎠ − a2 |Δ | j k 2 j i k∈Δi j=1 i=1 1 = aj aj cos 2π(jnk − j nk )x 2 1 ≤ j, j ≤ d, k, k ∈ Δi, (j, k) =( j ,k ) ≤| − |≤ 0 jnk j nk n(i−1)+ 1 + aj aj cos 2π(jnk − j nk )x 2 1 ≤ j, j ≤ d, k, k ∈ Δi | − | n(i−1)+ < jnk j nk

+Ri(x)

(2.4) = Ui(x)+Wi(x)+Ri(x). 2 2 Since Ri(x)isasumofatmost2d |Δi| trigonometric functions with coeﬃcients at most 1 and frequencies at least ni− , by Lemma 2.2, m((i−1)+) 2 2 2 (2.5) |E(Ri|Fi−1)|≤4d |Δi| ≤ C ni− and consequently M E |F ≤ (Ri i−1) CM. i=1

Ui and Wi are sums of at most Ci trigonometric functions with coeﬃcients at most 1. Indeed, the number of quadruples (j, j ,k,k )with1≤ j, j ≤ d, k, k ∈ Δi,for which |jnk − j nk |

(2.6) Ui ∞ ≤ Ci and Wi ∞ ≤ Ci. By (2.4), (2.5), Minkowski’s inequality and 2 − 2 ≤ − ≤ | | −2 | |≤ (2.7) Yi Ti ∞ Yi Ti ∞ Yi + Ti ∞ C Δi i Δi C, we get V − s M M 2 M ≤ E 2|F − 2| | (Ti i−1) p 2 Δi + CM i=1 2 M M ≤ E |F E |F (2.8) (Ui i−1) + (Wi i−1) + CM. i=1 2 i=1 2

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE LAW OF THE ITERATED LOGARITHM 5975 M E |F To estimate i=1 (Wi i−1) ,weobserve 2 M 2 E(Wi|Fi−1) ≤ 2 E(Wi|Fi−1)E(Wi |Fi−1). i=1 1≤i≤i≤M By (2.6), M M 2 2 3 (2.9) (E(Wi|Fi−1)) ≤ Ci ≤ CM . i=1 i=1 For i

≤ Wi ∞ |E(Wi |Fi−1)|

≤ Ci|E(Wi |Fi−1)| and thus (2.10) E E(Wi|Fi−1)E(Wi |Fi−1) ≤ Ci E |E(Wi |Fi−1)| .

Wi can be written as a trigonometric polynomial of the form n − i cu cos 2πux,

u=n − + (i 1) | |≤ | | where u cu C Δi . Thus using Lemma 2.2 with f(x)=cos2πx we get n − i −1 m((i−1)+) |E(Wi |Fi−1)|≤ |cu|u 2

u=n(i−1)+ 2 i n − + ≤ Ci (i 1) n(i−1)+ + + (2.11) ≤ Ci2 i q(i−1) −(i −1) ≤ Ci2 i q−(i −1). Combining the estimates (2.9), (2.10) and (2.11) we see that ⎛ ⎞1/2 M E |F ⎝ 3 3 −(i −1)⎠ ≤ 3/2 (2.12) (Wi i−1) = CM + Ci i q CM . i=1 1≤i

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5976 CHRISTOPH AISTLEITNER

which gives M M |F ≤ (2.13) E(Ui i−1) Ui + C log M. i=1 2 i=1 2 M The largest frequency of a trigonometric function in i=1 Ui is at most n(M−1)+ , so we can write n M (M−1)+ Ui(x)= cu cos 2πux, i=1 u=0 where by the Diophantine condition (1.8), M (|Δ | + |Δ |) | |≤ i=1 i i ≤ 2 −(1+ε) cu C 1+ε CM (log M) , M | | | | log i=1 Δi + Δi | |≤ M | |≤ 2 and u cu C i=1 Δi CM . Thus, using (2.13), we get ⎛ ⎞1/2 n(M−1)+ M |F ≤ ⎝ 2 ⎠ E(Ui i−1) cu + C log M i=1 2 u=0 1/2 ≤ CM2 M 2(log M)−(1+ε) + C log M (2.14) ≤ CM2(log M)−(1+ε)/2. Substituting the estimates (2.12) and (2.14) into (2.8), we obtain − ≤ 2 −(1+ε)/2 (2.15) VM sM 2 CM (log M) . We choose an α>0 such that ε −1 ε −1 (2.16) 1+ <α< 1+ 2 4 and deﬁne numbers (lα) Ml = 2 ,l≥ 0, and sets s S = x ∈ (0, 1) : |V − s | > M ,l≥ 0. l M M (log M)ε/4 Ml≤M≤Ml+1 Then, since α<1,

(2.17) lim sM /sM =1 l→∞ l+1 l and α−1 ε/4 α−1 α ε/4 (2.18) l (log Ml) ≤ Cl (l ) → 0asl →∞. Since V and s are increasing in M, and since for l ≥ 1, M M s s − s = s Ml+1 − 1 Ml+1 Ml Ml s Ml 2(l+1)α 2 α−1 ≤ s C − 1 ≤ s C22αl − 1 ≤ Cs lα−1, Ml 22(lα) Ml Ml

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE LAW OF THE ITERATED LOGARITHM 5977

we get, using (2.17) and (2.18), that s s P(S ) ≤ P V − s <− Ml +P V −s > Ml l Ml Ml+1 ε/4 Ml+1 Ml ε/4 (log Ml+1) (log Ml+1) s ≤ P V − s

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5978 CHRISTOPH AISTLEITNER

it follows that max(M−1)+

3. Proof of Theorems 1.1 - 1.3 To prove Theorem 1.3 we ﬁrst show

Lemma 3.1. Let (nk)k≥1 be a sequence of positive integers satisfying the Hadamard gap condition, and let r(x) be a function of the form ∞ r(x)= (aj cos 2πjx + bj sin 2πjx), j=d+1 for some ﬁxed d ≥ 0,where −1 −1 |aj |≤j and |bj |≤j ,j≥ d +1.

Then N r(nkx) k=1 −1/4 lim sup √ ≤ Cqd a.e., N→∞ 2N log log N

where Cq is a positive number depending on q.Forq ≥ 2 we can choose Cq =5. Note that Lemma 3.1 is valid without any Diophantine conditions.

Proof. The proof of Lemma 3.1 can be easily modelled after [19] and [16]. Com- bining the argument in [19, p. 104] and Lemma 3 of [16, p. 246] proves N k=1 r(nkx) √ lim sup √ ≤ 3 2B a.e., N→∞ N log log N where −1 1/2 B = q − 1 r 2. Takahashi’s paper considers only Lipschitz continuous functions, but the proof re- mains valid without any change for functions satisfying the conditions of Lemma 3.1. Clearly ∞ ∞ 1 r 2 = (a2 + b2) ≤ j−2 ≤ d−1, 2 2 j j j=d+1 j=d+1 completing the proof.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE LAW OF THE ITERATED LOGARITHM 5979

Proof of Theorem 1.3. Without loss of generality we can assume Var[0,1](f) ≤ 2. Letting ∞ f(x) ∼ (aj cos 2πjx + bj sin 2πjx) j=1 denote the Fourier series of f, we have (see Zygmund [21, p. 48])

|aj |≤1/j, |bj |≤1/j, j ≥ 1. Thus for any ﬁxed d ≥ 1, f(x) can be written as a sum of a trigonometric polynomial p(x)oforderd and a remainder function r(x) satisfying the conditions of Lemma 3.1. Applying Lemma 2.4 and Lemma 3.1 we get N f(nkx) k=1 −1/2 −1/2 lim sup √ ≤ p 2 + Cqd ≤ f 2 + Cqd a.e. N→∞ 2N log log N and N f(nkx) k=1 −1/2 −1/2 lim sup √ ≥ p 2 − Cqd ≥ f 2 − (Cq +1)d a.e., N→∞ 2N log log N since −1/2 p 2 ≥ f 2 − r 2 ≥ f 2 − d . Since d can be chosen arbitrarily, Theorem 1.3 follows.

To prove Theorems 1.1 and 1.2 we deﬁne, for r ≥ 0, N ≥ 1and(x1,...,xN ) ∈ RN , N −r I (x ) (≤2 ) k=1 [a,b) k DN (x1,...,xN )= sup , 0≤a

and N −r −r ∗ (≥2 ) k=1 I[0,a12 )(xk) D (x1,...,xN )= max , N r a1∈Z,0

where I[a,b) denotes the indicator of the interval [a, b),extendedwithperiod1and centered at expectation, i.e.

I[a,b)(x)=1[a,b)(x) − (b − a). It is easy to see that

− − − (≥2 r ) ≤ ≤ (≥2 r) (≤2 r) (3.1) DN DN DN +2DN and − ∗ (≥2 r ) ≤ ∗ ≤ (3.2) DN DN DN .

(≤2−r ) (≥2−r) The idea to split the discrepancy DN into two parts DN and DN to prove an exact LIL for the discrepancy of (nkx)isduetoFukuyama[9].

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5980 CHRISTOPH AISTLEITNER

Lemma 3.2. Let (nk)k≥1 be a sequence of positive integers satisfying the Hadamard gap condition and the Diophantine condition (1.8) for any d ≥ 1. Then for any ﬁxed r ≥ 0, ≥ −r ND( 2 )(n x) 1 lim sup √ N k = a.e. N→∞ 2N log log N 2 and ≥ −r ND∗ ( 2 )(n x) 1 lim sup √ N k = a.e. N→∞ 2N log log N 2

Proof. Clearly, I[a,b)(x), 0 ≤ a

Lemma 3.3. Let (nk)k≥1 be a sequence of positive integers satisfying the Hadamard gap condition. Then for any r ≥ 1, (≤2−r ) NDN (nkx) −1 lim sup √ ≤ Cqr a.e., N→∞ 2N log log N

where Cq is a positive number depending on q.Inparticular, ≤ −r ND( 2 )(n x) lim lim sup √ N k =0 a.e. →∞ r N→∞ 2N log log N Similarly to Lemma 3.1, this lemma is valid without any Diophantine conditions. Proof. The proof of this lemma can be modelled after [16]. It remains completely the same up to the end of page 249. On page 250, Philipp shows that N k=1 I[0,a)(nkx) lim sup sup √ ≤ Cq a.e., N→∞ a∈[0,1) N log log N

where Cq is a positive number depending on q. Under the additional assumption that a ≤ 2−r for ﬁxed r ≥ 1, it is easy to see that in the ﬁrst equation on page 250 it suﬃces to sum over h from r to H instead of 1 to H.Thusweget N ∞ k=1 I[0,a)(nkx) −h/8 lim sup sup √ ≤ Cq 2 a.e. N→∞ ∈ −r N log log N a [0,2 ) h=r Using the same method it is easy to see that a similar result holds for “shifted intervals”, i.e. N ∞ − − k=1 I[l2 r,l2 r +a)(nkx) −h/8 (3.3) lim sup sup √ ≤ Cq 2 a.e., N→∞ ∈ −r N log log N a [0,2 ) h=r for any integer l satisfying 0 ≤ l ≤ 2r − 1, and, using the same argument in a mirror-inverted way, that N ∞ − − k=1 I[l2 r −a,l2 r )(nkx) −h/8 (3.4) lim sup sup √ ≤ Cq 2 a.e., N→∞ ∈ −r N log log N a [0,2 ) h=r

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE LAW OF THE ITERATED LOGARITHM 5981

where 1 ≤ l ≤ 2r.Thosex ∈ [0, 1), for which (3.3) and (3.4) are valid for all possible values of l,satisfy N ∞ k=1 I[a,b)(nkx) −h/8 (3.5) lim sup sup √ ≤ 2Cq 2 a.e. N→∞ ≤ − ≤ −r N log log N 0 a

Proof of Theorem 1.1. Theorem 1.1 follows from Lemma 3.2 and Lemma 3.3, to- gether with (3.1) and (3.2).

Proof of Theorem 1.2. Theorem 1.2 follows from Lemma 2.4, Lemma 3.1 and Lemma 3.3. Assume (nk) is a sequence of positive integers such that

nk+1/nk >q≥ 2,k≥ 1. Set d = q/2. Then for ﬁxed 1 ≤ j, j ≤ d, the sequence

(mk)k≥1 =(mk(j, j )k≥1 =(jnk)k≥1 ∪ (j nk)k≥1,

i.e. the sequence consisting of the elements of the set-theoretic union of (jnk)k≥1 and (j nk)k≥1, sorted in increasing order, is lacunary. Now by a well-known prop- erty of lacunary sequences (see Zygmund [21, p. 203]), this implies that the number of solutions of the Diophantine equation

mk − mk = ν, k, k ≥ 1 is bounded by a number C(j, j), uniformly in ν ∈ Z, ν = 0. Also, the number of solutions of jnk − j nk =0,k,k≥ 1 is bounded by C(j, j ). Thus the sequence (nk)k≥1 satisﬁes (1.8) for this choice of d. By Lemma 2.4 and Lemma 3.1, for any function f(x) satisfying (1.10), writing p(x)forthed-th partial sum of f(x), N f(nkx) k=1 −1/4 −1/4 lim sup √ ≤ p 2 +5d ≤ f 2 +6q a.e., N→∞ 2N log log N and similarly N f(nkx) k=1 −1/4 −1/4 −1/4 lim sup √ ≥ p 2 − 5d ≥ f 2 − 6d ≥ f 2 − 8q a.e. N→∞ 2N log log N Theorem 1.2 is a consequence of these equations, together with Lemma 3.3.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5982 CHRISTOPH AISTLEITNER

References

[1] C. Aistleitner and I. Berkes. On the central limit theorem for f(nkx). Probab. Theory Relat. Fields, 146 (2010), no. 1-2, 267-289. MR2550364 [2] R. C. Baker. Metric number theory and the large sieve. J. London. Math. Soc. 24, 34-40, 1981. MR623668 (83a:10086) [3] I. Berkes. On the asymptotic behaviour of f(nkx). I. Main theorems, II. Applications. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 34:319-345; 347-365, 1976. [4] I. Berkes. On the central limit theorem for lacunary trigonometric series. Anal. Math. 4:159- 180, 1978. MR514757 (80c:60032) [5] I. Berkes and W. Philipp. An a.s. invariance principle for lacunary series f(nkx). Acta Math. Acad. Sci. Hungar. 34:141-155, 1979. MR546729 (80i:60042) [6] I. Berkes and W. Philipp. The size of trigonometric and Walsh series and uniform distribution mod 1. J. London Math. Soc. (2), 50:454-464, 1994. MR1299450 (96e:11099) [7] J. W. S. Cassels. Some metrical theorems in Diophantine approximation III. Proc. Cambridge Philos. Soc. 46:219-225, 1950. MR0036789 (12:162d) [8] P. Erd˝os and J. F. Koksma. On the uniform distribution modulo 1 of sequences (f(n, θ)). Proc. Kon. Nederl. Akad. Wetensch. 52:851-854, 1949. MR0032690 (11:331f) [9] K. Fukuyama. The law of the iterated logarithm for discrepancies of {θnx}. Acta Math. Hungar. 118:155-170, 2008. MR2378547 (2008m:60049) [10] V. F. Gaposhkin. Lacunary series and independent functions (in Russian). Uspehi Mat. Nauk 21/6:3-82,1966. MR0206556 (34:6374) [11] V. F. Gaposhkin. On the central limit theorem for some weakly dependent sequences (in Russian). Teor. Verojatn. Prim. 15:666-684, 1970. MR0282394 (43:8106) [12] M. Kac. On the distribution of values of sums of the type f(2kt). Ann. of Math. (2) 47:33-49, 1946. MR0015548 (7:436f) [13] M. Kac. Probability methods in some problems of analysis and number theory. Bull. Amer. Math. Soc. 55:641-665, 1949. MR0031504 (11:161b) [14] H. Kesten. The discrepancy of random sequences {kx}, Acta Arith. 10:183-213, 1964/65. MR0168546 (29:5807) [15] L. Kuipers and H. Niederreiter. Uniform Distribution of Sequences. Wiley, New York, 1974. MR0419394 (54:7415) [16] W. Philipp. Limit theorems for lacunary series and uniform distribution mod 1. Acta Arith. 26:241-251, 1975. MR0379420 (52:325) [17] R. Shorack and J. Wellner. Empirical processes with applications to statistics. Wiley, New York, 1986. MR838963 (88e:60002) [18] V. Strassen. Almost sure behavior of sums of independent random variables and martingales. Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Vol. II: Contributions to Probability Theory:315-343, 1967. MR0214118 (35:4969) [19] S. Takahashi. An asymptotic property of a gap sequence. Proc. Japan Acad. 38:101-104, 1962. MR0140865 (25:4279) [20] H. Weyl. Uber¨ die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77:313-352, 1916. MR1511862 [21] A. Zygmund. Trigonometric series. Vols. I, II. Third edition. Cambridge Mathematical Li- brary. Cambridge University Press, Cambridge, 2002. MR1963498 (2004h:01041)

Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria E-mail address: [email protected]

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use