diff --git "a/INE5T4oBgHgl3EQfWw9u/content/tmp_files/2301.05561v1.pdf.txt" "b/INE5T4oBgHgl3EQfWw9u/content/tmp_files/2301.05561v1.pdf.txt"
new file mode 100644--- /dev/null
+++ "b/INE5T4oBgHgl3EQfWw9u/content/tmp_files/2301.05561v1.pdf.txt"
@@ -0,0 +1,3668 @@
+arXiv:2301.05561v1  [math.NT]  13 Jan 2023
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND
+NUMBER THEORY
+CHRISTOPH AISTLEITNER, ISTV´AN BERKES AND ROBERT TICHY
+Abstract. In this paper we present the theory of lacunary trigonometric sums
+and lacunary sums of dilated functions, from the origins of the subject up to re-
+cent developments. We describe the connections with mathematical topics such
+as equidistribution and discrepancy, metric number theory, normality, pseudo-
+randomness, Diophantine equations, and the subsequence principle. In the final
+section of the paper we prove new results which provide necessary and sufficient
+conditions for the central limit theorem for subsequences, in the spirit of Nikishin’s
+resonance theorem for convergence systems. More precisely, we characterize those
+sequences of random variables which allow to extract a subsequence satisfying a
+strong form of the central limit theorem.
+Contents
+1.
+Introduction
+1
+2.
+Uniform distribution and discrepancy
+4
+3.
+Arithmetic effects: Diophantine equations and sums of common divisors
+7
+4.
+The central limit theorem for lacunary sequences
+12
+5.
+The law of the iterated logarithm for lacunary sequences
+16
+6.
+Normality and pseudorandomness
+20
+7.
+Random sequences
+24
+8.
+The subsequence principle
+28
+9.
+New results: Exact criteria for the central limit theorem for subsequences 36
+Acknowledgments
+49
+References
+50
+1. Introduction
+The word “lacunary” has its origin in the Latin lacuna (ditch, gap), which is a
+diminutive form of lacus (lake).
+Accordingly, a lacunary sequence is a sequence
+with gaps, and a lacunary trigonometric sum is a sum of trigonometric functions
+with gaps between the frequencies of consecutive summands.
+The origin of the
+theory of lacunary sums might lie in Weierstrass’ famous example of a continuous,
+nowhere differentiable function (1872). Since then the subject has evolved into many
+very different directions, reflecting for example the emergence of modern measure
+theory and axiomatic probability theory in the early twentieth century, profound
+1
+
+2
+C. AISTLEITNER, I. BERKES AND R. TICHY
+developments in harmonic analysis and Diophantine approximation, the establish-
+ment of ergodic theory as one of the key instruments of number theory, or the
+interest in notions of pseudo-randomness which are associated with the evolution
+of theoretical computer science. Throughout this paper we will be concerned with
+convergence/divergence properties of infinite trigonometric series
+∞
+�
+k=1
+ck cos(2πnkx)
+or
+∞
+�
+k=1
+ck sin(2πnkx),
+as well as with the asymptotic order and the distributional behavior of finite trigono-
+metric sums
+N
+�
+k=1
+ck cos(2πnkx)
+or
+N
+�
+k=1
+ck sin(2πnkx)
+(the latter often in the simple case where ck ≡ 1), and with their generalizations
+∞
+�
+k=1
+ckf(nkx)
+and
+N
+�
+k=1
+ckf(nkx).
+Here (ck)k≥1 is a sequence of coefficients, and (nk)k≥1 is a sequence of positive in-
+tegers (typically increasing), which satisfies some gap property such as the classical
+Hadamard gap condition
+nk+1
+nk
+> q > 1,
+k ≥ 1,
+or the “large gap condition” (also called “super-lacunarity property”)
+nk+1
+nk
+→ ∞,
+k → ∞.
+Furthermore, f is a 1-periodic function which is usually assumed to satisfy some reg-
+ularity properties (such as being of bounded variation, being Lipschitz-continuous,
+etc.), and which for simplicity is usually assumed to be centered such that
+� 1
+0 f(x) dx =
+0.
+Early appearances of such lacunary sums include the following.
+• Sums of the form �N
+k=1 f(2kx), where f is an indicator function of a dyadic
+sub-interval of [0, 1], extended periodically with period 1. Borel used such
+sums in 1909 to show that almost all reals are “normal”; more on this topic
+is contained in Section 6 below.
+• Uniform distribution of sequences ({nkx})k≥1 in Weyl’s seminar paper of
+1916; more on this in Section 2. Here and throughout the paper, we write
+{·} for the fractional part function.
+• Kolmogorov’s theorem on the almost everywhere convergence of lacunary
+trigonometric series if the sequence of coefficients is square-summable (1924),
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+3
+a result related to his Three Series Theorem for the almost sure conver-
+gence of series of independent random variables. Later it turned out that
+Kolmogorov’s convergence theorem for trigonometric series actually remains
+true without any gap condition whatsoever, a result which was widely be-
+lieved to be “too good to be true” before being established by Carleson [77]
+in 1966. More on this in Section 2.
+• Foundational work on the distribution of normalized lacunary trigonometric
+sums, in particular the central limit theorems of Kac (1946) and Salem and
+Zygmund (1947), and the laws of the iterated logarithms of Salem and Zyg-
+mund (1950) and of Erd˝os and G´al (1955). More on this in Sections 4 and 5.
+A fundamental observation is that the unit interval, equipped with Borel sets and
+Lebesgue measure, forms a probability space, and that consequently a sequence of
+functions such as (cos(2πnkx))k≥1 or (f(nkx))k≥1 can be viewed as a sequence of
+random variables over this space; if f is 1-periodic and if (nk)k≥1 is a sequence
+of positive integers then these random variables are identically distributed, but in
+general they are not independent. However, under appropriate circumstances the
+gap condition which is imposed upon (nk)k≥1 can ensure that these random vari-
+ables have a low degree of stochastic dependence. Consequently lacunary sums often
+mimic the behavior of sums of independent and identically distributed random vari-
+ables. This viewpoint was in particular taken by Steinhaus, Kac, and Salem and
+Zygmund in their fundamental work on the subject. In a particularly striking situa-
+tion, the dyadic functions considered by Borel actually turn out to be a version of a
+sequence of Bernoulli random variables which are truly stochastically independent;
+accordingly, Borel’s result on the normality of almost all reals is nowadays usually
+read as the historically very first version of the strong law of large numbers in prob-
+ability theory.
+When taking this probabilistic viewpoint, the theory of lacunary sums could be seen
+as a particular segment of the much wider field of the theory of weakly dependent
+random systems in probability theory, which is associated with notions such as mix-
+ing, martingales, and short-range dependence. However, it should be noted that
+the precise dependence structure in a lacunary function system (f(nkx))k≥1 is con-
+trolled by the analytic properties of the function f, in conjunction with arithmetic
+properties of the sequence (nk)k≥1. It is precisely this interplay between probabilis-
+tic, analytic and arithmetic aspects which makes the theory of lacunary sums so
+interesting, so challenging and so rewarding. In the following sections we want to
+illustrate some instances of these phenomena in more detail.
+
+4
+C. AISTLEITNER, I. BERKES AND R. TICHY
+2. Uniform distribution and discrepancy
+Let (xn)n≥1 be a sequence of real numbers in the unit interval. Such a sequence is
+called uniformly distributed modulo one (in short: u.d. mod 1) if
+(1)
+1
+N
+N
+�
+n=1
+1A(xn) = λ(A)
+for all sub-intervals A ⊂ [0, 1] of the unit interval. The word “equidistributed” is
+also used for this property, synonymously with “uniformly distributed modulo one”.
+In this definition, and in the sequel,
+1 denotes an indicator function, and λ denotes
+Lebesgue measure. In informal language, this definition means that a sequence is
+u.d. mod 1 if every interval A asymptotically receives its fair share of elements of the
+sequence, which is proportional to the length of the interval. Note that (for example
+as a consequence of the Glivenko–Cantelli theorem) for a sequence of independent,
+uniformly (0, 1)-distributed random variables (Un)n≥1 one has
+1
+N
+N
+�
+n=1
+1A(Un) = λ(A)
+almost surely
+for all intervals A ⊂ [0, 1], so that in a vague sense uniform distribution of a deter-
+ministic sequence can be interpreted in the sense that the sequence shows “random”
+behavior; more on this aspect in Section 6 below. Uniform distribution theory can
+be said to originate with Kronecker’s approximation theorem and with work of Bohl,
+Sierpi´nski and Weyl on the sequence ({nα})n≥1 for irrational α. However, the theory
+only came into its own with Hermann Weyl’s [222] seminal paper of 1916. Among
+many other fundamental insights, Weyl realized that Definition (1), which in ear-
+lier work had only be read in terms of counting points in certain intervals, can be
+interpreted in a “functional” way and can equivalently be written as
+(2)
+lim
+N→∞
+1
+N
+N
+�
+n=1
+f(xn) =
+� 1
+0
+f(x) dx
+for all continuous functions f. This viewpoint suggests that uniformly distributed
+sequences can be used as quadrature points for numerical integration; in the multi-
+dimensional setting and together with quantitative error estimates this observation
+forms the foundation of the so-called Quasi-Monte Carlo integration method, a con-
+cept which today forms a cornerstone of numerical methods in quantitative finance
+and other fields of applied mathematics (more on this below). Furthermore, Weyl
+realized that the indicator functions in (1) or the continuous functions in (2) could
+also be replaced by complex exponentials, as a consequence of the Weierstrass ap-
+proximation theorem; thus by the famous Weyl Criterion a sequence is u.d. mod 1
+if and only if
+lim
+N→∞
+1
+N
+N
+�
+n=1
+e2πihxn = 0
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+5
+for all fixed non-zero integers h, thereby tightly connecting uniform distribution the-
+ory with the theory of exponential sums.
+For the particular sequence ({nα})n≥1 it can be easily seen from the Weyl criterion
+that this sequence is u.d. mod 1 if and only if α ̸∈ Q. However, for other parametric
+sequences of the form ({nkα})k≥1 the situation is much more difficult, and in general
+it is completely impossible to determine whether for some particular value of α the
+sequence is u.d. or not. It turns out that in a metric sense the situation is quite
+different. Metric number theory arose after the clarification of the concept of real
+numbers, the realization that the reals drastically outnumber the integers and the
+rationals, and the development of modern measure theory. Loosely speaking, the
+purpose of metric number theory is to determine properties which hold for a set
+of reals which is “typical” with respect to a certain measure; here “typical” means
+that the measure of the complement is small. In the present paper the measure
+under consideration will always be the Lebesgue measure, and a set of reals will
+be considered typical if its complement has vanishing Lebesgue measure; however,
+metric number theory has for example also been intensively studied with respect to
+the Hausdorff dimension or other fractal measures.
+Returning to Weyl’s results, what he proved in the metric setting is the following.
+For every sequence of distinct integers (nk)k≥1, the sequence ({nkα})k≥1 is u.d. mod
+1 for (Lebesgue-) almost all reals α. In other words, even if we cannot specify the
+set of α’s for which uniform distribution holds, at least we know that the set of such
+α’s has full Lebesgue measure. It is amusing that after formulating the result, Weyl
+continues to write:
+Wenn ich nun freilich glaube, daß man den Wert solcher S¨atze, in
+denen eine unbestimmte Ausnahmemenge vom Maße 0 auftritt, nicht
+eben hoch einsch¨atzen darf, m¨ochte ich diese Behauptung hier doch
+kurz begr¨unden.1
+One should recall that Weyl’s paper was written in a time of intense conflict of
+formalists vs. constructivists (with Weyl favoring the latter ones), and only very
+briefly after the notion of a set of zero (Lebesgue) measure had been introduced
+at all. Today, Weyl’s theorem is seen as one of the foundational results of metric
+number theory, together with the work of Borel, Koksma, Khinchin and others.
+While uniform distribution modulo one is a qualitative asymptotic property, it is
+natural that one is also interested in having a corresponding quantitative concept
+which applies to finite sequences (or finite truncations of infinite sequences). Such
+1Even if I think that the value of theorems, which contain an unspecified exceptional set of
+measure zero, is not particularly high, I still want to give a short justification.
+
+6
+C. AISTLEITNER, I. BERKES AND R. TICHY
+a concept is the discrepancy of a sequence, which is defined by
+DN(x1, . . . , xN) = sup
+A⊂[0,1]
+�����
+1
+N
+N
+�
+n=1
+1A(xn) − λ(A)
+����� .
+Here the supremum is taken over all sub-intervals A ⊂ [0, 1], and it is easy to
+see that an infinite sequence (xn)n≥1 is u.d. mod 1 if and only if the discrepancy
+DN(x1, . . . , xN) tends to 0 as N → ∞. With a slight abuse of notation, we will
+write throughout the paper DN(xn) = DN(x1, . . . , xN) for the discrepancy of the
+first N elements of an infinite sequence (xn)n≥1. From a probabilistic perspective,
+the discrepancy is a variant of the (two-sided) Kolmogorov–Smirnov statistic, where
+one tests the empirical distribution of the point set x1, . . . , xN against the uniform
+distribution on [0, 1]. Without going into details, we note that DN(x1, . . . , xN) can
+be bounded above in terms of exponential sums by the Erd˝os–Tur´an inequality, and
+that the error when using x1, . . . , xN as a set of quadrature points to approximate
+� 1
+0 f(x) dx by 1
+N
+�N
+n=1 f(xn) can be bounded above by Koksma’s inequality in terms
+of the variation of f and the discrepancy DN; for details see the monographs [96, 161],
+which contain all the basic information on uniform distribution theory and discrep-
+ancy. See also [181] for a discussion of equidistribution and discrepancy from the
+viewpoint of analytic number theory, and [164, 165, 192] for expositions which put
+particular emphasis on the numerical analysis aspects.
+Weyl’s metric result from above can be written as
+lim
+N→∞ DN({nkα}) = 0
+for almost all α,
+for any sequence (nk)k≥1 of distinct itegers. Strikingly, the precise answer to the
+corresponding quantitative problem is still open more than a hundred years later.
+It is known that for every strictly increasing sequence of integers (nk)k≥1 one has
+(3)
+DN({nkα}) = O
+�(log N)3/2+ε
+√
+N
+�
+for almost all α.
+This is a result of R.C. Baker [38], who improved earlier results of Cassels [78] and
+of Erd˝os and Koksma [104] by using Carleson’s celebrated convergence theorem in
+the form of the Carleson–Hunt inequality [140]. In his paper Baker wrote that
+[. . . ] probably the exponent 3/2 + ε could be replaced by ε [. . . ]
+but it turned out that this is not actually the case. Instead, Berkes and Philipp [64]
+constructed an example of an increasing integer sequence (nk)k≥1 for which
+(4)
+lim sup
+N→∞
+���
+�N
+k=1 cos(2πnkx)
+���
+√N log N
+= +∞
+for almost every x.
+By the Erd˝os–Tur´an inequality this gives a corresponding lower bound for the dis-
+crepancy, which implies that the optimal exponent of the logarithmic term in an
+upper bound of the form (3) has to be at least 1/2. But the actual size of this opti-
+mal exponent, one of the most fundamental problems in metric discrepancy theory,
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+7
+still remains open. Note that for pure cosine-sums �N
+k=1 cos(2πnkx) it is easily seen
+that one has a metric upper bound with exponent 1/2 + ε in the logarithmic term;
+this follows from the orthogonality of the trigonometric system, together with Car-
+leson’s inequality and the Chebyshev inequality. Thus, in connection with (4), the
+optimal upper bound in a metric estimate for pure cosine sums is known. For sums
+�N
+k=1 f(nkx) with f being a 1-periodic function of bounded variation, the optimal
+exponent also is 1/2 + ε, but this is a much deeper result than the one for the pure
+cosine case, and was established only recently in [15, 169]. By Koksma’s inequality,
+an upper bound for the discrepancy implies an upper bound for sums of function
+values for a (fixed) function of bounded variation, but the opposite is not true. So
+while the case of a fixed function f is solved and is an important test case for the
+discrepancy, the problem of the discrepancy itself (which requires a supremum over
+a whole class of test functions) is more involved and remains open.
+3. Arithmetic effects: Diophantine equations and sums of common
+divisors
+One of the most classical tools of probability theory is the calculation of expectations,
+variances, and higher moments of sums of random variables. Due to trigonometric
+identities such as
+(5)
+cos a cos b = cos(a + b) + cos(a − b)
+2
+,
+the calculation of moments of sums of trigonometric functions (with integer frequen-
+cies) reduces to a counting of solutions of certain Diophantine equations. Indeed,
+while the first and second moments
+� 1
+0
+N
+�
+k=1
+cos(2πnkx) dx = 0
+and
+� 1
+0
+� N
+�
+k=1
+cos(2πnkx)
+�2
+dx = N
+2
+are trivial and do not depend on the particular sequence (nk)k≥1 (as long as the
+elements of the sequence are assumed to be distinct), interesting arithmetic effects
+come into play when one has to compute higher moments, and it can be clearly seen
+how the presence of a gap condition leads to a behavior of the moments which is
+similar to that of sums of independent random variables. More precisely, assume
+that we try to calculate
+� 1
+0
+� N
+�
+k=1
+cos(2πnkx)
+�m
+dx
+for some integer m ≥ 3. By (5) this can be written as a sum
+2−m �
+±
+�
+1≤k1,...,km≤N
+1 (±nk1 ± · · · ± nkm = 0) .
+Here the first sum is meant as a sum over all positive combinations of “+” and “-”
+signs inside the indicator function at the end. Now assume that, for simplicity, we
+consider the particular sequence nk = 2k, k ≥ 1, which is a prototypical example
+
+8
+C. AISTLEITNER, I. BERKES AND R. TICHY
+of a sequence satisfying the Hadamard gap condition. Then it is not difficult to see
+that
+±nk1 ± · · · ± nkm = 0
+is only possible if the elements of the sum cancel out in a pairwise way; that is, after a
+suitable re-ordering of the indices, we need to have k1 = k2, k3 = k4, . . . , km−1 = km,
+and it only remains to count how many such re-orderings are possible. The result
+is a combinatorial quantity, and it is exactly the same that arises when calculating
+an m-th moment of a sum of independent random variables. Thus the moments
+of the trigonometric lacunary sum converge to those of a suitable Gaussian distri-
+bution, which gives rise to the classical limit theorems for lacunary trigonometric
+sums. The situation is more delicate if one only has the Hadamard gap condition
+nk+1/nk > q > 1 rather than exact exponential growth, and again more delicate if
+one considers a sum of dilated functions � f(nkx) instead of a pure trigonometric
+sum, but the principle described here is very powerful also in these more general
+situations. For a long time this was the key ingredient in most of the proofs of
+limit theorems for lacunary sums; see for example [103, 145, 193, 201, 214, 221]. A
+different method is based on the approximation of a lacunary sum by a martingale
+difference; here the “almost independent” behavior is not captured by controlling
+the moments of the sum, but in the fact that later terms of the sum (functions with
+high frequency) oscillate quickly in small regions where earlier summands (functions
+with much lower frequency) are essentially constant.
+As far as we can say, this
+method was first used in the context of lacunary sums by Berkes [53] and, indepen-
+dently, by Philipp and Stout [196]. We will come back to this topic in Section 4.
+Broadly speaking, the “almost independent” behavior of sums of dilated functions
+breaks down when the lacunarity condition is relaxed. Many papers have been de-
+voted to this effect; see in particular [57, 59, 102, 184]. In order to maintain the
+“almost independent” behavior of the sum, there are two natural routes to take. On
+the one hand, one could randomize the construction of the sequence (nk)k≥1, and
+assume that the undesired effects disappear almost surely with respect to the under-
+lying probability measure – it turns out that this is a very powerful method, and we
+will come back to it in Section 7 below. On the other hand, when adapting the view-
+point that the “almost independence” property is expressed in the small number of
+solutions of certain Diophantine equations, one could try to compensate the weaker
+growth assumption by stronger arithmetic assumptions. A prominent example of a
+class of sequences for which the latter approach has been very successfully used are
+the so-called Hardy–Littlewood–P´olya sequences, which consist of all the elements of
+the multiplicative semigroup generated by a finite set of primes, sorted in increasing
+order. These sequences are in several ways a natural analogue of lacunary sequences;
+note that the sequence (2k)k≥1 actually also falls into this framework by consisting
+of all elements of the semigroup generated by a single prime. Such sequences gener-
+ated by a finite set of primes have attracted the attention of number theorists again
+and again, a particularly interesting instance being F¨urstenberg’s [126] paper on
+disjointness in ergodic theory. It is known that Hardy–Littlewood–P´olya sequences
+(if generated by two or more primes) grow sub-exponentially, and the precise (only
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+9
+slightly sub-exponential) growth rates are known (Tijdeman [216]). What is more
+striking (and a much deeper fact) is that also the number of solutions of the rel-
+evant linear Diophantine equations can be bounded efficiently – this is Schmidt’s
+celebrated Subspace Theorem [207] in a quantitative form such as that of Evertse,
+Schlickewei and Schmidt [107] or Amoroso and Viada [30]. By a combination of the
+(slightly weaker) growth condition with the (strong) arithmetic information avail-
+able for Hardy–Littlewood–P´olya sequences, much of the machinery that is used for
+Hadamard lacunary sequences can be rescued for this generalized setup; see [194] as
+well as [19, 65, 123].
+We briefly come back to the case of sums of dilated functions � f(nkx) without
+the presence of a growth condition on (nk)k≥1.
+We assume for simplicity that
+� 1
+0 f(x) dx = 0, so trivially
+� 1
+0
+N
+�
+k=1
+f(nkx) dx = 0,
+but already the calculation of the variance
+(6)
+� 1
+0
+� N
+�
+k=1
+f(nkx)
+�2
+dx
+is in general quite non-trivial. If f(x) = cos(2πx), then one can simply use the
+orthogonality of the trigonometric system. If f is a more general function, then one
+can still express f by its Fourier series, expand the square and integrate, and thus
+translate the problem of calculating (6) into a problem of counting the solutions
+of certain linear Diophantine equations. When carrying out this approach, one is
+naturally led to the problem of estimating a certain sum involving greatest common
+divisors. For example, assume that f(x) = {x}−1/2. In this case a classical formula
+(first stated by Franel and first proved by Landau) asserts that
+� 1
+0
+f(mx)f(nx) dx = 1
+12
+(gcd(m, n))2
+mn
+,
+and consequently
+� 1
+0
+� N
+�
+k=1
+({nkx} − 1/2)
+�2
+dx = 1
+12
+�
+1≤k,ℓ≤N
+(gcd(nk, nℓ))2
+nknℓ
+.
+The sum on the right-hand side of this equation is called a GCD sum. A similar
+identity holds for example for the Hurwitz zeta function ζ(1 − α, ·), where
+� 1
+0
+ζ(1 − α, {mx})ζ(1 − α, {nx}) dx = 2Γ(α)2 ζ(2α)
+(2π)2α
+(gcd(m, n))2α
+(mn)α
+for α > 1/2, thus leading to a GCD sum
+�
+1≤k,ℓ≤N
+(gcd(nk, nℓ))2α
+(nknℓ)α
+
+10
+C. AISTLEITNER, I. BERKES AND R. TICHY
+with parameter α. If f(x) is a general 1-periodic function, then one usually does not
+obtain such a nice exact representation of the variance of a sum of dilated function
+values, but typically the variance (6) can be bounded above by a GCD sum, which
+together with Chebyshev’s inequality and the Borel–Cantelli lemma allows to make
+a statement on the almost everywhere asymptotic behavior of a sum of dilated func-
+tion values.
+This connection between sums of dilated functions and GCD sums is explained in
+great detail in Chapter 3 in Harman’s monograph on Metric Number Theory [136],
+where mainly the context of metric Diophantine approximation is treated (see also [127,
+155]). Recently this connection has also led to a solution of the problem of the al-
+most everywhere convergence of series of dilated functions. Recall that Carleson’s
+theorem [77] asserts that the series
+∞
+�
+k=1
+ck cos(2πnkx)
+is almost everywhere convergent provided that �
+k c2
+k < ∞. It is natural to ask
+which assumption on the sequence of coefficients (ck)k≥1 is necessary to ensure the
+almost everywhere convergence of the more general series
+(7)
+∞
+�
+k=1
+ckf(nkx),
+under some regularity assumptions on f. Gaposhkin [129, 130] obtained some partial
+results, but a satisfactory understanding of the problem was only achieved very
+recently, when the connection with GCD sums was fully understood and optimal
+upper bounds for such sums were obtained. Exploiting this connection with GCD
+sums, it was shown in [15, 169] that for 1-periodic f which is of bounded variation
+on [0, 1] the series (7) is almost everywhere convergent provided that
+∞
+�
+k=3
+c2
+k(log log k)γ < ∞
+for some γ > 2, and this result is optimal in the sense that the same assumption
+with γ = 2 would not be sufficient. In [16] it was shown that for the class Cα of
+1-periodic square integrable functions f with Fourier coefficients aj, bj satisfying
+aj = O(j−α),
+bj = O(j−α)
+for 1/2 < α < 1, a sharp criterion for the almost everywhere convergence of (7) is
+that
+(8)
+∞
+�
+k=1
+c2
+k exp
+�K(log k)1−α
+(log log k)α
+�
+< ∞
+with a suitable K = K(α).
+In the case of 1-periodic Lipschitz α functions f,
+Gaposhkin [130] proved that for α > 1/2, the series (7) converges a.e. under
+�
+k c2
+k < ∞ (just like in the case of Carleson’s theorem) and Berkes [60] showed
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+11
+that this result is sharp, i.e. for α = 1/2 the exact analogue of Carleson’s theorem is
+not valid. No sharp convergence criteria exists in the case 0 < α ≤ 1/2; for sufficient
+criteria for see Gaposhkin [129]. See also [4, 68, 128] for a general discussion and
+several further results for the convergence of series �
+k ckf(nkx).
+For general periodic f ∈ L2 the direct connection between the integral (6) and GCD
+sums breaks down, but upper bounds for (6) as well as for
+(9)
+� 1
+0
+� N
+�
+k=1
+ckf(nkx)
+�2
+dx
+can be given in terms of the coefficients ck, of the Fourier coefficients of f, and
+arithmetic functions such as d(n) = �
+d|n 1, σs(n) = �
+d|n ds, or the Erd˝os-Hooley
+function ∆(n) = supu∈R
+�
+d|n,u≤d≤eu 1. See Koksma [156, 157], Weber [220], and
+Berkes and Weber [69, 70]. A typical example (see [220]) is the bound
+� 1
+0
+��
+k∈H
+ckf(kx)
+�2
+dx ≤
+� ∞
+�
+ν=1
+a2
+ν∆(ν)
+� �
+k∈H
+c2
+kd(k)
+valid for any set H of disjoint positive integers lying in some interval [er, er+1], r ≥ 1.
+Here ak are the complex Fourier coefficients of f. Using standard methods, such
+bounds lead easily to a.e. convergence criteria for sums �
+k ckf(kx), see the papers
+cited above.
+In Wintner [223] it was proved that if f is a periodic L2 function with Fourier
+coefficients ak, bk, then the series �
+k ckf(kx) converges in L2 norm for all coefficient
+sequences (ck)k≥1 satisfying �
+k c2
+k < ∞ if and only if the functions defined by the
+Dirichlet series
+∞
+�
+k=1
+akk−s,
+∞
+�
+k=1
+bkk−s,
+are bounded and regular in the half plane ℜ(s) > 0. There is also a remarkable con-
+nection between the maximal order of magnitude of GCD sums with the order of ex-
+treme values of the Riemann zeta function in the critical strip; see [72, 94, 138, 209].
+Naturally, estimating the integral (6) provides important information also on the
+asymptotic behavior of averages
+(10)
+1
+N
+N
+�
+k=1
+f(nkx).
+By the Weyl equidistribution theorem, for any 1-periodic f with bounded variation
+in (0, 1) we have
+(11)
+lim
+N→∞
+1
+N
+N
+�
+k=1
+f(kx) =
+� 1
+0
+f(x) dx
+a.e.
+
+12
+C. AISTLEITNER, I. BERKES AND R. TICHY
+(actually for every irrational x). Khinchin [150] conjectured that (11) holds for every
+1-periodic Lebesgue integrable f as well. This conjecture remained open for nearly
+50 years and was finally disproved by Marstrand [178]. An example for a periodic
+integrable f and a sequence (nk)k≥1 of positive integers such that the averages (10)
+do not converge almost everywhere had already been given earlier by Erd˝os [100].
+On the other hand, Koksma [157] proved that (11) holds if f ∈ L2 and the Fourier
+coefficients ak, bk of f satisfy
+∞
+�
+k=1
+
+(a2
+k + b2
+k)
+�
+d|k
+1
+d
+
+ < ∞,
+and Berkes and Weber [70] proved that the last condition is optimal. No similarly
+sharp criteria are known in the case f ∈ L1.
+For further results related to the
+Khinchin conjecture, see [37, 50, 70, 74, 185].
+4. The central limit theorem for lacunary sequences
+Salem and Zygmund [201] proved the first central limit theorem (CLT) for lacunary
+trigonometric sums. More specifically, they showed that for any integer sequence
+(nk)k≥1 satisfying the Hadamard gap condition one has
+lim
+N→∞ λ
+�
+x ∈ (0, 1) :
+N
+�
+k=1
+cos(2πnkx) ≤ t
+�
+N/2
+�
+= Φ(t),
+where Φ denotes the standard normal distribution. Note that
+� 1
+0
+� N
+�
+k=1
+cos(2πnkx)
+�2
+dx = N
+2 ,
+so the result above contains the “correct” variance for the limit distribution, exactly
+as it should also be expected in the truly independent case. This result has been
+signi���cantly strengthened since then; for example, Philipp and Stout [196] showed
+that under the Hadamard gap condition the function
+S(t, x) =
+�
+k≤t
+cos(2πnkx),
+considered as a stochastic process over the space ([0, 1], B[0, 1], λ), is a small per-
+turbation of a Wiener process, a characterization which allows to deduce many fine
+asymptotic results for this sum. It is also known that the central limit theorem
+for pure trigonometric lacunary sums remains valid under a slightly weaker gap
+condition than Hadamard’s: as Erd˝os [102] proved, it is sufficient to assume that
+nk+1/nk ≥ 1 + ck−α, α < 1/2, while such an assumption with α = 1/2 is not suffi-
+cient.
+The whole situation becomes very different when the cosine-function is replaced
+by a more general 1-periodic function, even if it is such a well-behaved one as a
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+13
+trigonometric polynomial. For example, consider
+(12)
+f(x) = cos(2πx) − cos(4πx),
+nk = 2k, k ≥ 1.
+In this case the lacunary sum is telescoping, and it can be immediately seen that
+there cannot be a non-trivial limit distribution. A more delicate example is attrib-
+uted to Erd˝os and Fortet2, and goes as follows. Let
+f(x) = cos(2πx) + cos(4πx),
+nk = 2k − 1, k ≥ 1.
+Then it can be shown that N−1/2 �N
+k=1 f(nkx) does indeed have a limit distribution,
+but one which is actually non-Gaussian. More precisely, for this example one has
+lim
+N→∞ λ
+�
+x ∈ (0, 1) :
+N
+�
+k=1
+f(nkx) ≤ t
+�
+N/2
+�
+=
+1
+√π
+� 1
+0
+� t/2| cos(πs)|
+−∞
+e−u2duds.
+Thus the limit distribution in this case is a so-called “variance mixture Gaussian”,
+which can be seen as a normal distribution whose variance is a function rather than
+a constant. This limiting behavior can be explained from the observation that
+(13)
+f(nkx) = cos((2k+1 − 2)πx) + cos((2k+2 − 4)πx)
+and
+(14)
+f(nk+1x) = cos((2k+2 − 2)πx) + cos((2k+3 − 4)πx).
+Combining the second term on the right-hand side of (13) with the first term on the
+right-hand side of (14) we obtain
+cos((2k+2 − 4)πx) + cos((2k+2 − 2)πx) = 2 cos(πx) cos((2k+2 − 3)πx),
+so the whole lacunary sum �N
+k=1 f(nkx) can essentially be written as 2 cos(πx)
+multiplied with a pure cosine lacunary sum. This is exactly what the “variance
+mixture Gaussian” indicates: the limit distribution is actually that of 2 cos(πx)
+independently multiplied with a Gaussian. The failure a of Gaussian central limit
+theorem in the example above can be seen as a consequence of the fact that the
+Diophantine equation
+nk+1 − 2nk = 1
+possesses many solutions k for this particular choice of sequence. Equipped with
+this observation, one could readily construct similar examples with other trigono-
+metric polynomials f, and other variance mixture Gaussians as limit distributions,
+by creating situations where there are many solutions k, ℓ to
+(15)
+ank − bnℓ = c
+2The Erd˝os–Fortet example is first mentioned in print in a paper of Salem and Zygmund [202].
+They mention the example without proof, and write: “This remark is essentially due to Erd˝os.”.
+Later the example was mentioned in a paper of Kac [146], who wrote: “It thus came as a surprise
+when simultaneously and independently of each other, Erd˝os and Fortet constructed an example
+showing that the limit [. . . ] need not be Gaussian”, with a footnote: “In Salem and Zygmund this
+example is erroneously credited to Erd˝os alone.” No proof is given in Kac’s paper either, but he
+writes: “Details will be given in [a forthcoming] paper by Erd˝os, Ferrand, Fortet and Kac”. Such
+a joint paper never appeared.
+
+14
+C. AISTLEITNER, I. BERKES AND R. TICHY
+for some fixed a, b, c. However, interestingly, a special role is played by such equations
+when c has the particular value c = 0; very roughly speaking, solutions of the
+equation for c = 0 effect only the limiting variance (in a Gaussian distribution),
+but not the structure of the limiting distribution itself. This is visible in a paper
+of Kac [145], who studied the sequence nk = 2k, k ≥ 1, where indeed the only
+equations that have many solution are of the form 2mnk − nℓ = 0 for some m (the
+solutions being ℓ = k + m). Kac proved that for this sequence and any 1-periodic f
+of bounded variation and zero mean one has
+(16)
+lim
+N→∞ λ
+�
+x ∈ (0, 1) :
+N
+�
+k=1
+f(nkx) ≤ tσf
+√
+N
+�
+= Φ(t)
+with a limiting variance σ2
+f, provided that
+(17)
+σ2
+f :=
+� 1
+0
+f 2(x) dx + 2
+∞
+�
+m=1
+� 1
+0
+f(x)f(2mx) dx ̸= 0.
+Thus in this case the limit distribution is always a Gaussian, and the failure of the
+trivial example in (12) to produce such a Gaussian limit comes from the fact that
+the limiting variance is degenerate.
+These observations show that there is a delicate interplay between arithmetic, an-
+alytic and probabilistic effects; in particular, it is obviously not only the order of
+growth of (nk)k≥1 which is responsible for the fine probabilistic behavior of a la-
+cunary sum. Takahashi [211] proved a CLT (with pure Gaussian limit) under the
+assumption that nk+1/nk → ∞, and Gaposhkin [128] proved that a CLT (with pure
+Gaussian limit) holds when nk+1/nk is an integer for all k, or if nk+1/nk → α for
+some α such that αr ̸∈ Q, r = 1, 2, . . . (and if additionally the variance does not
+degenerate). A general framework connecting Diophantine equations and the dis-
+tribution of lacunary sums was established in Gaposhkin’s profound paper [131],
+where he proved that a CLT (with pure Gaussian limit) holds if for all fixed positive
+integers a, b the number of solutions k, ℓ of the Diophantine equation (15) is bounded
+by a constant which is independent of c (where only c ̸= 0 needs to be considered,
+provided that the variance does not degenerate). One can check the validity of this
+general condition for sequences satisfying the assumptions mentioned earlier in this
+paragraph, such as nk+1/nk → ∞ or nk+1/nk → α for αr ̸∈ Q. Finally, an optimal
+result was established in [13]: For (nk)k≥1 satisfying the Hadamard gap condition,
+the limit distribution of N−1/2 �N
+k=1 f(nkx) is Gaussian provided that the number
+of solutions (k, ℓ) of (15), subject to k, ℓ ≤ N, is of order o(N) (for all fixed a, b,
+uniformly in c ̸= 0). If, on the other hand, for some a, b, c the number of solutions
+is Ω(N), then the CLT generally fails to hold. If the number of solutions with c = 0
+also is of order o(N), then the CLT has the “correct” variance
+� 1
+0 f(x)2dx, in perfect
+accordance with the independent case. Even if the number of solutions is of order
+Ω(N) for some a, b, c, then the deviation of the distribution of N−1/2 �N
+k=1 f(nkx)
+from the Gaussian distribution can be quantified in terms of the ratio “(number of
+solutions)/N”. This shows for example that while the CLT generally fails in the
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+15
+case nk+1/nk → p/q, one obtains an “almost CLT” if both p and q are assumed to be
+large. Another example for such an “almost CLT” is when the growth constant in
+the Hadamard gap condition is assumed to be very large. See the statement of [13,
+Theorem 1.3] and the subsequent discussion for more details.
+Note that Gaposhkin’s condition implies that the CLT also holds for all subsequences
+that are picked out of (nk)k≥1. This is not the case under the assumptions from [13],
+where one might be able to extract a subsequence along which the CLT fails (by
+choosing a subsequence which allows a large number of solutions of the relevant
+Diophantine equations). It is interesting that the probabilistic behavior of lacunary
+sums might change when one passes to a subsequence of the original sequence –
+this is in clear contrast to the bevahior of sums of independent random variables,
+where any subsequence of course is independent as well. A similar remark holds for
+permutations of lacunary sums resp. permutations of sums of independent random
+variables. These phenomena have received strong attention during the last years; see
+for example [17, 18, 19, 21, 114]. To give only one sample result, in [17] the following
+is shown. As noted above, the CLT is true for pure trigonometric sums under the
+Erd˝os gap condition nk+1/nk ≥ 1 + ck−α for some α < 1/2. However, this is only
+true for the unpermuted sequence (i.e. sorted in increasing order). If permutations
+of the sequence are allowed, then this gap condition is not sufficient anymore for the
+validity of the CLT, as is no other gap conditon weaker than Hadamard’s. More
+precisely, for any sequence (εk)k≥1 with εk → 0 there exists a sequence of positive
+integers satisfying nk+1/nk ≥ 1 + εk, together with a permutation σ : N �→ N, such
+that the permuted (pure trigonometric) sum N−1/2 �N
+k=1 cos(2πnσ(k)x) converges in
+distribution to a non-Gaussian limit. One can also construct such examples where
+the norming sequence N−1/2 has to be replaced by (log N)1/2N−1/2 and the limit is a
+Cauchy distribution, and examples where no limit distribution exists at all. See [17]
+for details on this particular result, and Chapter 3 of [61] for a detailed discussion
+of permutation-invariance of limit theorems for lacunary (trigonometric) systems.
+We close this section with some further references. For Hadamard lacunary (nk)k≥1,
+the limit distribution of N1/2DN(nkx) was calculated in [14]; under suitable Dio-
+phantine assumptions it coincides with the Kolmogorov distribution, which is the
+distribution of the range of a Brownian bridge. A central limit theorem for Hardy–
+Littlewood–P´olya sequences was established in [124]. In [87] the Erd˝os–Fortet ex-
+ample was revisited from the perspective of ergodic theory, and was interpreted in
+terms of the limiting behavior of certain modified ergodic sums, and generalized to
+cases such as expanding maps, group actions, and chaotic dynamical systems under
+the assumption of multiple decorrelation. See also [86, 88]. The limit distribution
+of N−1/2 �N
+k=1 cos(2πnkx) for the special sequence nk = k2, k ≥ 1, was determined
+by Jurkat and Van Horne in [141, 142, 143], and turned out to have finite moments
+of order < 4, but not of order 4. The theory of such sums is closely related to theta
+sums, and goes back to Hardy and Littlewood [135]. For further related results,
+see [80, 108, 219]. For non-Gaussian limit distributions of N−1/2 �N
+k=1 cos(2πnkx)
+
+16
+C. AISTLEITNER, I. BERKES AND R. TICHY
+near the Erd˝os gap condition nk+1/nk ≥ 1 + ck−1/2 see [58]. For a multidimensional
+generalization of Kac’s results see [112, 120], and for a multidimensional generaliza-
+tion of the CLT for Hardy–Littlewood–P´olya sequences (considering a semi-group
+generated by powers of matrices instead) see [167, 168]. See also [85, 90] for general-
+izations of the CLT for Hardy–Littlewood–P´olya sequences to a very general setup
+of sums over powers of transformations/automorphisms.
+5. The law of the iterated logarithm for lacunary sequences
+Together with the law of large numbers (LLN) and the central limit theorem (CLT),
+the law of the iterated logarithm (LIL) is one of the fundamental results of prob-
+ability theory. Very roughly speaking, the (strong) law of large numbers says that
+when scaling by N−1 one has almost sure convergence of a sum of random variables,
+and the central limit theorem says that when scaling by N−1/2 one has a (Gaussian)
+limit distribution. The law of the iterated logarithm operates between these two
+other asymptotic limit theorems; in its simplest form, it says that for a sequence
+(Xn)n≥1 of centered i.i.d. random variables (under suitable extra assumptions, such
+as boundedness) one has
+lim sup
+N→∞
+�N
+n=1 Xn
+√2N log log N = σ
+almost surely,
+where σ is the standard deviation. Heuristically, the law of the iterated logarithm
+identifies the threshold between convergence in distribution and almost sure conver-
+gence for sums of i.i.d. random variables; indeed, while
+�N
+n=1 Xn
+√2N log log N converges to 0 in
+distribution by the CLT, it does not converge to 0 almost surely by the LIL. The
+first version of the LIL was given by Khinchin in 1924, and a more general variant
+was established by Kolmogorov in 1929. Note that the law of large numbers for
+trigonometric sums or sums of dilated functions is rather unproblematic: for any
+sequence of distinct integers (nk)k≥1 one has
+(18)
+lim
+N→∞
+1
+N
+N
+�
+k=1
+f(nkx) =
+� 1
+0
+f(x) dx,
+as long as one can assume a bit of regularity for f (such as f being a trigonometric
+polynomial, being Lipschitz-continuous, being of bounded variation on [0, 1], etc.).
+Only if one is not willing to impose any regularity assumptions upon f the situation
+becomes quite different; see the remarks at the end of Section 3.
+The most basic law of the iterated logarithm for lacunary systems is
+(19)
+lim sup
+N→∞
+�N
+n=1 cos(2πnkx)
+√2N log log N
+= 1
+√
+2
+a.e.
+under the Hadamard gap condition on (nk)k≥1; this was obtained by Salem and
+Zygmund (upper bound) [203] and Erd˝os and G´al (lower bound) [103]. Generally
+speaking, as often in probability theory the lower bound is more difficult to estab-
+lish than the upper bound, since the latter can be proved by an application of the
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+17
+first Borel–Cantelli lemma (convergence part), while the former is proved by the
+second Borel–Cantelli lemma (divergence part, which needs some sort of stochastic
+independence as an extra assumption). Note that (19) is a perfect analogue of (18)
+with the “correct” constant on the right-hand side.
+As in the case of the CLT, replacing pure trigonometric sums by sums of more
+general 1-periodic functions makes the situation much more delicate.
+As in the
+previous section, a key role is played by Diophantine equations. However, while
+for the CLT it is crucial that the number of solutions of Diophantine equations
+“stabilizes” in some way to allow for a limit distribution (albeit a potentially non-
+Gaussian one), no such property is necessary for the validity of a form of the LIL
+(since, as noted above, this is defined as a lim sup, not as a lim). Instructive examples
+are the following. In all examples, we assume that f is 1-periodic with mean zero
+and bounded variation on [0, 1].
+• If nk+1/nk → ∞ as k → ∞, then
+lim sup
+N→∞
+�N
+n=1 f(nkx)
+√2N log log N =
+�� 1
+0
+f 2(x) dx
+�1/2
+a.e.
+• If nk = 2k, k ≥ 1, then
+lim sup
+N→∞
+�N
+n=1 f(nkx)
+√2N log log N = σf
+a.e.,
+with
+σ2
+f =
+� 1
+0
+f 2(x)dx + 2
+∞
+�
+m=1
+� 1
+0
+f(x)f(2mx) dx.
+• Assume that nk+1/nk ≥ q > 1, k ≥ 1. Then there exists a constant C
+(depending on f and on q) such that
+(20)
+lim sup
+N→∞
+�N
+n=1 f(nkx)
+√2N log log N ≤ C
+a.e.
+• If nk = 2k − 1, k ≥ 1, and if f(x) = cos(2πx) + cos(4πx), then
+(21)
+lim sup
+N→∞
+�N
+n=1 f(nkx)
+√2N log log N =
+√
+2| cos(πx)|
+a.e.
+The first result in this list (due to Takahashi [213]) is in perfect accordance with
+the LIL for truly independent random sums, in accordance with the fact that
+also the CLT holds in the “truly independent” form under the large gap condi-
+tion nk+1/nk → ∞. The second result is an analogue of Kac’s CLT in Equations
+(16) and (17): as with the CLT, also the LIL holds for the sequence (2k)k≥1, but
+the limiting variance deviates from the one in the “truly independent” case. Note
+that in contrast to the CLT case we now do not need to require that σf ̸= 0 for the
+validity of the statement. The third result (Takahashi [212]) asserts that there is an
+upper-bound version of the LIL for lacunary sums (even for sequences where there is
+no convergence of distributions, and any form of the CLT fails). Finally, the fourth
+
+18
+C. AISTLEITNER, I. BERKES AND R. TICHY
+result (the Erd˝os–Fortet example for the LIL instead of the CLT) shows the remark-
+able fact that the lim sup in the LIL for Hadamard lacunary sums might actually be
+non-constant – this is very remarkable, and a drastic deviation from what one can
+typically observe for sequences of independent random variables. In particular this
+example shows that under the Hadamard gap condition an upper-bound version of
+the LIL is in general the best that one can hope for. Not very surprisingly, the source
+of all these phenomena are (as in the previous section) Diophantine equations such
+as (15), and their number of solutions within the sequence (nk)k≥1. So in the LIL
+there is again a complex interplay between probabilistic, analytic and arithmetic
+aspects which controls the fine asymptotic behavior of lacunary sums.
+In probability theory there is a version of the LIL for the Kolmogorov–Smirnov
+statistic of an empirical distribution. This is called the Chung–Smirnov LIL, and
+in the special case of a sequence (Xn)n≥1 of i.i.d. random variables having uniform
+distribution on [0, 1] (where the Kolmogorov–Smirnov statistic coincides with the
+discrepancy) it asserts that
+lim sup
+N→∞
+NDN(Xn)
+√2N log log N = 1
+2
+almost surely.
+Here the number 1/2 on the right-hand side arises essentially as the maximal L2
+norm (“standard deviation”) of a centered indicator function of an interval A ⊂ [0, 1]
+(namely the indicator function of an interval of length 1/2). Based on the princi-
+ple that lacunary sequences tend to “imitate” the behavior of truly independent
+sequences, it was conjectured that an analogue of the Chung–Smirnov LIL should
+also hold for the discrepancy of ({nkx})k≥1, where (nk)k≥1 is a Hadamard lacunary
+sequence. This was known as the Erd˝os–G´al conjecture, and was finally solved by
+Philipp [193], who proved that for any q > 1 there exists a constant Cq such that
+for (nk)k≥1 satisfying nk+1/nk ≥ q we have
+(22)
+1
+√
+32 ≤ lim sup
+N→∞
+NDN({nkx})
+√2N log log N ≤ Cq
+a.e.
+An admissible value of Cq was specified in [193] as Cq = 166/
+√
+2 + 664/(√2q −
+√
+2).
+The first inequality in (22) follows from (a complex version of) Koksma’s inequality
+together with (19), so the novelty is the second inequality (upper bound). Note also
+that the upper bound in (22) implies Takahashi’s “upper bound” LIL in (20), again
+as a consequence of Koksma’s inequality.
+Philipp’s result has been extended and refined into many different directions. The
+most precise results, many of which were obtain by Fukuyama, show again a fasci-
+nating interplay between arithmetic, analytic and probabilistic effects. As a sample
+we state the following results (all from Fukuyama’s paper [113]):
+Let nk = θk, k ≥ 1. Then
+(23)
+lim sup
+N→∞
+NDN({nkx})
+√2N log log N
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+19
+exists and is constant (almost everywhere). Denoting the value of this lim sup by
+Σθ, we have:
+• If θr ̸∈ Q for all r = 1, 2, . . . , then Σθ = 1/2 a.e.
+• If r denotes the smallest positive integer such that θr = p/q for some coprime
+p, q, then 1/2 ≤ Σθ ≤
+�
+(pq + 1)/(pq − 1)/2 a.e.
+• If θr = p/q as above and both p and q are odd, then Σθ =
+�
+(pq + 1)/(pq − 1)/2
+a.e.
+• If θ = 2, then Σθ =
+√
+42/9 a.e.
+• If θ > 2 is an even integer, then Σθ =
+�
+(p + 1)p(p − 2)/(p − 1)3/2 a.e.
+• If θ = 5/2, then Σθ =
+√
+22/9 a.e.
+All these results were obtained by very delicate calculations involving Fourier anal-
+ysis and Diophantine equations. The calculations from [113] were continued by the
+same author and his group in [119, 121, 122, 125], so that now we have a relatively
+comprehensive picture on the behavior of these lim sup’s in the case when (nk)k≥1
+is (exactly) a geometric progression.
+In [7, 8] for general Hadamard lacunary sequences (nk)k≥1 a direct connection was
+established which links the number of solutions of (15) with the value of the lim sup
+in the LIL, in the same spirit as this was done before in [13] for the CLT (as de-
+scribed in the previous section). In particular, if the number of solutions of (15) is
+sufficiently small, then the LIL holds with the constant 1/2 on the right-hand side,
+exactly as in the truly independent case. Another interesting observation is that
+if (nk)k≥1 is Hadamard lacunary with growth factor q > 1, and if Σ denotes the
+value of the lim sup in (23), then the difference |Σ − 1/2| can be quantified in terms
+of q and tends to zero a.e. as q → ∞. Thus there is a smooth transition towards
+the “truly independent” LIL as the growth factor q increases, and under the large
+gap condition nk+1/nk → ∞ the value of Σ actually equals 1/2. Another remark-
+able fact is that there exist Hadamard lacunary sequences for which the lim sup
+in the LIL for the discrepancy is not a constant almost everywhere, but rather a
+function of x, similar to what happened in (21) for the LIL for � f(nkx). In some
+cases the limit functions in the LIL for the discrepancy can be explicitly calculated,
+and are “surprisingly exotic” (in the words of Ben Green’s MathSciNet review of [6]).
+As noted above, Philipp’s LIL for the discrepancy has been extended into many
+different directions. For example, while it is known that the result can fail as soon
+as the Hadamard gap condition is relaxed to any sub-exponential growth condition,
+it turns out to be possible to obtain an LIL for the discrepancy when a weaker
+growth condition is compensated by stronger arithmetic assumptions. In particu-
+lar, an analogue of Philipp’s result has been proved for Hardy–Littlewood–P´olya
+sequences [195]; see also [5, 65, 123, 215]. As a closing remark concerning the LIL,
+it is interesting that the optimal value of the lower bound in (22) is still unknown;
+cf. [25] for more context.
+
+20
+C. AISTLEITNER, I. BERKES AND R. TICHY
+While much effort has been spent towards understanding the probabilistic behavior
+of lacunary sums at the scales of the CLT and LIL, it seems that investigations at
+other scales (such as in particular at the large deviations scale) were only started
+recently. The few results which are currently available point once again towards an
+intricate connection between probabilistic, analytic and arithmetic effects; see the
+very recent papers [22, 111].
+6. Normality and pseudorandomness
+Normal numbers were introduced by Borel [73] in 1909. From the very beginning
+the concept of normality of real numbers was associated with “randomness”. While
+normality of real numbers was originally defined in terms of counting the number
+of blocks of digits, it is not difficult to see3 that a number x is normal in base b if
+and only if the sequence ({bnx})n≥1 is equidistributed. As Borel proved, Lebesgue-
+almost all real numbers are normal in a (fixed) integer base b ≥ 2, and thus almost
+all reals are normal in all bases b ≥ 2 (such numbers are called absolutely normal).
+While normal numbers are ubiquitous from a measure-theoretic perspective,4 it is
+difficult to construct normal numbers. The most fundamental construction is due
+to Champernowne, who proved (using combinatorial arguments) that the number
+0.1 2 3 4 5 6 7 8 9 10 11 12 13 14 . . . ,
+which is obtained by a concatenation of the decimal expansions of the integers, is
+normal in base 10. The idea of creating normal numbers by a concatenation of the
+(b-ary) expansions of the values of (simple) functions at integers (or primes) is still
+the most popular, and probably most powerful, method in this field. We only note
+that Copeland and Erd˝os [89] proved that
+0.2 3 5 7 11 13 17 19 23 . . . ,
+which is obtained by concatenating the decimal expansions of the primes, is normal
+in base 10, and refer to [92, 93, 171, 174, 186, 187] for more results of this flavor.
+It should be noted, however, that there have been earlier constructions of a con-
+ceptually very different nature, such as that of Sierpinski [208] in 1917. See [43]
+for an exposition of Sierpinski’s construction and more context, and see also [44] on
+an early (unpublished) algorithm of Turing for the construction of normal numbers.
+We finally mention a very recent idea for the construction of a normal number by
+Drmota, Mauduit and Rivat [95], which is not based on the concatenation of deci-
+mal blocks as above, but rather on the evaluation of an automatic sequence along
+a subsequence of the index set (in this particular case, the Thue–Morse sequences
+evaluated along the squares); see also [183, 210].
+3Probably first explicitly mentioned by D.D. Wall in his PhD thesis, 1949.
+4Interestingly, while the set of normal numbers is large from a measure-theoretic point of view, it
+turns out to be small from a topological point of view. More precisely, the set of normal numbers is
+meager (of first Baire category), see e.g. [133]. In the words of Edmund Hlawka [139, p. 78]: “Thus
+whereas the normal numbers almost force themselves on to the measure theorist, the topologist is
+apt to overlook them entirely.”
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+21
+While Lebesgue-almost all numbers are normal and there are some constructions
+of normal numbers, it is generally considered to be completely hopeless to prove
+that natural constants such as π, e,
+√
+2, . . . are normal in a given base (although
+the experimental evidence clearly points in that direction: [189, 218, 224]). Many
+such open problems “for the next millennium” are contained in Harman’s survey
+article [137]; see also [33]. However, it is quite clear that the mathematical machin-
+ery which would be necessary to prove the normality of
+√
+2 or other such constants
+is completely lacking; compare the rather deplorable current state of knowledge on
+the binary digits of
+√
+2 as given in [34, 98, 217]. A small spark of hope is provided
+by the very remarkable formulas of Bailey, Borwein and Plouffe (now widely known
+as BBP formulas), which allow to calculate deep digits of π (and other constants)
+without the need of computing all previous digits. See [51] for a very comprehen-
+sive “source book” covering computational aspects of π, and see [35] for a very rare
+example of a possible strategy of what a proof of the normality of π could possibly
+like (cf. also [162]).
+Since normality of x in a base b can be expressed in terms of the equidistribution of
+the sequence ({bnx})n≥1, it is very natural to consider the discrepancy of DN({bnx})
+and call this (with a slight abuse of language) the discrepancy of x (as a normal
+number, with respect to a base b).
+Remarkably, it is still unknown how small
+the discrepancy of a normal number can be (this is known as Korobov’s problem).
+Levin [166] constructed (for given base b) a number x such that
+DN({bnx}) = O
+�(log N)2
+N
+�
+;
+by Schmidt’s general lower bound the exponent of the logarithm cannot be reduced
+below 1, but the optimal size of this exponent remains open.
+One of the most interesting, and most difficult, aspects of normal numbers is nor-
+mality with respect to two or more different bases. Extending work of Cassels [79],
+Schmidt [206] characterized when normality with respect to a certain base im-
+plies normality with respect to another base, and when this is not the case. See
+also [48, 76]. However, generally speaking it is very difficult to construct numbers
+which are normal with respect to several different bases, and the “constructions”
+are much less explicit than the ones of Champernowne and Copeland–Erd˝os men-
+tioned above. The problem of the minimal order of the discrepancy of normal num-
+bers seems to be very difficult when different bases are considered simultaneously.
+Aistleitner, Becher, Scheerer and Slaman [12] constructed a number x such that
+DN({bnx}) = Ob
+�
+N−1/2�
+for all integer bases b ≥ 2; this is considered to be an “unexpectedly small” order
+of the discrepancy by Bugeaud in his MathSciNet review of [12]. It is not known if
+the exponent −1/2 of N in this estimate is optimal or not; indeed, no non-trivial
+lower bounds whatsoever (beyond the general lower bound (log N)/N of Schmidt)
+
+22
+C. AISTLEITNER, I. BERKES AND R. TICHY
+are known for this problem, but it is quite possible that whenever simultaneous nor-
+mality with respect to different (multiplicatively independent) bases is considered,
+there must be at least one base for which the discrepancy is “large”.
+In a recent years, there has been a special focus on algorithmic aspects of the con-
+struction of normal numbers. A particularly striking contribution was a polynomial-
+time algorithm for the construction of absolutely normal numbers due to Becher,
+Heiber and Slaman [45]. See also [29, 47, 205]. Related to such algorithmic and
+computational problems are questions on the complexity of the set of normal num-
+bers from the viewpoint of descriptive set theory in mathematical logic; in this
+framework, the rank of the set of normal numbers [152] and absolutely normal num-
+bers [46] within the Borel hierarchy has been determined.
+The notion of normality can be extended in a natural way to many other situa-
+tions, where it is always understood that normality should be the typical behaviour.
+For example, one can consider normal continued fractions, where the “expected”
+number of occurences of each partial quotient is prescribed by the Gauss–Kuzmin
+measure; see for example [1, 49]. Other generalizations consider for example normal-
+ity with respect to β-expansions [39, 173], a numeration system which generalizes
+the b-ary expansion to non-integral bases β, or normality with respect to Cantor ex-
+pansions [3, 109, 175], a numeration system which allows a different set of “digits”
+at each position. For a particularly general framework, see [172]. Interestingly, in
+such generalized numeration systems there can be more than one natural definition
+of normality, using as starting point for exampe either the idea of counting blocks
+of digits, or the idea of equidistribution of an associated system. The relation be-
+tween such different (sometimes conflicting) notions of normality has been studied
+in particular detail for Cantor expansions [2, 170, 176].
+Normal numbers feature prominently in the chapter on random numbers in Volume 2
+of Knuth’s celebrated series on The Art of Computer Programming [153]. There he
+tries to come to terms with the notion of “random” sequences of numbers, and intro-
+duces an increasingly restrictive scheme of “randomness” of deterministic sequences.
+The concept of normality is also the starting point for one of the (quantitative) mea-
+sures of pseudorandomness, which were introduced by Mauduit and S´ark¨ozy [179]
+and then studied in a series of papers. Note in this context that the transformation
+T : x �→ bx mod 1, which is at the foundation of the concept of normal numbers,
+can in some sense be seen as the continuous analogue of the recursive formula which
+defines a linear congruential generator (LCG), one of the most classical devices for
+pseudorandom number generation. For this connection between normal numbers
+and pseudo-random number generators, see for example [36]. Another very fruitful
+aspect of normal numbers is the connection with ergodic theory, which comes from
+the observation that the sequence ({bnx})n≥1 is the orbit of x under the transforma-
+tion T from above, and that this transformation is measure-preserving (with respect
+to the Lebesgue measure) and ergodic. We will not touch upon this connection in
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+23
+any detail, and instead refer to [91, 149].
+Another sequence which is often associated with “randomness” is the sequence
+({xn})n≥1 for real x > 1, or more generally ({ξxn})n≥1 for ξ ̸= 0 and x > 1.
+This looks quite similar to a (Hadamard) lacunary sequence such as (bnx)n≥1, but
+its metric theory is of a very different nature in several respects. Both sequences are
+variants of a geometric progression, but while in the lacunary sequence the base b is
+fixed and x is assumed to be a “parameter”, now ξ is assumed to be fixed and the
+base x of the geometric progression is the parameter. While (bnx)n≥1 can in many
+ways be easily interpreted in terms of harmonic analysis, digital expansions, ergodic
+theory, etc., such simple interpretations fail for ({xn})n≥1. Note in particular that in
+contrast to lacunary sequences there now is no periodicity when replacing x �→ x+1,
+there is no “orthogonality”, and the calculation of moments of sums � f(xn) does
+not simply reduce to the counting of solutions of Diophantine equations. Still, what
+is preserved from the setup of lacunary sequences is that xn (as a function of x)
+oscillated quickly on intervals where xm is essentially constant, provided that n is
+significantly larger than m, and there are good reasons to consider systems such as
+(cos(2πxn))n≥1 to be “quasi-orthogonal” and “almost independent” in some appro-
+priate sense.
+One of the most fundamental results on this type of sequence is due to Koksma [154]:
+assuming that ξ ̸= 0 is fixed, the sequence ({ξxn})n≥1 is uniformly distributed mod
+1 for almost all x > 1.
+In particular, when ξ = 1, the sequence ({xn})n≥1 is
+uniformly distributed mod 1 for almost all x > 1.
+In very sharp contrast with
+Koksma’s metric result is the fact that until today not a single example of a number
+x is known for which ({xn})n≥1 is indeed uniformly distributed. This problem is
+related with Mahler’s problem on the range of ({(3/2)n})n≥1, which also seems to be
+completely out of reach for current methods (cf. [97, 110]). The sequence ({xn})n≥1
+is discussed at length in Knuth’s book, where it is conjectured that this sequence is
+a good candidate to pass several very strict pseudorandomness criteria for almost
+all x. For example, Knuth conjectured that for all sequences of distinct integers
+(sn)n≥1 the sequence ({xsn})n≥1 (a subsequence of the original sequence) has a strong
+equidistribution property called complete uniform distribution, for almost all x > 1;
+this was indeed established by Niederreiter and Tichy [188]. It is also known that
+({xn})n≥1 satisfies an law of the iterated logarithm in the “truly independent” form
+lim sup
+N→∞
+NDN({xn})
+√2N log log N = 1
+2
+a.e.,
+and similarly satisfies a central limit theorem which is perfectly analogous to the one
+for truly independent systems [9]. Note that Knuth’s assertion that the sequence
+({xn})n≥1 shows good pseudo-random behavior for almost all x > 1 is of limited
+practical use, as long as no such value of x is found. The discrete analogue would
+be to study the pseudo-randomness properties of an mod q for n = 1, 2, . . . , where a
+and q are fixed integers. Investigations on the pseudo-randomness properties of such
+sequence were carried out for example by Arnol’d [32], who experimentally observed
+
+24
+C. AISTLEITNER, I. BERKES AND R. TICHY
+good pseudorandom behavior; cf. also [10].
+To close this section, we note that equidistribution is of course just one property
+which can be used to characterize “pseudorandom” behavior (essentially by analogy
+with the Glivenko–Cantelli theorem). There are many other statistics which could
+be applied to a sequence in [0, 1] to determine whether it behaves in a “random”
+way or not. One class of such statistics are gap statistics at the level of the average
+gap (which is of order 1/N when considering the first N elements of a sequence
+in [0, 1]), such as the distribution of nearest-neighbor gaps, or the pair correlation
+statistics. We do not give formal definitions of these concepts here, but note that
+they are inspired by investigations of the statistics of quantum energy eigenvalues in
+the context of the Berry–Tabor conjecture in theoretical physics; see [177] for more
+context. Pseudorandom behavior with respect to such statistics is called “Poisso-
+nian”, since it agress with the corresponding statistics for the Poisson process. The
+general principle that lacunary systems show pseudorandom behavior is also valid
+in this context. For example, Rudnick and Zaharescu [199] showed that for (nk)k≥1
+satisfying the Hadamard gap condition the sequence ({nkx})k≥1 is Poissonian for
+almost all x, and Aistleitner, Baker, Technau and Yesha [11] showed that the same
+holds for ({xn})n≥1 for almost all x > 1.
+This section on normal numbers and sequences of the form ({ξxn})n≥1 gives of course
+only a very brief overview of the subject, and has to leave out many interesting
+aspects. For a much more detailed exposition we refer the reader to the book of
+Bugeaud [75].
+7. Random sequences
+In the previous sections we have illustrated the philosophy that gap sequences ex-
+hibit many probabilistic properties which are typical for sequences of i.i.d. random
+variables. In many cases the large gap condition nk+1/nk → ∞ gives “true” random
+limit theorems, the Hadamard gap condition nk+1/nk ≥ q > 1 is a critical transi-
+tion point where a mixture of probabilistic, analytic and arithmetic effects comes
+into play, and the “almost independent” behavior is lost when the gap condition is
+relaxed below Hadamard’s. There are results which hold under weaker gap condi-
+tions such as the Erd˝os gap condition nk+1/nk ≥ 1 + ck−α, 0 < α < 1/2, or under
+additional arithmetic assumptions, but as a whole the Hadamard gap condition is
+the critical point where the “almost independent” behavior of systems of dilated
+functions starts to break down.
+However, while almost independent behavior is generally lost under a weaker gap
+condition (without strong arithmetic information), there is another possible per-
+spective on the problem. As noted, for a fixed sequence (nk)k≥1 one cannot ex-
+pect “almost independent” behavior of ({nkx})k≥1, say, without assuming a strong
+growth condition on (nk)k≥1. However, even without such a growth condition one
+might expect that ({nkx})k≥1 shows independent behavior for “typical” sequences
+(nk)k≥1. Here the word “typical” of course implies that the sequence has to be taken
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+25
+from a generic set in some appropriate space which possesses a measure, so quite
+naturally this idea leads to considering “random” sequences (nk)k≥1 = (nk(ω))k≥1
+which are constructed in a randomized way over some probability space.
+Of course there are many possible ways how a random sequence can be constructed.
+From results of Salem and Zygmund [204] for trigonometric sums with random signs
+it follows easily that if we define a sequence (nk)k≥1 by flipping a coin (independently)
+for every positive integer to decide whether it should be contained in the sequence
+or not, and let P denote the probability measure on the space over which the “coins”
+are defined, then for P-almost all sequences as defined above one has
+(24)
+1
+√
+N
+N
+�
+k=1
+cos(2πnkx)
+D
+−→ N(0, 1/4)
+and
+(25)
+lim sup
+N→∞
+1
+√2N log log N
+N
+�
+k=1
+cos(2πnkx) = 1
+2
+for almost all x,
+where N(0, σ2) denotes the normal distribution with mean 0 and variance σ2 and
+D
+−→ denotes convergence in distribution. Note that (24) and (25) are not exactly
+matching with the truly independent case, where the limit distribution would be
+N(0, 1/2) and the limsup in the LIL would be 1/
+√
+2. The “loss” on the right-hand
+sides of (24) and (25) comes from the fact that a Dirichlet kernel is “hiding” in this
+linearly growing sequence, and this kernel is highly localized near 0 and 1 so that its
+contribution is lost in the CLT and LIL. By the strong law of large numbers (SLLN)
+clearly nk ∼ 2k as k → ∞, P-almost surely, so the sequences constructed here are
+very far from satisfying any substantial gap condition; in contrast, their (typical)
+order of growth is only linear. It should be noted that the gaps nk+1 − nk in this
+sequence are not bounded: with full P-probability, nk+1 −nk = 1 for infinitely many
+k (roughly, in half of the cases), but for infinitely many k, the gap nk+1 − nk has
+order of magnitude c log k; this follows from the “pure heads” theorem of Erd˝os and
+R´enyi, see [198].
+We call an increasing sequence (nk)k≥1 of positive integers a B2 sequence if there
+exists a constant C > 0 such that for any integer ν > 0 the number of representations
+of ν in the form ν = nk ±nℓ, k > ℓ ≥ 1, is at most C. By a result of Gaposhkin [131]
+already mentioned in Section 4, the sequence (f(nkx))k≥1 satisfies the CLT for all
+Hadamard lacunary (nk)k≥1 and all 1-periodic Lipschitz continuous f if and only if
+for any m ≥ 1, the set-theoretic union of the sequences (nk)≥1, (2nk)≥1, . . . , (mnk)≥1
+satisfies the B2 condition.5
+No similarly complete result is known for sequences
+(nk)k≥1 growing slower than exponentially, but Berkes [54] proved that if (nk)k≥1 is
+5Note that the definition of the B2 property used in [131] is slightly different from the standard
+usage in number theory (see e.g. [134]) requiring that the number of solutions of ν = nk + nℓ, k >
+ℓ ≥ 1, is bounded by C, but this does not affect the discussion below.
+
+26
+C. AISTLEITNER, I. BERKES AND R. TICHY
+a B2 sequence satisfying the gap condition
+(26)
+nk+1/nk ≥ 1 + ck−α,
+k ≥ 1,
+for some c > 0, α > 0, then (cos(2πnkx))k≥1 satisfies the CLT and LIL. To verify
+the B2 property for a concrete sequence (nk)k≥1 is generally a difficult problem, but
+the situation is quite different for random constructions. Let I1, I2, . . . be disjoint
+blocks of consecutive integers and let n1, n2, . . . be independent random variables
+on some probability space (Ω, F, P) such that nk is uniformly distributed over Ik.
+Clearly, the number of different sums ±nk1 ± nk2 ± nk3, 1 ≤ k1, k2, k3 ≤ k − 1, is
+at most 8(k − 1)3, and thus if the size of Ik is ≥ k5, then the probability that nk
+is equal to any of these sums is ≤ 8k3k−5 = O(k−2). Thus by the Borel-Cantelli
+lemma, with P-probability 1, such a coincidence can occur only for finitely many k.
+Thus the equation
+±nk1 ± nk2 ± nk3 ± nk4 = 0,
+k1 ≤ k2 ≤ k3 < k4
+has only finitely many solutions, which implies that (nk)k≥1 is a B2 sequence.
+Let us recall now that by a result of Erd˝os [102], (cos(2πnkx))k≥1 satisfies the CLT
+with limit distribution N(0, 1/2), provided that (26) holds with α < 1/2, and this
+result is sharp, i.e. there exists a sequence (nk)k≥1 satisfying (26) with α = 1/2 such
+that the CLT fails for (cos(2πnkx))k≥1. Note that the counterexample is irregular:
+while nk+1/nk − 1 is of the order O(k−1/2) for most k, there is also a subsequence
+along which nk+1/nk → ∞. One may therefore wonder if regular behavior of nk+1/nk
+implies the CLT; in particular, Erd˝os [102] conjectured that the CLT holds for
+(cos(2πnkx))k≥1 if nk = ⌊e(kβ)⌋ for some β in the range 0 < β ≤ 1/2. (Note that for
+β > 1/2 condition (26) is satisfied with α < 1/2, so the CLT follows from Erd˝os’
+result.) This conjecture was proved by Murai [184] for β > 4/9, but for smaller β the
+problem is still open. Random constructions provide here important information.
+Kaufman [148] proved that if c is chosen at random, with uniform distribution on a
+finite interval (a, b) ⊂ (0, ∞), then (cos(2πnkx))k≥1 with nk = e(ckβ) satisfies the CLT
+with probability 1 for any fixed β > 0. An even wider class of random sequences with
+the CLT property is obtained by choosing the blocks Ik in the random construction
+above as the integers in the interval
+(27)
+Jk =
+�
+e(ckβ)(1 − rk), e(ckβ)(1 + rk)
+�
+,
+rk = o(k−(1−β)).
+A simple calculation shows that these intervals are disjoint for k ≥ k0 and for nk ∈ Jk
+we have (26) with α = 1 − β, in fact we even have
+nk+1/nk = 1 + c1(1 + o(1))/k1−β
+with some constant c1 > 0. Now if rk decreases like a negative power of k, then the
+length of Jk will be ≥ k5 and thus the constructed random sequence (nk)k≥1 will
+be a B2 sequence with probability 1, so (cos(2πnkx))k≥1 satisfies the CLT. In other
+words, the CLT for (cos(2πnkx))k≥1 holds for a huge class of sequences nk ∼ e(ckβ)
+for any c > 0, β > 0.
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+27
+Concerning B2 sequences, it is worth pointing out that Erd˝os [101] proved, decades
+before Carleson’s convergence theorem, that �∞
+k=1(ak cos(2πnkx) + bk sin(2πnkx))
+is almost everywhere convergent if (nk)k≥1 is a B2 sequence. The question of how
+slowly B2 sequences can grow is a much investigated problem of number theory,
+see e.g. Halberstam and Roth [134], Chapters II and III. It is easily seen that
+a B2 sequence (nk)k≥1 cannot be o(k2) and Erd˝os and R´enyi [105] proved by a
+random construction that for any ε > 0 there exists a B2 sequence (nk)k≥1 with
+nk = O(k2+ε). Changing the B2 property slightly and requiring that all numbers
+nk ± nℓ, k > ℓ, are actually different, makes the problem considerably harder. The
+“greedy algorithm” yields a B2 sequence (nk)k≥1 with nk = O(k3), see [180], and
+it took nearly 40 years to improve this to nk = o(k3), see [26]. The best currently
+known (random) construction is due to Ruzsa [200], and satisfies nk = k1/(
+√
+2−1)+o(1).
+Let (ωn)n≥1 be a nondecreasing sequence of positive integers tending to +∞ and let
+us divide the set of positive integers into disjoint blocks I1, I2, . . . such that the cardi-
+nality of Ik is ωk. Using these blocks in the random construction above, the resulting
+random sequence (nk)k≥1 cannot be a B2 sequence if (ωn)n≥1 grows slower than any
+power of n, but it is proved in Berkes [55] that with P-probability 1, (cos(2πnkx))k≥1
+still satisfies the CLT and LIL. The limit distribution here is N(0, 1/2) and the lim-
+sup in the LIL is 1/
+√
+2, so that the “loss of mass” phenomenon observed in the
+case of the random sequence (nk)k≥1 in the Salem-Zygmund paper [204] does not
+occur here. The gaps in this sequence satisfy nk+1 − nk ≤ 2ωk+1, i.e. they can grow
+arbitrarily slowly. An LIL for the discrepancy of ({nkx})k≥1 under the same gap
+condition was given in Fukuyama [118]. In [55] the question was raised if there
+exists a sequence (nk)k≥1 with bounded gaps nk+1 − nk = O(1) such that the CLT
+holds. Bobkov and G¨otze [71] showed that if we want no loss of mass in the CLT,
+the answer is negative: if (nk)k≥1 is any increasing sequence of positive integers
+with nk+1 − nk ≤ L, k ≥ 1, such that N−1/2 �N
+k=1 cos(2πnkx) has a Gaussian limit
+distribution N(0, σ2), then necessarily σ2 < 1/2 and L ≥ 1/(1 − 2σ2). On the other
+hand, Fukuyama [115, 116, 117] showed that for any σ2 < 1/2 there exists indeed
+a random subsequence (cos(2πnkx))k≥1 of the trigonometric system satisfying the
+CLT with limit distribution N(0, σ2) and with bounded gaps nk+1 − nk ≤ L with
+L ∼ 4/(1 − 2σ2) as σ2 → 1/2. This shows that the result of Bobkov and G¨otze is
+optimal up to a factor 4. This remarkable result is the “small gaps” counterpart of
+Erd˝os’ central limit theorem [102]: the latter determines the smallest gap sizes in
+(nk)k≥1 implying the CLT for (cos(2πnkx))k≥1, while Fukuyama’s result determines
+the smallest gap size which still allows a CLT with limit distribution N(0, σ2) to hold.
+It is worth pointing out that the bounded gap sequences in [115, 116, 117] are
+obtained by rather complicated random constructions, while using the previously
+discussed simple construction and choosing the nk as independent random variables
+uniformly distributed over adjoining blocks Ik with equal length results in a random
+
+28
+C. AISTLEITNER, I. BERKES AND R. TICHY
+sequence (nk)k≥1 satisfying almost surely
+(28)
+1
+√
+N
+N
+�
+k=1
+cos(2πnkx)
+D
+−→ N(0, Y ),
+where Y ≥ 0 is a random variable and N(0, Y ) is a “variance mixture” normal
+distribution with characteristic function E exp(−Y t2/2), see [71]. We also note that
+there is generally no “loss of mass” phenomenon for the LIL for trigonometric series
+with bounded gaps, see [23, 24]. For further results for trigonometric series with
+bounded/random gaps, see [42, 41, 62].
+8. The subsequence principle
+The purpose of the previous sections was to illustrate the principle that thin subse-
+quences of the trigonometric system, or thin subsequences of a more general system
+of dilated functions, exhibit properties which are typical for sequences of indepen-
+dent random variables.
+However, an analogous principle holds in a much wider
+framework: it is known that, under suitable technical assumptions, sufficiently thin
+subsequences of general systems of random variables behave like genuine indepen-
+dent sequences, in the sense that a general sequence of random variables allows
+to extract a subsequence showing independent behavior. For example, Gaposhkin
+[128, 132] and Chatterji [83, 84] proved that if (Xn)n≥1 is any sequence of random
+variables satisfying supn EX2
+n < ∞, then there exist a subsequence (Xnk)k≥1 and
+random variables X ∈ L2, Y ∈ L1, Y ≥ 0, such that
+(29)
+1
+√
+N
+�
+k≤N
+(Xnk − X)
+D
+−→ N(0, Y )
+and
+(30)
+lim sup
+N→∞
+1
+√2N log log N
+�
+k≤N
+(Xnk − X) = Y 1/2
+a.s.,
+where as at the end of the previous section N(0, Y ) denotes the “variance mix-
+ture”normal distribution with characteristic function E exp(−Y t2/2), and where
+again
+D
+−→ denotes convergence in distribution. A functional (Strassen type) ver-
+sion of (30) was proved by Berkes [52].
+By a result of Koml´os [159], from any
+sequence (Xn)n≥1 of random variables satisfying supn E|Xn| < ∞ one can select a
+subsequence (Xnk)k≥1 such that
+(31)
+lim
+N→∞
+1
+N
+�
+k≤N
+Xnk = X
+a.s.
+for some X ∈ L1. Chatterji [81] proved that if (Xn)n≥1 is a sequence of random vari-
+ables satisfying supn E|Xn|p < ∞ for some 0 < p < 2, then there exist a subsequence
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+29
+(Xnk)k≥1 and a random variable X with E|X|p < ∞ such that
+(32)
+lim
+N→∞
+1
+N1/p
+�
+k≤N
+(Xnk − X) = 0
+a.s.
+These results establish the analogues of the central limit theorem (CLT), the law
+of the iterated logarithm (LIL), the strong law of large numbers (SLLN) and Mar-
+czinkiewicz’ strong law for subsequences (Xnk)k≥1. Note the mixed (or randomized)
+character of (29)–(32): the limit X in the strong law of large numbers, the cen-
+tering factor X in Marczinkiewicz’ strong law, and the limiting variance Y in the
+CLT (which also determines the limsup in the LIL) all become random. For fur-
+ther limit theorems for subsequences of arbitrary random variable sequences, see
+Gaposhkin [128]. On the basis of these and several other examples, Chatterji [82]
+formulated the following heuristic principle:
+Subsequence Principle. Let T be a probability limit theorem valid for all se-
+quences of i.i.d. random variables belonging to an integrability class L defined by
+the finiteness of a norm ∥ ·∥L. Then if (Xn)n≥1 is an arbitrary (dependent) sequence
+of random variables satisfying supn ∥Xn∥L < +∞ then there exists a subsequence
+(Xnk)k≥1 satisfying T in a mixed form.
+In a profound study, Aldous [27] proved the validity of the subsequence principle
+for all distributional and almost sure limit theorems subject to minor technical
+conditions. To formulate his results, let M denote the class of probability measures
+on the Borel sets of R, equipped with the L´evy metric. By [27], a subset A ⊂ M×R∞
+is called a limit statute if:
+(a) P((λ, X1(ω), X2(ω), . . .) ∈ A) = 1 provided X1, X2, . . . are i.i.d. random vari-
+ables with distribution λ.
+(b) (λ, x1, x2, . . .) ∈ A and � |xi − x′
+i| < ∞ implies that (λ, x′
+1, x′
+2, . . .) ∈ A.
+An a.s. limit theorem can thus be identified with a limit statute, where the analytic
+statement of the theorem is expressed by (a), while relation (b) means that a small
+perturbation of the sequence X1, X2, . . . does not change the validity of the limit
+theorem. Let us give two examples of limit statutes representing the strong law of
+large numbers and the law of the iterated logarithm:
+A1 =
+�
+(λ, x) ∈ A : limN→∞ N−1 �N
+k=1 xk = |λ|1
+�
+∪ {(λ, x) : |λ|1 = ∞},
+A2 =
+�
+(λ, x) ∈ A : lim supN→∞(2N log log N)−1/2 ��N
+k=1 xk − N|λ|1
+�
+= |λ|2
+�
+∪ {(λ, x) : |λ|2 = ∞}.
+Here |λ|1 and |λ|2 denote the mean and variance of λ provided they are finite, and
+we write |λ|1 = ∞, resp. |λ|2 = ∞ if
+�
+R |x|dλ(x) = ∞, resp.
+�
+R |x|2dλ(x) = ∞.
+
+30
+C. AISTLEITNER, I. BERKES AND R. TICHY
+On the other hand, by the definitions in [27], a weak limit theorem for i.i.d. random
+variables is a system
+T = (f1, f2, . . . , {Gλ, λ ∈ M0})
+where
+(a) M0 is a measurable subset of M.
+(b) For each k ≥ 1, fk = fk(λ, x1, x2, . . .) is a real function on M × R∞, measurable
+in the product topology, satisfying the smoothness condition
+|fk(λ, x) − fk(λ, x′)| ≤
+∞
+�
+k=1
+ck,i|xi − x′
+i|
+where 0 ≤ ck,i ≤ 1 and limk→∞ ck,i = 0 for each i.
+(c) For each λ ∈ M0, Gλ is a probability distribution on the real line such that the
+map λ → Gλ is measurable (with respect to the Borel σ-field in M0).
+(d) If λ ∈ M0 and X1, X2, . . . are independent random variables with common
+distribution λ then
+fk(λ, X1, X2, . . . , )
+D
+−→ Gλ
+as k → ∞.
+For example, the central limit theorem corresponds to the case when M0 is the class
+of distributions with mean 0 and finite variance,
+(33)
+fk(λ, x1, x2, . . .) = x1 + . . . + xk − kE(λ)
+√
+k
+and Gλ = N(0, Var(λ)).
+Let now (Xn)n≥1 be a sequence of random variables with supn ∥Xn∥L < ∞ with any
+norm ∥ · ∥L on R. Then (Xn)n≥1 is bounded in probability, i.e.
+lim
+K→∞ P(|Xn| > K) = 0
+uniformly in n.
+By an extension of the Helly–Bray theorem (see e.g. [66]), (Xn)n≥1 has a subsequence
+(Xnk)k≥1 having a limit distribution conditionally on any event in the probability
+space with positive probability, i.e. for any A ⊂ Ω with P(A) > 0 there exists a
+distribution function FA such that
+lim
+k→∞ P(Xnk ≤ t | A) = FA(t)
+for all continuity points t of FA.
+According to the terminology of [66], such a
+subsequence is called determining. Thus when investigating asymptotic properties
+of sufficiently thin subsequences of sequences (Xn)n≥1 with bounded norms, we can
+assume, without loss of generality, that (Xn)n≥1 itself is determining. As is shown
+in [27, 66], for any determining sequence (Xn)n≥1 there exists a random measure µ
+(i.e. a measurable map from the underlying probability space (Ω, F, P) to M) such
+that for any A with P(A) > 0 and any continuity point t of FA we have
+(34)
+FA(t) = EA(µ(−∞, t])
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+31
+where EA denotes conditional expectation given A. This measure µ is called the
+limit random measure of (Xn)n≥1; see Section 9 below for more details.
+With these preparations, we are now in a position to formulate the subsequence
+theorems of Aldous.
+Theorem A (Aldous [27]). Let (Xn)n≥1 be a determining sequence with limit ran-
+dom measure �� and let A be a limit statute. Then there exists a subsequence (Xnk)k≥1
+such that for any further subsequence (Xmk)k≥1 ⊂ (Xnk)k≥1 we have
+P((λ(ω), Xm1(ω), Xm2(ω), . . .) ∈ A) = 1.
+Theorem B (Aldous [27]). Let (Xn)n≥1 be a determining sequence with limit ran-
+dom measure µ and let
+T = (f1, f2, . . . , {Gλ, λ ∈ M0})
+be a weak limit theorem. Assume that P(µ ∈ M0) = 1. Then there exists a sub-
+sequence (Xnk)k≥1 such that for any further subsequence (Xmk)k≥1 ⊂ (Xnk)k≥1 we
+have
+lim
+k→∞ P(fk(Xm1(ω), Xm2(ω), . . . µ(ω)) ≤ t) = EGµ(ω)(t)
+at all continuity points t of the distribution function on the right hand side.
+Writing out Theorem A and B in the case of the limit statutes A1, A2 above and
+the weak limit theorem defined by (33), we get the CLT, LIL and SLLN for thin
+subsequences of determining sequences, as stated in (29), (30), (31) above.
+The proof of Koml´os’ result (31) exemplifies the technique used in the field of sub-
+sequence behavior before Aldous’ paper [27], and in particular in proving the results
+(29)–(32) mentioned above. As Koml´os showed, if (Xn)n≥1 is a sequence of random
+variables with bounded L1 norms, then its sufficiently thin subsequences (Xnk)k≥1
+are, after a random centering and small perturbation, an identically distributed
+martingale difference sequence with finite means and thus, by classical martingale
+theory, they satisfy the SLLN. Martingale versions of the CLT and LIL yield also
+relations (29), (30) and their functional versions. While this method yields several
+further limit theorems for lacunary sequences, martingale difference sequences cer-
+tainly do not satisfy all i.i.d. limit theorems in a randomized form and thus the
+general subsequence principle cannot be proved in such a way. The proof of Theo-
+rems A and B in [27] uses a different way and utilizes near exchangeability properties
+of subsequences of general sequences of random variables. Let (Xn)n≥1 be a deter-
+mining sequence with limit random measure µ and let (Yn)n≥1 be a sequence of
+random variables, defined on the same probability space as the Xn’s, conditionally
+i.i.d. with respect to µ, with conditional distribution µ. (For the construction of
+such an (Yn)n≥1 one may need to enlarge the probability space.) Clearly, (Yn)n≥1
+is exchangeable, i.e. for any permutation σ : N → N of the positive integers, the
+sequence (Yσ(n))n≥1 has the same distribution as (Yn)n≥1, and it satisfies limit the-
+orems of i.i.d. random variables in a mixed form. For example, if EY 2
+1 < ∞ and
+
+32
+C. AISTLEITNER, I. BERKES AND R. TICHY
+Y = E(Y1 | µ), Z = Var (Y1 | µ), then
+N−1/2
+N
+�
+k=1
+(Yk − Y )
+D
+−→ N(0, Z)
+and
+lim sup
+N→∞
+(2N log log N)−1/2
+N
+�
+k=1
+(Yk − Y ) = Z1/2
+a.s.
+This principle holds in full generality, i.e. for all a.s. and distributional limit theorems
+in the above formalization. Indeed, if the Yn are conditionally i.i.d. with respect to
+µ and with conditional distribution µ (a random probability measure on R) and if
+A is a limit statute, then
+(35)
+P((µ, Y1, Y2, . . .) ∈ A|µ)(ω) = P(µ(ω), Y ∗
+1 , Y ∗
+2 , . . .) ∈ A)
+a.e.
+where (Y ∗
+n )n≥1 is an i.i.d. sequence with marginal distribution µ(ω). By the definition
+of limit statute, the last probability in (35) equals 1 and taking expectations we get
+P((µ, Y1, Y2, . . .) ∈ A) = 1,
+which is exactly our claim. Specializing to the case of the limit statutes A1 and A2
+above, we get relations (29) and (30). A similar argument works for distributional
+limit theorems. Now, as is shown in [27], for every k ≥ 1 we have
+(36)
+(Xn1, Xn2, . . . Xnk)
+D
+−→ (Y1, Y2, . . . , Yk)
+as
+n1 < n2 < . . . < nk, n1 → ∞.
+In other words, for large indices the finite dimensional distributions of the sequence
+(Xnk)k≥1 are close to those of the limiting exchangeable sequence (Yk)k≥1 and thus
+one may expect that limit theorems of (Yk)k≥1 (which, as we have just seen, are
+mixed versions of i.i.d. limit theorems) continue to hold for sufficiently thin subse-
+quences (Xnk)k≥1 as well. Of course, a limit theorem for (Xnk)k≥1 can describe a
+complicated analytic property of the infinite vector (Xn1, Xn2, . . . , Xnk, . . .) which
+does not follow from the weak convergence of the finite dimensional distributions
+of the sequence, but with a suitable thinning procedure and delicate analytic ar-
+guments, Aldous showed an infinite dimensional extension of (36), leading to the
+validity of Theorems A and B.
+Although the theorems of Aldous are of exceptional generality, there are important
+results for lacunary sequences which are not covered by them. As was shown by
+Gaposhkin [128], for every uniformly bounded sequence (Xn)n≥1 of random variables
+there exists a subsequence (Xnk)k≥1 and bounded random variables X and Y ≥ 0
+such that for any numerical sequence (an)n≥1 satisfying
+(37)
+AN :=
+N
+�
+k=1
+a2
+k → ∞,
+aN = o(A1/2
+N )
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+33
+we have
+(38)
+1
+AN
+N
+�
+k=1
+ak(Xnk − X)
+D
+−→ N(0, Y ),
+and if the second relation of (37) is replaced by
+(39)
+aN = o(AN/(log log AN)1/2)
+then we have
+(40)
+lim sup
+N→∞
+1
+�
+2A2
+N log log AN
+N
+�
+k=1
+ak(Xnk − X) = Y 1/2
+a.s.
+The difference of these results from (29) and (30) is that in the CLT and LIL we have
+weighted sums �N
+k=1 ak(Xnk −X) instead of ordinary sums �N
+k=1(Xnk −X). For ev-
+ery fixed coefficient sequence (an)n≥1 the CLT and LIL in (38) and (39) follow from
+Theorems A and B, but the subsequence (Xnk)k≥1 provided by the proofs depends
+on (ak)k≥1 and it is not clear that we can select a subsequence (Xnk)k≥1 satisfying
+(38) and (39) simultaneously for all considered coefficient sequences (ak)k≥1.
+Another important situation not covered by Aldous’ general theorems is when we
+investigate permutation-invariance of limit theorems for subsequences.
+Since the
+asymptotic properties of an exchangeable sequence (Yn)n≥1 do not change after any
+permutation of its terms, it is natural to expect that the conclusions in Theorem A
+and B remain valid after an arbitrary permutation of the subsequence (Xnk)k≥1
+in the theorems. However, the proofs of Theorem A and B are not permutation-
+invariant and it does not follow that, e.g., any sequence (Xn)n≥1 of random variables
+with bounded L1 norms contains a subsequence (Xnk)k≥1 satisfying the strong law
+of large numbers after any permutation of its terms. Using ad hoc methods, the
+latter result has been proved by Berkes [56] and another classical case, namely the
+unconditional a.e. convergence of series � ck(Xnk − X) under � c2
+k < ∞ for subse-
+quences (Xnk)k≥1 of L2 bounded sequences (Xn)n≥1, has been settled by Koml´os [160]
+(see [27] for another proof via exchangeability). It clearly would be desirable to pro-
+vide further general results in this direction.
+We now formulate some structure theorems for lacunary sequences enabling one to
+handle problems of the kind discussed above. Recall that if (Xn)n≥1 is a determin-
+ing sequence with limit random measure µ and (Yn)n≥1 is a sequence conditionally
+i.i.d. with respect to the σ-algebra generated by µ and with conditional marginal
+distributions µ, then there exists a subsequence (Xnk)k≥1 such that (36) holds. This
+shows that, in some sense, for large indices the sequence (Xnk)k≥1 resembles the
+sequence (Yk)k≥1, but this property is far too weak to deduce limit theorems for
+(Xnk)k≥1 from those valid for the exchangeable sequence (Yk)k≥1. The following
+theorem, proved by Berkes and P´eter [63], shows that with a suitable choice of the
+subsequence (nk)k≥1, the variables (Xnk)k≥1 can be chosen to be close to the Yk in
+a pointwise sense. We call a sequence (Xn)n≥1 of random variables ε-exchangeable
+
+34
+C. AISTLEITNER, I. BERKES AND R. TICHY
+if on the same probability space there exists an exchangeable sequence (Yn)n≥1 such
+that P(|Xn − Yn| ≥ ε) ≤ ε for all n. Then we have
+Theorem C (Berkes and P´eter [63]). Let (Xn)n≥1 be a sequence of random variables
+bounded in probability, and let (εn)n≥1 be a sequence of positive reals tending to zero.
+Then, if the underlying probability space is large enough, thee exists a subsequence
+(Xnk)k≥1 such that, for all l ≥ 1, the sequence Xnl, Xnl+1, . . . is εl-exchangeable.
+Note that Theorem C provides a different approximating exchangeable sequence
+(Y (l)
+j )j≥1 for each tail sequence (Xnl, Xnl+1, . . .), with termwise approximating error
+εl. The following theorem describes precisely the structure of the the sequences
+(Y (l)
+j )j≥1.
+Theorem D (Berkes and P´eter [63]). Let (Xn)n≥1 be a determining sequence of
+random variables, and let (εn)n≥1 be a sequence of positive reals. Then there exists
+a subsequence (Xmk)k≥1 and a sequence (Yk)k≥1 of discrete random variables such
+that
+(41)
+P
+�
+|Xmk − Yk| ≥ εk
+�
+≤ εk
+k = 1, 2 . . . ,
+and for each k > 1 the atoms of the finite σ-field σ{Y1, . . . , Yk−1} can be divided into
+two classes Γ1 and Γ2 such that the following holds. Firstly,
+(42)
+�
+B∈Γ1
+P(B) ≤ εk.
+Secondly, for any B ∈ Γ2 there exist PB-independent random variables {Z(B)
+j
+, j =
+k, k + 1, . . . } defined on B with common distribution function FB such that
+(43)
+PB
+�
+|Yj − Z(B)
+j
+| ≥ εk
+�
+≤ εk,
+j = k, k + 1, . . .
+Here FB denotes the limit distribution of (Xn)n≥1 relative to B and PB denotes
+conditional probability given B.
+We now give applications of Theorem D to the problems discussed above. First we
+note that using Theorem D it is a simple exercise to prove, for suitable subsequences
+of a uniformly bounded sequences (Xn)n≥1, the weighted CLT and LIL in (38),
+(40) simultaneously for all permitted coefficient sequences (an)n≥1. Next we give a
+permutation-invariant form of Theorem B for distributional limit theorems.
+Definition 1. We call the weak limit theorem T = (f1, f2, . . . , S, {Gµ, µ ∈ M0})
+regular if there exist sequences pk ≤ qk of positive integers tending to +∞ and a
+sequence ωk → +∞ such that
+(i) fk(λ, x1, x2, . . .) depends only on λ, xpk, . . . , xqk.
+(ii) fk satisfies the Lipschitz condition
+|fk(λ, xpk, . . . , xqk) − fk(λ′, x′
+pk, . . . , x′
+qk)| ≤
+≤ 1
+ωk
+qk
+�
+i=pk
+|xi − x′
+i|α + ̺∗(λ, λ′)
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+35
+for some 0 < α ≤ 1, where ̺∗ is a metric on M0 generating the same topology
+as the Prohorov metric ̺.
+For example, the central limit theorem can be formalized by the functions
+fk(λ, x[k1/4], . . . , xk) = x[k1/4] + . . . + xk − kE(λ)
+√
+k
+,
+leading to a regular limit theorem. Note that originally we formalized the CLT with
+the functions fk in (33) containing all variables x1, x2, . . ., but under bounded second
+moments the first k1/4 terms here are irrelevant and hence we can always switch to
+the present version. The same procedure applies in the general case.
+Theorem E (Aistleitner, Berkes and Tichy [20]). Let (Xn)n≥1 be a determining
+sequence with limit random measure ˜µ. Let T = (f1, f2, . . . , S, {Gµ, µ ∈ M0}) be a
+regular weak limit theorem and assume that P(˜µ ∈ M0) = 1. Then there exists a
+subsequence (Xnk)k≥1 such that for any permutation (X∗
+k)k≥1 of (Xnk)k≥1 we have
+(44)
+fk(X∗
+1, X∗
+2, . . . , ˜µ) →d
+�
+G˜µdP.
+In case of the CLT formalized above, assuming supn EX2
+n < +∞ implies easily that
+˜µ has finite variance almost surely, and thus denoting its mean and variance by X
+and Y , respectively, we see that the integral in (44) is the distribution N(0, Y ).
+Hence (44) states in the present case that
+1
+√
+N
+N
+�
+k=1
+(X∗
+k − X)
+D
+−→ N(0, Y ),
+which is the permutation-invariant form of the CLT.
+Concerning a.s. limit theorems, a permutation-invariant form of the strong law of
+large numbers for subsequences of an L1-bounded sequence was proved, as already
+mentioned, in Berkes [56], and a similar argument yields the corresponding result for
+the LIL. No permutation-invariant version of the general result in Theorem A has
+been proved in the literature, but there is no need for that, since a.s. limit theorems
+can be reformulated in a distributional form and thus the proof of Theorem B applies
+with obvious changes. For illustration, we give here the reformulation of the LIL:
+Theorem F. Let (Xn)n≥1 be a sequence of random variables with E|Xn| ≤ 1, n =
+1, 2, . . . Put Sn = �n
+i=1 Xi, Sk,l = �l
+i=k+1 Xi, and L(N) = (2N log log N)1/2. Then
+lim supN→∞ SN/L(N) = 1 a.s. iff for any ε > 0 there exists a sequence m1 < m2 <
+· · · of positive integers such that mk ≥ 5k and
+P
+�
+max
+mk≤j≤mk+1
+Sk,j
+L(j) > 1 + ε
+�
+≤ 2−k,
+k ≥ k0,
+and
+P
+�
+max
+mk≤j≤mk+1
+Sk,j
+L(j) < 1 − ε
+�
+≤ 2−k,
+k ≥ k0.
+
+36
+C. AISTLEITNER, I. BERKES AND R. TICHY
+It is worth pointing out that given a sequence (Xn)n≥1 of random variables, find-
+ing a subsequence (Xnk)k≥1 satisfying the permutation-invariant form of some limit
+theorem generally requires a much faster growing sequence (nk)k≥1 than to find a
+subsequence to satisfy the original limit theorem. This is a phenomenon which also
+occurs for lacunary trigonometric sums or lacunary sums of dilated functions; com-
+pare the last paragraph of Section 4 above.
+In conclusion we note that if (Xn)n≥1 is a sequence of random variables with fi-
+nite means over the probability space (0, 1) equipped with the Borel σ-algebra and
+Lebesgue measure such that for all n ≥ 1 and (a1, . . . , an) ∈ Rn we have
+(45)
+C1
+� n
+�
+k=1
+|ak|p
+�1/p
+≤ E
+�����
+n
+�
+k=1
+akXk
+����� ≤ C1
+� n
+�
+k=1
+|ak|p
+�1/p
+for some p ≥ 1 and positive constants C1, C2, then the closed subspace of L1(0, 1)
+spanned by the Xn is isomorphic with the ℓp space (Hilbert space if p = 2). Relation
+(45) holds, in particular, if the Xn are i.i.d. symmetric p-stable random variables with
+p > 1, i.e. their characteristic function (Fourier transform) is given by exp(−c|t|p)
+with some c > 0. Thus applying the subsequence principle to the “limit theorem”
+(45) provides important information on the subspace structure of L1(0, 1). Using this
+method, Aldous [28] proved the famous conjecture that every infinite dimensional
+closed subspace of L1(0, 1) contains an isomorphic copy of ℓp for some 1 ≤ p ≤ 2. For
+a further application of this method, see an improvement of the classical theorem of
+Kadec and Pe�lczy´nski [147] on the subspace structure on Lp, p > 2, in Berkes and
+Tichy [67].
+9. New results: Exact criteria for the central limit theorem for
+subsequences
+By the classical resonance theorem of Landau [163], for a real sequence (xn)n≥1
+the series �∞
+n=1 anxn converges for all (an)n≥1 ∈ ℓp (1 ≤ p ≤ ∞) if and only if
+(xn)n≥1 ∈ ℓq, where 1/p + 1/q = 1. A deep extension of this result to the case of
+function series was given by Nikishin [190]. We call a sequence (fn)n≥1 of measurable
+functions on (0, 1) a convergence system in measure for ℓp if for any real sequence
+(an)n≥1 ∈ ℓp the series �∞
+n=1 anfn converges in measure. In the case p = 2 Nikishin
+proved the following result.
+Theorem G (Nikishin [190, 191]). A function system (fn)n≥1 over (0, 1) is a conver-
+gence system in measure for ℓ2 if and only if for any ε > 0 there exists a measurable
+set Aε ⊂ (0, 1) with measure exceeding 1 − ε and a constant Kε > 0 such that for all
+N ≥ 1 and all (a1, . . . , aN) ∈ RN we have
+(46)
+�
+Aε
+� N
+�
+n=1
+anfn
+�2
+dx ≤ Kε
+N
+�
+n=1
+a2
+n.
+The sufficiency of (46) is obvious, so the essential (and highly remarkable) state-
+ment is the converse: if a sequence (fn)n≥1 is a convergence system in measure for
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+37
+ℓ2, then, except for a subset of (0, 1) with arbitrary small measure, (fn)n≥1 behaves
+like an orthonormal sequence.
+In the previous section we discussed the subsequence principle stating that suffi-
+ciently thin subsequences of arbitrary sequences of random variables, subject to
+mild boundedness conditions, satisfy “all” limit theorems for i.i.d. random variables
+in a mixed (randomized) form. A typical special case of this principle is the following
+result:
+Theorem H (Gaposhkin [132]). Let (Xn)n≥1 be a sequence of random variables
+satisfying
+(47)
+sup
+n
+EX2
+n < +∞.
+Then there exists a subsequence (Xnk)k≥1 together with random variables X and
+Y ≥ 0 such that for any further subsequence (Xmk)k≥1 of (Xnk)k≥1 we have
+(48)
+1
+√
+N
+N
+�
+k=1
+(Xmk − X)
+D
+−→ N(0, Y ),
+where N(0, Y ) denotes the “variance mixture” normal distribution with characteris-
+tic function E exp(−Y t2/2).
+If (X2
+n)n≥1 is uniformly integrable then by well-known compactness results (see e.g.
+[99]) there exist a subsequence (Xmk)k≥1 and random variables X ∈ L2 and Y ∈ L1/2,
+Y ≥ 0, such that
+(49)
+Xmk → X weakly in L2,
+(Xmk − X)2 → Y 2 weakly in L1.6
+As Gaposhkin [128] showed, in this case the random variables X, Y in (48) can be
+chosen as in (49).
+In Theorem H, condition (47) is not necessary: simple examples show (see below)
+that there exist sequences (Xn)n≥1 of random variables without any finite moments,
+but having subsequences satisfying (48). The purpose of this section is to give nec-
+essary and sufficient conditions for the existence of subsequences (Xnk)k≥1 satisfying
+the randomized CLT (48), and it will turn out that our conditions have the same
+character as Nikishin’s conditions for the existence of a subsequence being a con-
+vergence system, i.e. “nice” behavior of the sequence on subsets of the probability
+space with measure as close to 1 as we wish.
+To formulate our results, call a sequence (Xn)n≥1 of random variables nontrivial if
+it has no subsequence converging with positive probability. It is easily seen that
+for non-degenerate sequences the random variable Y in Theorem H is almost surely
+6A sequence (ξn)n≥1 of random variables in Lp, p ≥ 1, is said to converge weakly to ξ ∈ Lp
+if E(ξnη) → E(ξη) for any η ∈ Lq, where 1/p + 1/q = 1. This type of convergence should not
+be confused with weak convergence of probability measures and distributions, called generally
+convergence in distribution, and denoted by
+D
+−→.
+
+38
+C. AISTLEITNER, I. BERKES AND R. TICHY
+positive and Gaposhkin’s theorem can be rewritten in a form involving a pure (i.e.
+not mixed) Gaussian limit distribution.
+Theorem J. Let (Xn)n≥1 be a nontrivial sequence of random variables satisfying
+(47). Then there exists a subsequence (Xnk)k≥1 and random variables X, Y with
+Y > 0 such that for all subsequences (Xmk)k≥1 of (Xnk)k≥1 and for any set A in the
+probability space with P(A) > 0 we have
+(50)
+PA
+��N
+k=1(Xmk − X)
+Y
+√
+N
+< t
+�
+→ Φ(t)
+for all t.
+Here PA denotes the conditional probability with respect to A, and Φ is the cumulative
+distribution function of the standard normal distribution.
+The nontriviality of (Xn)n≥1 is assumed here to avoid degenerate cases. If Xnk → X
+on some set A with positive probability then for any sufficiently thin subsequence
+(Xmk)k≥1 of (Xnk)k≥1 we have � |Xmk − X| < +∞ a.s. on A, and consequently
+a−1
+N
+N
+�
+k=1
+(Xmk − X) → 0
+a.s. on A
+for any norming sequence aN → ∞ (random or not). Since for any sequence (Xn)n≥1
+satisfying (47) (and in fact any tight sequence (Xn)n≥1) there is a subsequence
+(Xnk)k≥1 and a measurable partition A ∪ B of the probability space such that Xnk
+converges on A and is nontrivial on B, there is no loss of generality in assuming that
+(Xn)n≥1 is nontrivial.
+Clearly, if (Xn)n≥1 satisfies the conclusion of Theorem J, then so does the sequence
+(Xn + 2−nZ)n≥1 for any a.s. finite random variable Z, and thus the assumption (47)
+is, as stated above, not necessary in Theorem J. Below we will give necessary and suf-
+ficient condition for the CLT for lacunary subsequences of a given sequence (Xn)n≥1
+of random variables without any moment assumption on (Xn)n≥1. To formulate our
+results, let us note that if all subsequences (Xmk)k≥1 of a sequence (Xn)n≥1 satisfy
+(50) for some random variables X, Y , then (Xn)n≥1 is bounded in probability (see
+Lemma 2 below). As mentioned in the previous section, every sequence (Xn)n≥1 of
+random variables bounded in probability has a subsequence (Xnk)k≥1 which has a
+limit distribution relative to every set A of the probability space with P(A) > 0.
+Such a sequence was called determining. This concept is the same as that of stable
+convergence, introduced by R´enyi [197]; our terminology follows that of functional
+analysis. Hence in our investigations we can assume without loss of generality that
+the original sequence (Xn)n≥1 is determining. Now if (Xn)n≥1 is determining and FA
+denotes its limit distribution relative to the set A, then, as we noted in the previous
+section, there exists a random measure µ (called the limit random measure of (Xn))
+such that
+(51)
+FA(t) = EA(µ(−∞, t])
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+39
+for any continuity point t of FΩ, where EA denotes conditional expectation relative
+to A. Let F• denote the distribution function of µ; we shall call it the limit random
+distribution of (Xn)n≥1. We can state now our first new theorem.
+Theorem 1. Let (Xn)n≥1 be a nontrivial sequence of random variables. Then the
+following statements are equivalent:
+A) There exist a subsequence (Xnk)k≥1 and random variables X, Y with Y > 0 such
+that (50) holds for all subsequences (Xmk)k≥1 of (Xnk)k≥1 and for any set A ⊂ Ω
+with P(A) > 0.
+B) For every ε > 0 there is a subsequence (Xnk)k≥1 and a set Aε ⊂ Ω with P(Aε) ≥
+1 − ε such that
+(52)
+sup
+k
+�
+Aε
+X2
+nkdP < +∞.
+If (Xn)n≥1 is determining, then two further equivalent statements are:
+C) We have
+(53)
++∞
+�
+−∞
+x2dF•(x) < +∞
+almost surely.
+D) For every ε > 0 there exists a set Aε ⊂ Ω with P(Aε) ≥ 1 − ε such that
+(54)
++∞
+�
+−∞
+x2dFAε(x) < +∞.
+Our second new theorem characterizes sequences (Xn)n≥1 for which (50) holds with
+X ∈ L2, Y ∈ L1/2.
+Theorem 2. Let (Xn)n≥1 be a nontrivial sequence of random variables defined on an
+atomless probability space (Ω, F, P). Then the following statements are equivalent:
+A) There exists a subsequence (Xnk)k≥1 and random variables X, Y with Y > 0,
+X ∈ L2, Y ∈ L1/2 such that (50) holds for all subsequences (Xmk)k≥1 of (Xnk)k≥1
+and all sets A ⊂ Ω with P(A) > 0.
+B) There exists a subsequence (Xnk)k≥1 and sequences (Yk)k≥1, (τk)k≥1 of random
+variables satisfying
+(55)
+Xnk = Yk + τk,
+where
+(56)
+sup
+k
+EY 2
+k < +∞,
+�
+k
+|τk| < +∞
+a.s.
+
+40
+C. AISTLEITNER, I. BERKES AND R. TICHY
+If (Xn)n≥1 has a limit distribution F, then a third equivalent statement is:
+C) We have
+(57)
++∞
+�
+−∞
+x2dF(x) < +∞.
+In other words, for the validity of (50) with X ∈ L2, Y ∈ L1/2, assumption (47) is
+necessary and sufficient after a small perturbation of (Xn)n≥1, and for identically
+distributed (Xn)n≥1 even this perturbation is not needed. A particularly simple case
+when X ∈ L2, Y ∈ L1/2 is satisfied is when X, Y are nonrandom.
+A trivial example showing the difference between condition (D) of Theorem 1 and
+condition (C) of Theorem 2 is the following. Let {Hk, k ≥ 1} be a partition of
+the probability space with P(Hk) = 2−k for k = 1, 2, . . . , and let (Xn)n≥1 be a
+sequence of random variables on this space which is conditionally i.i.d. given each
+Hk with mean 0 and variance 2k.
+Then (Xn)n≥1 is nontrivial, determining and
+clearly satisfies condition (D) of Theorem 1, but since it is identically distributed
+(in fact exchangeable) and since EX2
+1 = +∞, condition (C) of Theorem 2 is not
+satisfied.
+9.1. Some lemmas. The key for the proof of our theorems is a general structure
+theorem for lacunary sequences which was proved in [63], and which was stated as
+Theorem D in the previous section. Furthermore, we need the following lemmas.
+Lemma 1. Let (Xn)n≥1 be a sequence of random variables such that for some ran-
+dom variables X, Y with Y > 0 and for all subsequences (Xnk)k≥1 we have
+(58)
+N�
+k=1
+(Xnk − X)
+Y
+√
+N
+D
+−→ N(0, 1).
+Then (Xn)n≥1 is bounded in probability.
+Proof. Clearly (58) implies that the sequence (XnN − X)/(Y
+√
+N) is bounded in
+probability as N → ∞, and thus XnN/
+√
+N is bounded in probability for any sub-
+sequence (nk)k≥1. If (Xn)n≥1 were not bounded in probability then one could find
+a subsequence (mk)k≥1 and a constant c > 0 such that P(|Xmk| ≥ k) ≥ c for
+k = 1, 2, . . . , i.e. Xmk/
+√
+k would not be bounded in probability, a contradiction.
+□
+Lemma 2. Let (Xn)n≥1 be a sequence of random variables and assume that for some
+random variables X and Y > 0 and all sets A ⊂ Ω with P(A) > 0 we have
+(59)
+PA
+
+
+
+
+
+N�
+k=1
+(Xk − X)
+Y
+√
+N
+< t
+
+
+
+
+ → Φ(t)
+for all t.
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+41
+Assume further that (59) remains valid if we replace X, Y by some random variables
+X∗ and Y ∗ > 0. Then X = X∗ a.s. and Y = Y ∗ a.s.
+Proof. From the assumption it follows that the sequences
+N−1/2
+N
+�
+k=1
+(Xk − X)
+and
+N−1/2
+N
+�
+k=1
+(Xk − X∗)
+are bounded in probability, and thus the same holds for their difference
+√
+N(X−X∗),
+whence X = X∗ a.s. To prove Y = Y ∗, fix c > 1 and set A = {Y ∗ ≥ cY }. If
+P(A) > 0 then clearly we cannot have both (59) and the analogous relation with
+Y replaced by Y ∗. Thus P(A) > 0 for all c > 1 whence Y ∗ ≤ Y a.s. The same
+argument yields Y ≤ Y ∗ a.s., completing the proof.
+□
+Lemma 3. Let X1, X2, . . . , Xn be i.i.d. random variables with distribution function
+F and set Sn = X1 + · · · + Xn. Then for any t > 0 we have
+(60)
+P(|Sn| ≤ 2t) ≤ A t
+√n
+
+
+
+�
+|x|≤t
+x2dF(x) − 2
+
+
+
+�
+|x|≤t
+xdF(x)
+
+
+
+2
+
+
+−1/2
+,
+provided the difference on the right-hand side is positive and
+�
+|x|≤t
+dF(x) ≥ 1/2. Here
+A is an absolute constant.
+Proof. Let F ∗ denote the distribution function obtained from F by symmetrization.
+From a well-known concentration function inequality of Esseen [106, Theorem 2] it
+follows that the left-hand side of (60) cannot exceed
+A1
+t
+√n
+
+
+
+�
+|x|≤2t
+x2dF ∗(x)
+
+
+
+−1/2
+,
+where A1 is an absolute constant.
+Hence to prove (60) it suffices to show that
+�
+|x|≤t
+dF(x) ≥ 1/2 implies
+(61)
+�
+|x|≤2t
+x2dF ∗(x) ≥
+�
+|x|≤t
+x2dF(x) − 2
+
+
+
+�
+|x|≤t
+xdF(x)
+
+
+
+2
+.
+Let ξ and η be independent random variables with distribution function F, and set
+C = {|ξ − η| ≤ 2t},
+D = {|ξ| ≤ t, |η| ≤ t}.
+Then
+�
+|x|≤2t
+x2dF ∗(x) =
+�
+C
+(ξ − η)2dP
+
+42
+C. AISTLEITNER, I. BERKES AND R. TICHY
+≥
+�
+D
+(ξ − η)2dP
+= 2
+�
+|ξ|≤t
+ξ2dP · P(|η| ≤ t) − 2
+
+
+
+�
+|ξ|≤t
+ξdP
+
+
+
+2
+≥
+�
+|ξ|≤t
+ξ2dP − 2
+
+
+
+�
+|ξ|≤t
+ξdP
+
+
+
+2
+,
+provided P(|η| ≤ t) ≥ 1/2. Thus (61) is valid.
+□
+Lemma 4. Let (Ω, F, P) be an atomless probability space and X1, X2, . . . a se-
+quence of random variables on (Ω, F, P) with limit distribution F. Then there exist
+a subsequence (Xnk)k≥1 and sequences (Yk)k≥1 and (τk)k≥1 of random variables on
+(Ω, F, P) such that Xnk = Yk + τk, k = 1, 2, . . . , such that the random variables Yk
+have distribution function F, and such that �
+k |τk| < +∞ a.s.
+Proof. Let ( ˆXn)n≥1 be discrete random variables such that P(|Xn − X′
+n| ≥ 2−n) ≤
+2−n, n = 1, 2, . . . , and denote by Fn the distribution function of Xn. Clearly Fn → F
+and thus εn := ̺(Fn, F) → 0, where ̺ denotes the Prohorov distance. By a theorem
+of Strassen [8] there exists a probability measure µn on R2 with marginals Fn and
+F such that
+µn((x, y) : |x − y| ≥ εn) ≤ εn.
+Let c be a possible value of ˆXn. Since the probability space restricted to A = { ˆXn =
+c} is atomless, there exists a random variable Vn on this space such that
+PA(Vn < t) = µn((x, y) : x = c, y < t)
+µn((x, y) : x = c)
+for all t. Carrying out this construction for all possible values of c in the range of
+ˆXn, we get a random variable Vn defined on the whole probability space such that
+the joint distribution of ˆXn and Vn is µn. Clearly the distribution of Vn is F and
+P(| ˆXn − Vn| ≥ εn) ≤ εn. Choosing (nk)k≥1 so that εnk ≤ 2−k we get
+P
+�
+| ˆXnk − Vnk| ≥ 2−k�
+≤ 2−k,
+i.e.
+P
+�
+|Xnk − Vnk| ≥ 2 · 2−k�
+≤ 2 · 2−k
+and thus �
+k |Xnk − Vnk| < +∞ a.s. by the Borel–Cantelli lemma. Thus the de-
+composition Xnk = Yk + τk, where Yk = Vnk and τk = Xnk − Vnk, satisfies the
+requirements.
+□
+Our final two lemmas concern the properties of the limit random distribution of
+determining sequences.
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+43
+Lemma 5. Let (Xn)n≥1 be a determining sequence of random variables with limit
+random distribution F. Then for any set A ⊂ Ω with P(A) > 0 we have
+(62)
+EA
+
+
++∞
+�
+−∞
+x2dF•(x)
+
+ =
++∞
+�
+−∞
+x2dFA(x),
+in the sense that if one side is finite then the other side is also finite and the two
+sides are equal. The statement remains valid if in (62) we replace the interval of
+integrations by (−t, t), provided t and −t are continuity points of FA.
+Proof. This lemma follows easily from (51) by integration by parts.
+□
+Lemma 6. Let X, X1, X2, . . . be random variables such that both sequences (Xn)n≥1
+and (Xn − X)n≥1 are determining; let F• and G• denote, respectively, their limit
+random distributions. Then
++∞
+�
+−∞
+x2dG•(x) < +∞ a.s. implies
++∞
+�
+−∞
+x2dF•(x) < +∞
+a.s. and conversely.
+Proof. Let ε > 0 and choose a set A ⊂ Ω such that P(A) ≥ 1 − ε and on A
+both X and
++∞
+�
+−∞
+x2dF•(x) are bounded. Let FA and GA denote the limit random
+distribution of (Xn)n≥1 resp. (Xn − X)n≥1 relative to A. Replacing Xn by Xn + τn
+where τn → 0 a.s. clearly does not change the limit distributions FA, GA, F•, G•, and
+thus by passing to a subsequence and using Lemma 4 we can assume, without loss
+of generality, that the Xn are identically distributed on A. Then
+EAX2
+1 = EAX2
+2 = · · · =
++∞
+�
+−∞
+x2dFA(x),
+where the last integral is finite by the boundedness of
++∞
+�
+−∞
+x2dF•(x) on A and
+Lemma 5. By Minkowski’s inequality and the boundedness of X on A it follows
+that EA((Xn − X)2) is also bounded, and thus Fatou’s lemma implies that
++∞
+�
+−∞
+x2dGA(x) ≤ lim inf
+n→∞ EA
+�
+(Xn − X)2�
+< +∞.
+Using Lemma 5 again it follows that
++∞
+�
+−∞
+x2dG•(x) < +∞ a.s. on A. As the measure
+of A can be chosen arbitrarily close to 1, we get
+�
+x2dG•(x) < +∞ a.s., as required.
+□
+9.2. Proof of the theorems. We begin with the proof of Theorem 1. Using diag-
+onalization and Chebyshev’s inequality it follows that if a sequence (Xn)n≥1 satisfies
+(B), then it has a subsequence bounded in probability and thus also a determining
+subsequence. By Lemma 1 the same conclusion holds if (Xn)n≥1 satisfies (A). Thus
+
+44
+C. AISTLEITNER, I. BERKES AND R. TICHY
+to prove our theorem it suffices to prove the equivalence of (A), (B), (C), (D) for
+determining sequences (Xn)n≥1. In what follows we shall prove the implications
+(A) =⇒ (C) =⇒ (D) =⇒ (B); since (B) =⇒ (A) follows easily from Theorem D in
+the previous section by diagonalization, this will prove Theorem 1.
+Assume that (Xn)n≥1 is a determining sequence satisfying (A), i.e. there exists a
+subsequence (Xnk)k≥1 and random variables X, Y with Y > 0 such that for any
+further subsequence (Xmk)k≥1 of (Xnk)k≥1 and any set A ⊂ Ω with P(A) > 0 we
+have
+(63)
+PA
+
+
+
+
+
+N�
+k=1
+(Xmk − X)
+Y
+√
+N
+< t
+
+
+
+
+ → Φ(t)
+for all t.
+We claim that (Xn)n≥1 satisfies (C). Clearly we can assume without loss of generality
+that (Xnk)k≥1 = (Xk)k≥1 and since (Xn−X)k≥1 contains a determining subsequence,
+we can assume also that (Xn − X)n≥1 itself is determining. Moreover, since (Xn −
+X)n≥1 satisfies (C) if and only if (Xn)n≥1 does (see Lemma 6), we can assume that
+X = 0. Assume indirectly that (Xn)n≥1 does not satisfy (C), i.e. there exists a set
+B ⊂ Ω with P(B) > 0 such that
+(64)
+lim
+t→∞
+t
+�
+−t
+x2dF•(x) = +∞
+on B.
+Then there exists a set B∗ ⊂ B with P(B∗) ≥ P(B)/2 such that on B∗ the random
+variable Y is bounded and (63) holds uniformly, i.e. there exists a constant K > 0
+and a numerical sequence Kt → +∞ such that
+t
+�
+−t
+x2dF•(x) ≥ Kt and Y ≤ K on B∗.
+Also, 1 − F•(t) + F•(−t) → 0 a.s. as t → ∞, and thus we can choose a set B∗∗ ⊂
+B∗ with P(B∗∗) ≥ P(B∗)/2 such that on B∗∗ the last convergence relation holds
+uniformly, i.e. there exists a positive numerical sequence εt ց 0 such that
+(65)
+1 − F•(t) + F•(−t) ≤ εt on B∗∗.
+We show that there exists a subsequence (Xmk)k≥1 of (Xnk)k≥1 such that (63) fails
+for A = B∗∗. Since our argument will involve the sequence (Xn)n≥1 only on the set
+B∗∗, in the sequel we can assume, without loss of generality, that B∗∗ = Ω. That is,
+we may assume that (65) holds on the whole probability space.
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+45
+Let C be an arbitrary set in the probability space with P(C) > 0. Integrating (65)
+and using (51) and Lemma 5 we get
+(66)
+t
+�
+−t
+x2dFC(x) ≥ Kt,
+1 − FC(t) + FC(−t) ≤ εt,
+t ∈ H,
+where H denotes the set of continuity points of FΩ. Choose t0 ∈ H so large that
+εt0 ≤ 1/16 and then choose t1 so large that
+K1/2
+t
+/4 ≥ 2t2
+0 for t ≥ t1, t ∈ H.
+Then for t ≥ t1, t ∈ H we have, using the second relation of (66),
+������
+t
+�
+−t
+xdFC(x)
+������
+≤
+2t2
+0 +
+�
+t0≤|x|≤t
+|x|dFC(x)
+≤
+2t2
+0 +
+
+
+
+�
+|x|≥t0
+dFC(x)
+
+
+
+1/2 
+
+
+�
+|x|≤t
+x2dFC(x)
+
+
+
+1/2
+≤
+2t2
+0 + 1
+4
+
+
+
+�
+|x|≤t
+x2dFC(x)
+
+
+
+1/2
+≤
+1
+2
+
+
+
+�
+|x|≤t
+x2dFC(x)
+
+
+
+1/2
+,
+and thus we proved that for any C ⊂ Ω with P(c) > 0 we have
+(67)
+t
+�
+−t
+x2dFC(x) − 2
+
+
+t
+�
+−t
+xdFC(x)
+
+
+2
+≥ 1
+2Kt,
+t ≥ t1, t ∈ H.
+Since (Xn)n≥1 is bounded in probability, there exists a function ψ(x) ր ∞ such
+that
+(68)
+sup
+n Eψ(Xn) ≤ 1
+(see [9]). Let (ak)k≥1 be a sequence of integers tending to +∞ so slowly that ak ≤
+log k and
+(69)
+δk := ak/ψ(k1/4) → 0.
+Let further (εn)n≥1 tend to 0 so rapidly that εak ≤ 2−k. By Theorem D there exists
+a subsequence (Xmk)k≥1 and a sequence (Yk)k≥1 of discrete random variables such
+
+46
+C. AISTLEITNER, I. BERKES AND R. TICHY
+that (41) holds and for each k > 1 the atoms of the finite σ-field σ{Y1, . . . , Yak} can
+be divided into two classes Γ1 and Γ2 such that
+(70)
+�
+B∈Γ1
+P(B) ≤ εak ≤ 2−k
+and for each B ∈ Γ2 there exist i.i.d. random variables Z(B)
+ak+1, . . . , Z(B)
+k
+on B with
+distribution FB such that
+(71)
+PB
+�
+|Yj − Z(B)
+j
+| ≥ 2−k�
+≤ 2−k,
+j = ak + 1, . . . , k.
+We show that
+(72)
+P
+
+
+
+
+
+N�
+k=1
+Xmk
+Y
+√
+N
+< t
+
+
+
+
+ → Φ(t)
+for all t
+cannot hold; this will complete our indirect proof of (A) =⇒ (C). Set
+S(B)
+ak,k =
+k
+�
+j=ak+1
+Z(B)
+j
+,
+B ∈ Γ2,
+Sak,k =
+�
+B∈Γ2
+S(B)
+ak,k
+1B,
+where
+1B denotes the indicator function of B. By (71),
+PB
+������
+k
+�
+j=ak+1
+Yj −
+k
+�
+j=ak+1
+Z(B)
+j
+����� ≥ 1
+�
+≤ 2−ak,
+B ∈ Γ2,
+and thus using (70) we get
+(73)
+P
+������
+k
+�
+j=ak+1
+Yj − Sak,k
+����� ≥ 1
+�
+≤ 2 · 2−k.
+By (68), (69) and the Markov inequality we have
+P
+������
+ak
+�
+j=1
+Xmj
+����� ≥ akk1/4
+�
+≤
+ak sup
+1≤j≤ak
+P
+�
+|xmj| ≥ k1/4�
+≤
+akψ(k1/4)−1
+=
+δk,
+which, together with (73) and (41), yields
+(74)
+P
+������
+k
+�
+j=1
+Xmj − Sak,k
+����� ≥ 2akk1/4
+�
+≤ 3 · 2−ak + δk.
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+47
+Applying Lemma 3 to the i.i.d. sequence {Z(B)
+j
+, ak + 1 ≤ j ≤ k} and using (67) we
+get
+PB
+������
+S(B)
+ak,k
+√
+k
+����� ≤ 1
+�
+≤ PB
+�
+S(B)
+ak,k
+√k − ak
+≤ 2
+�
+≤ const. K−1/2
+√
+k ,
+where the constant is absolute. Thus using (70) and Y ≤ K it follows that
+(75)
+P
+�����
+Sak,k
+Y
+√
+k
+���� ≤ 1
+K
+�
+≤ const. K−1/2
+√
+k
++ 2−k.
+If (72) were true then by (74) and ak ≤ log k we would also have
+Sk,ak/Y
+√
+k
+D
+−→ N(0, 1),
+which clearly contradicts (75) for large k, since the right-hand side tends to zero.
+This completes the proof of (A) =⇒ (C).
+The remaining implications (C) =⇒ (D) and (D) =⇒ (B) of Theorem 1 are easy.
+Assume first that (C) holds, then for any ε > 0 there exists a set A ⊂ Ω with
+P(A) ≥ 1 − ε and a constant K = Kε such that
++∞
+�
+−∞
+x2dF•(x) ≤ K on A.
+Integrating the last relation on A and using Lemma 5 we get (52), i.e. (D) holds.
+Assume now that (D) holds, i.e. for any ε > 0 there exists a set A ⊂ Ω with
+P(A) ≥ 1 − ε such that (52) is valid. Applying Lemma 4 for (Xn)n≥1 on the set
+A it follows that there exists a subsequence (Xnk)k≥1 and random variables Yk and
+τk, k = 1, 2, . . . , defined on A such that Xnk = Yk + τk, k = 1, 2, . . . , and such that
+the random variables Yk have distribution FA on A and τk → 0 a.s. on A. Choose
+a set B ⊂ A with P(B) ≥ 1 − 2ε such that τk → 0 uniformly on B. Then clearly
+(τn)n≥1 is uniformly bounded on B, and further
+�
+B
+Y 2
+k dP
+≤
+�
+A
+Y 2
+k dP
+=
+P(A)
++∞
+�
+−∞
+x2dFA(x)
+≤
++∞
+�
+−∞
+x2dFA(x) < +∞
+for each k ≥ 1 by the identical distribution of the Yk’s and (52). Thus on B the
+sequences (Yk)k≥1 and (τk)k≥1 have bounded L2 norms and thus the same holds for
+
+48
+C. AISTLEITNER, I. BERKES AND R. TICHY
+Xmk = Yk + τk, i.e.
+sup
+k
+�
+B
+X2
+mkdP < +∞.
+In view of P(B) ≥ 1 − 2ε this shows that (Xn)n≥1 satisfies statement (B). This
+completes the proof of Theorem 1.
+Proof of Theorem 2. Theorem 2 follows from Theorem 1 and a slightly sharper form
+of Theorems H and J which was proved in [147]. We already mentioned the fact that
+in Theorem H the random variables X, Y appearing in (50) actually satisfy X ∈ L2,
+Y ∈ L1/2. Moreover, if instead of (47) we make the slightly stronger assumption
+that the sequence (X2
+n)n≥1 is uniformly integrable then by the weak compactness
+criteria in L1 and L2 it follows that there exists a subsequence (Xnk) and random
+variables X ∈ L2, Y ∈ L1/2 such that
+(76)
+Xnk → X weakly in L2,
+(Xnk − X)2 → Y 2 weakly in L1.
+As is shown in [163], in this case (50) holds with the random variables X, Y de-
+termined by (76).
+We turn now to the proof of Theorem 2.
+As in the case of
+Theorem 1, it suffices to prove the equivalence of statements (A), (B), (C) in the
+case when (Xn)n≥1 is determining.
+Also, since replacing Xn by Xn + τn where
+� |τn| < +∞ a.s. does not affect the validity of (50), the conclusion (B) =⇒ (A)
+of Theorem (2) is contained in the stronger form of Theorem H mentioned above.
+Thus it suffices to verify the implications (A) =⇒ (C) and (C) =⇒ (B). To prove
+(A) =⇒ (C) let us assume that (Xn)n≥1 is determining with limit distribution F, and
+that there exist a subsequence (Xnk)k≥1 and random variables X ∈ L2, Y ∈ L1/2,
+Y > 0, such that for all subsequences (Xmk)k≥1 of (Xnk)k≥1 and any set A ⊂ Ω with
+P(A) > 0 equation (50) holds. We show
++∞
+�
+−∞
+x2dF(x) < +∞. Clearly we can assume
+without loss of generality that (Xnk)k≥1 < (Xk)k≥1. Fix ε > 0. By the implication
+(A) =⇒ (B) =⇒ (D) of Theorem 1 there is a set A ⊂ Ω with P(A) ≥ 1 − ε and a
+subsequence (Xnk)k≥1 such that
+(77)
+sup
+k
+�
+A
+X2
+nkdP < +∞ and
+�
+A
+x2dFA(x) < +∞,
+where FA is the limit distribution of (Xn)n≥1 on A. Applying Lemma 4 for (Xnk)k≥1
+on A it follows that there exists a subsequence (Xmk)k≥1 of (Xnk)k≥1 admitting the
+decomposition
+(78)
+Xmk = Yk + τk on A,
+where the Yk are identically distributed on A with distribution function FA and
+� |τk| < +∞ a.s. on A. Being an identically distributed sequence with finite expec-
+tation, the sequence (Y 2
+k )k≥1 is uniformly integrable on A, and thus by the sharper
+form of Theorem H mentioned above it follows that there exists a subsequence
+(Ypk)k≥1 of (Yk)k≥1 such that
+Ypk → U weakly in L2(A),
+(Ypk − U)2 → V 2 weakly in L1(A),
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+49
+and for any B ⊂ A with P(B) > 0 we have
+PB
+��N
+k=1(Ypk − U)
+V
+√
+N
+< t
+�
+→ Φ(t) for all t,
+where U, V are random variables such that U ∈ L2(A), V ∈ L1/2(A), V > 0. Thus
+by (78) and � |τk| < +∞ a.s. on A we get
+PB
+��N
+k=1(Xmpk − U)
+V
+√
+N
+< t
+�
+→ Φ(t) for all t
+for any B ⊂ A with P(B) > 0. Comparing with (50) and using Lemma 2 we get
+U = X, V = Y a.s. on A, and thus we proved that
+Ypk → X weakly in L2(A),
+(Ypk − X)2 → Y 2 weakly in L1(A).
+Hence
+EAY 2 = lim
+k→∞ EA(Ypk − X)2 = lim
+k→∞(EAY 2
+pk − 2EAYpkX + EAX2)
+= lim
+k→∞ EAY 2
+pk − EAX2 =
++∞
+�
+−∞
+x2dFA(x) − EAX2,
+(79)
+where in the last step we used the fact that the Yk’s have distribution FA on A.
+Hence
+(80)
+P(A)−1
+�
+A
+Y 2dP =
++∞
+�
+−∞
+x2dFA(x) − P(A)−1
+�
+A
+X2dP.
+Since X ∈ L2(Ω), Y 2 ∈ L1(Ω), the left-hand side of (80) and the second term on
+the right-hand side approach finite limits as P(A) → 1 and thus
+� +∞
+−∞ x2dFA(x) also
+converges to a finite limit. On the other hand, FA → F as P(A) → 1 and thus
+Fatou’s lemma implies
++∞
+�
+−∞
+x2dF(x) ≤ lim inf
+P (A)→1
++∞
+�
+−∞
+x2dFA(x) < +∞,
+proving the implication (A) =⇒ (C). Now if (C) holds then by Lemma 4 there exists
+a subsequence (Xnk)k≥1 permitting the decomposition (55) where � |τk| < +∞ a.s.
+and Yk are identically distributed with distribution F; since F has finite variance
+by (C), the first relation of (56) holds. Thus (Xn)n≥1 satisfies (B) and the proof of
+Theorem 2 is completed.
+Acknowledgments
+Christoph Aistleitner is supported by the Austrian Science Fund (FWF), projects F-
+5512, I-3466, I-4945, I-5554, P-34763, P-35322 and Y-901. Istvan Berkes is supported
+by Hungarian Foundation NKFI-EPR No. K-125569.
+
+50
+C. AISTLEITNER, I. BERKES AND R. TICHY
+References
+[1] R. Adler, M. Keane, and M. Smorodinsky. A construction of a normal number for the con-
+tinued fraction transformation. J. Number Theory, 13(1):95–105, 1981.
+[2] D. Airey and B. Mance. Normality of different orders for Cantor series expansions. Nonlin-
+earity, 30(10):3719–3742, 2017.
+[3] D. Airey, B. Mance, and J. Vandehey. Normal number constructions for Cantor series with
+slowly growing bases. Czechoslovak Math. J., 66(141)(2):465–480, 2016.
+[4] C. Aistleitner. Convergence of � ckf(kx) and the Lip α class. Proc. Amer. Math. Soc.,
+140(11):3893–3903, 2010.
+[5] C. Aistleitner. Diophantine equations and the LIL for the discrepancy of sublacunary se-
+quences. Illinois J. Math., 53(3):785–815, 2009.
+[6] C. Aistleitner. Irregular discrepancy behavior of lacunary series II. Monatsh. Math.,
+161(3):255–270, 2010.
+[7] C. Aistleitner. On the law of the iterated logarithm for the discrepancy of lacunary sequences.
+Trans. Amer. Math. Soc., 362(11):5967–5982, 2010.
+[8] C. Aistleitner. On the law of the iterated logarithm for the discrepancy of lacunary sequences
+II. Trans. Amer. Math. Soc., 365(7):3713–3728, 2013.
+[9] C. Aistleitner. Quantitative uniform distribution results for geometric progressions. Israel J.
+Math., 204(1):155–197, 2014.
+[10] C. Aistleitner. On some questions of V. I. Arnold on the stochasticity of geometric and
+arithmetic progressions. Nonlinearity, 28(10):3663–3675, 2015.
+[11] C. Aistleitner, S. Baker, N. Technau, and N. Yesha. Gap statistics and higher correla-
+tions for geometric progressions modulo one. Math Ann., to appear. Preprint available at
+arXiv:2010.10355.
+[12] C. Aistleitner, V. Becher, A.-M. Scheerer, and T. A. Slaman. On the construction of abso-
+lutely normal numbers. Acta Arith., 180(4):333–346, 2017.
+[13] C. Aistleitner and I. Berkes. On the central limit theorem for f(nkx). Probab. Theory Related
+Fields, 146(1-2):267–289, 2010.
+[14] C. Aistleitner and I. Berkes. Limit distributions in metric discrepancy theory. Monatsh.
+Math., 169(3-4):253–265, 2013.
+[15] C. Aistleitner, I. Berkes, and K. Seip. GCD sums from Poisson integrals and systems of
+dilated functions. J. Eur. Math. Soc. (JEMS), 17(6):1517–1546, 2015.
+[16] C. Aistleitner, I. Berkes, K. Seip and M. Weber. Convergence of series of dilated functions
+and spectral norms of GCD matrices. Acta Arithmetica 168:221–246, 2015.
+[17] C. Aistleitner, I. Berkes, and R. Tichy. On the asymptotic behavior of weakly lacunary series.
+Proc. Amer. Math. Soc. 139(7):2505–2517, 2011.
+[18] C. Aistleitner, I. Berkes, and R. F. Tichy. Lacunary sequences and permutations. In Depen-
+dence in probability, analysis and number theory, pages 35–49. Kendrick Press, Heber City,
+UT, 2010.
+[19] C. Aistleitner, I. Berkes, and R. F. Tichy. On permutations of Hardy-Littlewood-P´olya se-
+quences. Trans. Amer. Math. Soc. 363(12):6219–6244, 2011.
+[20] C. Aistleitner, I. Berkes, and R. F. Tichy. On permutations of lacunary series. RIMS
+Kˆokyˆuroku Bessatsu, B34 (2012), 1–25.
+[21] C. Aistleitner and C. Elsholtz. The central limit theorem for subsequences in probabilistic
+number theory. Canad. J. Math. 64(6):1201–1221, 2012.
+[22] C. Aistleitner, N. Gantert, Z. Kabluchko, J. Prochno, and K. Ramanan. Large deviation
+principles for lacunary sums. Trans. Amer. Math. Soc. 376(1):507–553, 2023.
+[23] C. Aistleitner, K. Fukuyama. On the law of the iterated logarithm for trigonometric series
+with bounded gaps. Probab. Theory Related Fields 154(3-4):607–620, 2012.
+[24] C. Aistleitner, K. Fukuyama. On the law of the iterated logarithm for trigonometric series
+with bounded gaps II. J. Th´eor. Nombres Bordeaux 28(2):391–416, 2016.
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+51
+[25] C. Aistleitner, K. Fukuyama, and Y. Furuya. Optimal bound for the discrepancies of lacunary
+sequences. Acta Arith. 158(3):229–243, 2013.
+[26] M. Ajtai, J. Koml´os, E. Szemer´edi. A dense infinite Sidon sequence. European J. Combin.
+2(1):1-11, 1981.
+[27] D. J. Aldous. Limit theorems for subsequences of arbitrarily-dependent sequences of random
+variables. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 40(1):59–82, 1977.
+[28] D. J. Aldous. Subspaces of L1, via random measures. Trans. Amer. Math. Soc. 267(2):445–
+463, 1981.
+[29] J. I. Almarza and S. Figueira. Normality in non-integer bases and polynomial time random-
+ness. J. Comput. System Sci. 81(7):1059–1087, 2015.
+[30] F. Amoroso and E. Viada. Small points on subvarieties of a torus. Duke Math. J. 150(3):407–
+442, 2009.
+[31] J. Angst and G. Poly. Variations on Salem–Zygmund results for random trigonometric poly-
+nomials. Application to almost sure nodal asymptotics. Electron. J. Probab. 26:1–36, 2021.
+[32] V. I. Arnol’d. Stochastic and deterministic statistics of orbits in chaotically looking dynamical
+systems. Tr. Mosk. Mat. Obs. 70:46–101, 2009.
+[33] D. H. Bailey and J. M. Borwein. Pi Day is upon us again and we still do not know if pi is
+normal. Amer. Math. Monthly 121(3):191–206, 2014.
+[34] D. H. Bailey, J. M. Borwein, R. E. Crandall, and C. Pomerance. On the binary expansions
+of algebraic numbers. J. Th´eor. Nombres Bordeaux 16(3):487–518, 2004.
+[35] D. H. Bailey and R. E. Crandall. On the random character of fundamental constant expan-
+sions. Experiment. Math. 10(2):175–190, 2001.
+[36] D. H. Bailey and R. E. Crandall. Random generators and normal numbers. Experiment.
+Math. 11(4):527–546, 2002.
+[37] R. C. Baker. Khinchin’s conjecture and Marstrand’s theorem. Mathematika 21:248–260, 1974.
+[38] R. C. Baker. Metric number theory and the large sieve. J. London Math. Soc. (2) 24(1):34–40,
+1981.
+[39] S. Baker and D. Kong. Numbers with simply normal β-expansions. Math. Proc. Cambridge
+Philos. Soc. 167(1):171–192, 2019.
+[40] P. Balister, B. Bollob´as, R. Morris, J. Sahasrabudhe, and M. Tiba. Flat Littlewood polyno-
+mials exist. Ann. of Math. (2) 192(3):977–1004, 2020.
+[41] A. Bazarova, I. Berkes, and M. Raseta. On trigonometric sums with random frequencies.
+Studia Sci. Math. Hungar. 55(1):141–152, 2018.
+[42] A. Bazarova, I. Berkes, and M. Raseta. Strong approximation of lacunary series with random
+gaps. Monatsh. Math. 186(3):393–403, 2018.
+[43] V. Becher and S. Figueira. An example of a computable absolutely normal number. Theoret.
+Comput. Sci. 270(1-2):947–958, 2002.
+[44] V. Becher, S. Figueira, and R. Picchi. Turing’s unpublished algorithm for normal numbers.
+Theoret. Comput. Sci. 377(1-3):126–138, 2007.
+[45] V. Becher, P. A. Heiber, and T. A. Slaman. A polynomial-time algorithm for computing
+absolutely normal numbers. Inform. and Comput. 232:1–9, 2013.
+[46] V. Becher, P. A. Heiber, and T. A. Slaman. Normal numbers and the Borel hierarchy. Fund.
+Math. 226(1):63–78, 2014.
+[47] V. Becher, P. A. Heiber, and T. A. Slaman. A computable absolutely normal Liouville num-
+ber. Math. Comp. 84(296):2939–2952, 2015.
+[48] V. Becher and T. A. Slaman. On the normality of numbers to different bases. J. Lond. Math.
+Soc. (2) 90(2):472–494, 2014.
+[49] V. Becher and S. A. Yuhjtman. On absolutely normal and continued fraction normal numbers.
+Int. Math. Res. Not. IMRN (19):6136–6161, 2019.
+[50] J. Beck. From Khinchin’s conjecture on strong uniformity to superuniform motions. Mathe-
+matika 61(3):591–707, 2015.
+
+52
+C. AISTLEITNER, I. BERKES AND R. TICHY
+[51] L. Berggren, J. Borwein, and P. Borwein. Pi: a source book. Springer-Verlag, New York,
+third edition, 2004.
+[52] I. Berkes. The law of the iterated logarithm for subsequences of random variables. Z.
+Wahrscheinlichkeitstheorie und Verw. Gebiete 30:209–215, 1974.
+[53] I. Berkes. On the asymptotic behaviour of � f(nkx). I. Main theorems. II. Applications. Z.
+Wahrscheinlichkeitstheorie und Verw. Gebiete 34(4):319–345 & 347–365, 1976.
+[54] I. Berkes. On the central limit theorem for lacunary trigonometric series. Analysis Math.
+4:159-180, 1978.
+[55] I. Berkes. A central limit theorem for trigonometric series with small gaps. Z. Wahrsch. Verw.
+Gebiete 47(2):157–161, 1979.
+[56] I. Berkes. An extension of the Koml´os subsequence theorem. Acta Math. Hungar. 55(1-2):103–
+110, 1990.
+[57] I. Berkes. A note on lacunary trigonometric series. Acta Math. Hungar. 57(1-2):181–186,
+1991.
+[58] I. Berkes. Non-Gaussian limit distributions of lacunary trigonometric series. Canad. J. Math.
+43(5):948–959, 1991.
+[59] I. Berkes. Critical LIL behavior of the trigonometric system. Trans. Amer. Math. Soc.
+338(2):553–585, 1993.
+[60] I. Berkes. On the convergence of � cnf(nx) and the Lip 1/2 class. Trans. Amer. Math. Soc.
+349(10):4143–4158, 1997.
+[61] I. Berkes. On the uniform theory of lacunary series. In Number theory—Diophantine problems,
+uniform distribution and applications, pages 137–167. Springer, Cham, 2017.
+[62] I. Berkes and B. Borda. On the law of the iterated logarithm for random exponential sums.
+Trans. Amer. Math. Soc. 371(5):3259–3280, 2019.
+[63] I. Berkes and E. P´eter. Exchangeable random variables and the subsequence principle. Probab.
+Theory Related Fields 73(3):395–413, 1986.
+[64] I. Berkes and W. Philipp. The size of trigonometric and Walsh series and uniform distribution
+mod 1. J. London Math. Soc. (2) 50(3):454–464, 1994.
+[65] I. Berkes, W. Philipp, and R. F. Tichy. Empirical processes in probabilistic number theory:
+the LIL for the discrepancy of (nkω) mod 1. Illinois J. Math. 50(1-4):107–145, 2006.
+[66] I. Berkes and H. P. Rosenthal. Almost exchangeable sequences of random variables. Z.
+Wahrsch. Verw. Gebiete 70(4):473–507, 1985.
+[67] I. Berkes and R. F. Tichy. The Kadec-Pe�lczy´nski theorem in Lp, 1 ≤ p < 2. Proc. Amer.
+Math. Soc. 144(5):2053–2066, 2016.
+[68] I. Berkes and M. Weber. On the convergence of � ckf(nkx). Memoirs of the AMS 201: no.
+943, 2009.
+[69] I. Berkes and M. Weber. On series of dilated functions. Quarterly J. Math. 65:25–52, 2014.
+[70] I. Berkes and M. Weber. On series � ckf(kx) and Khinchin’s conjecture. Israel J. Math.
+201:591–609, 2014.
+[71] S. G. Bobkov and F. G¨otze. Concentration inequalities and limit theorems for randomized
+sums. Probab. Theory Related Fields 137(1-2):49–81, 2007.
+[72] A. Bondarenko and K. Seip. Large greatest common divisor sums and extreme values of the
+Riemann zeta function. Duke Math. J. 166(9):1685–1701, 2017.
+[73] E. Borel. Les probabilit´es denombrables et leurs applications arithm´etiques. Rend. Circ. Mat.
+Palermo 27:247–271, 1909.
+[74] Z. Buczolich, B. Hanson, B. Maga, G. V´ertesy. Type 1 and 2 sets for series of translates of
+functions. Acta Math. Hungar. 158(2):271–293, 2019.
+[75] Y. Bugeaud. Distribution modulo one and Diophantine approximation, volume 193 of Cam-
+bridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2012.
+[76] Y. Bugeaud. On the expansions of a real number to several integer bases. Rev. Mat. Iberoam.
+28(4):931–946, 2012.
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+53
+[77] L. Carleson. On convergence and growth of partial sums of Fourier series. Acta Math. 116:135–
+157, 1966.
+[78] J. W. S. Cassels. Some metrical theorems of Diophantine approximation. III. Proc. Cambridge
+Philos. Soc. 46:219–225, 1950.
+[79] J. W. S. Cassels. On a problem of Steinhaus about normal numbers. Colloq. Math. 7:95–101,
+1959.
+[80] F. Cellarosi, J. Marklof. Quadratic Weyl sums, automorphic functions and invariance prin-
+ciples. Proc. Lond. Math. Soc. 113(6):775–828, 2016.
+[81] S. D. Chatterji. A general strong law. Invent. Math. 9:235–245, 1969/70.
+[82] S. D. Chatterji. Un principe de sous-suites dans la th´eorie des probabilit´es. In S´eminaire de
+Probabilit´es, VI (Univ. Strasbourg, ann´ee universitaire 1970-1971; Journ´ees Probabilistes de
+Strasbourg, 1971), pages 72–89. Lecture Notes in Math., Vol. 258. 1972.
+[83] S. D. Chatterji. A principle of subsequences in probability theory: the central limit theorem.
+Advances in Math. 13:31–54; correction, ibid. 14 (1974), 266–269, 1974.
+[84] S. D. Chatterji. A subsequence principle in probability theory. II. The law of the iterated
+logarithm. Invent. Math. 25:241–251, 1974.
+[85] G. Cohen and J.-P. Conze. CLT for random walks of commuting endomorphisms on compact
+abelian groups. J. Theoret. Probab. 30(1):143–195, 2017.
+[86] J.-P. Conze, S. Isola, and S. Le Borgne. Diffusive behavior of ergodic sums over rotations.
+Stoch. Dyn. 19(2):1950016, 26, 2019.
+[87] J.-P. Conze and S. Le Borgne. Limit law for some modified ergodic sums. Stoch. Dyn.
+11(1):107–133, 2011.
+[88] J.-P. Conze, S. Le Borgne, and M. Roger. Central limit theorem for stationary products of
+toral automorphisms. Discrete Contin. Dyn. Syst. 32(5):1597–1626, 2012.
+[89] A. H. Copeland and P. Erd¨os. Note on normal numbers. Bull. Amer. Math. Soc. 52:857–860,
+1946.
+[90] C. Cuny, J. Dedecker, and D. Voln´y. A functional CLT for fields of commuting transfor-
+mations via martingale approximation. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst.
+Steklov. (POMI) 441(Veroyatnost i Statistika. 22):239–262, 2015.
+[91] K. Dajani and C. Kraaikamp. Ergodic theory of numbers volume 29 of Carus Mathematical
+Monographs. Mathematical Association of America, Washington, DC, 2002.
+[92] H. Davenport and P. Erd¨os. Note on normal decimals. Canad. J. Math. 4:58–63, 1952.
+[93] J.-M. De Koninck and I. K´atai. On a problem on normal numbers raised by Igor Shparlinski.
+Bull. Aust. Math. Soc. 84(2):337–349, 2011.
+[94] R. de la Bret`eche and G. Tenenbaum. Sommes de G´al et applications. Proc. Lond. Math.
+Soc. (3), 119(1):104–134, 2019.
+[95] M. Drmota, C. Mauduit, and J. Rivat. Normality along squares. J. Eur. Math. Soc. (JEMS)
+21(2):507–548, 2019.
+[96] M. Drmota and R. F. Tichy. Sequences, discrepancies and applications, volume 1651 of Lec-
+ture Notes in Mathematics. Springer-Verlag, Berlin, 1997.
+[97] A. Dubickas. Powers of a rational number modulo 1 cannot lie in a small interval. Acta Arith.
+137(3):233–239, 2009.
+[98] A. Dubickas. On the size of a restricted sumset with application to the binary expansion of
+√
+d. Appl. Anal. Discrete Math. 13(2):346–360, 2019.
+[99] N. Dunford and J. T. Schwartz. Linear Operators. I. General Theory. Pure and Applied
+Mathematics, Vol. 7. Interscience Publishers, 1958.
+[100] P. Erd˝os. On the strong law of large numbers. Trans. Amer. Math. Soc. 67:51–56, 1949.
+[101] P. Erd˝os. On the convergence of trigonometric series. J. Math. Phys. Mass. Inst. Tech. 22:37–
+39, 1943.
+[102] P. Erd˝os. On trigonometric sums with gaps. Magyar Tud. Akad. Mat. Kutat´o Int. K¨ozl.
+7:37–42, 1962.
+
+54
+C. AISTLEITNER, I. BERKES AND R. TICHY
+[103] P. Erd¨os and I. S. G´al. On the law of the iterated logarithm. I, II. Nederl. Akad. Wetensch.
+Proc. Ser. A. 58 = Indag. Math. 17:65–76, 77–84, 1955.
+[104] P. Erd¨os and J. F. Koksma. On the uniform distribution modulo 1 of sequences (f(n, θ)).
+Nederl. Akad. Wetensch., Proc. 52:851–854 = Indagationes Math. 11, 299–302 (1949), 1949.
+[105] P. Erd˝os and A. R´enyi. Additive properties of random sequences of positive integers. Acta
+Arith. 6: 83–110, 1960.
+[106] C. G. Esseen. On the concentration function of a sum of independent random variables. Z.
+Wahrscheinlichkeitstheorie und Verw. Gebiete 9:290–308, 1968.
+[107] J.-H. Evertse, H. P. Schlickewei, and W. M. Schmidt. Linear equations in variables which lie
+in a multiplicative group. Ann. of Math. (2), 155(3):807–836, 2002.
+[108] H. Fiedler, W. Jurkat, O. K¨orner. Asymptotic expansions of finite theta series. Acta Arith.
+32(2):129–146, 1977.
+[109] F. Filip and J. ˇSustek. Normal numbers and Cantor expansions. Unif. Distrib. Theory
+9(2):93–101, 2014.
+[110] L. Flatto, J. C. Lagarias, and A. D. Pollington. On the range of fractional parts {ξ(p/q)n}.
+Acta Arith. 70(2):125–147, 1995.
+[111] L. Fr¨uhwirth, M. Juhos, and J. Prochno. The large deviation behavior of lacunary sums.
+Monatsh. Math. 199(1):113-133, 2022.
+[112] K. Fukuyama. The central limit theorem for Riesz-Raikov sums. Probab. Theory Related
+Fields 100(1):57–75, 1994.
+[113] K. Fukuyama. The law of the iterated logarithm for discrepancies of {θnx}. Acta Math.
+Hungar. 118(1-2):155–170, 2008.
+[114] K. Fukuyama. The law of the iterated logarithm for the discrepancies of a permutation of
+{nkx}. Acta Math. Hungar. 123(1-2):121–125, 2009.
+[115] K. Fukuyama. A central limit theorem and metric discrepancy result for sequences with
+bounded gaps. Dependence in Probability, Analysis and Number Theory, pp. 233–246.
+Kendrick Press 2010.
+[116] K. Fukuyama. A central limit theorem to trigonometric series with bounded gaps. Probab.
+Theory Related Fields 149(1-2):139–148, 2011.
+[117] K. Fukuyama. Pure Gaussian limit distributions of trigonometric series with bounded gaps.
+Acta Math. Hungar. 129(4):303–313, 2010.
+[118] K. Fukuyama. A metric discrepancy result for a lacunary sequence with small gaps. Monatsh.
+Math. 162(3):277–288, 2011.
+[119] K. Fukuyama. Metric discrepancy results for alternating geometric progressions. Monatsh.
+Math. 171(1):33–63, 2013.
+[120] K. Fukuyama. The central limit theorem for Riesz-Raikov sums II. Trans. Amer. Math. Soc.
+372(2):1193–1211, 2019.
+[121] K. Fukuyama and N. Hiroshima. Metric discrepancy results for subsequences of {θkx}.
+Monatsh. Math. 165(2):199–215, 2012.
+[122] K. Fukuyama and S. Miyamoto. Metric discrepancy results for Erd¨os-Fortet sequence. Studia
+Sci. Math. Hungar. 49(1):52–78, 2012.
+[123] K. Fukuyama and K. Nakata. A metric discrepancy result for the Hardy-Littlewood-P´olya
+sequences. Monatsh. Math. 160(1):41–49, 2010.
+[124] K. Fukuyama and B. Petit. Le th´eor`eme limite central pour les suites de R. C. Baker. Ergodic
+Theory Dynam. Systems 21(2):479–492, 2001.
+[125] K. Fukuyama and M. Yamashita. Metric discrepancy results for geometric progressions with
+large ratios. Monatsh. Math. 180(4):731–742, 2016.
+[126] H. Furstenberg. Disjointness in ergodic theory, minimal sets, and a problem in Diophantine
+approximation. Math. Systems Theory 1:1–49, 1967.
+[127] I. S. G´al. A theorem concerning Diophantine approximations. Nieuw Arch. Wiskunde (2),
+23:13–38, 1949.
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+55
+[128] V. F. Gaposhkin. Lacunary series and independent functions. Uspehi Mat. Nauk 21(6
+(132)):3–82, 1966.
+[129] V. F. Gaposhkin. On series with respect to the system {ϕ(nx)}. Mat. Sb. (N.S.) 69 (111):328–
+353, 1966.
+[130] V. F. Gaposhkin. Convergence and divergence systems. Mat. Zametki 4:253–260, 1968.
+[131] V. F. Gaposhkin. The central limit theorem for certain weakly dependent sequences. Teor.
+Verojatnost. i Primenen 15:666–684, 1970.
+[132] V. F. Gaposhkin. Convergence and limit theorems for subsequences of random variables.
+Teor. Verojatnost. i Primenen 17:401–423, 1972.
+[133] M. Goldstern, J. Schmeling, and R. Winkler. Further Baire results on the distribution of
+subsequences. Unif. Distrib. Theory 2(1):127–149, 2007.
+[134] H. Halberstam and K. F. Roth. Sequences, Vol. I. Clarendon Press, Oxford 1966.
+[135] G. H. Hardy ad J. Littlewood. Some problems in diophantine approximation. Acta Math.
+37:193–214, 1914.
+[136] G. Harman. Metric number theory, volume 18 of London Mathematical Society Monographs.
+New Series. The Clarendon Press, Oxford University Press, New York, 1998.
+[137] G. Harman. One hundred years of normal numbers. In Number theory for the millennium,
+II (Urbana, IL, 2000), pages 149–166. A K Peters, Natick, MA, 2002.
+[138] T. Hilberdink. An arithmetical mapping and applications to Ω-results for the Riemann zeta
+function. Acta Arith. 139(4):341–367, 2009.
+[139] E. Hlawka. The theory of uniform distribution. Academic Publishers, 1984.
+[140] R. A. Hunt. On the convergence of Fourier series. In Orthogonal Expansions and their Con-
+tinuous Analogues (Proc. Conf., Edwardsville, Ill., 1967), pages 235–255. Southern Illinois
+Univ. Press, Carbondale, Ill., 1968.
+[141] W. B. Jurkat, J. W. Van Horne. The proof of the central limit theorem for theta sums. Duke
+Math. J. 48(4):873–885, 1981.
+[142] W. B. Jurkat, J. W. Van Horne. On the central limit theorem for theta series. Michigan
+Math. J. 29(1):65–77, 1982.
+[143] W. B. Jurkat, J. W. Van Horne. The uniform central limit theorem for theta sums. Duke
+Math. J. 50(3):649–666, 1983.
+[144] M. Kac. Convergence of certain gap series. Ann. of Math. 44:411–415, 1943.
+[145] M. Kac. On the distribution of values of sums of the type � f(2kt). Ann. of Math. (2),
+47:33–49, 1946.
+[146] M. Kac. Probability methods in some problems of analysis and number theory. Bull. Amer.
+Math. Soc. 55:641–665, 1949.
+[147] M. I. Kadec and A. Pe�lczy´nski. Bases, lacunary sequences and complemented subspaces in
+the spaces Lp. Studia Math. 21:161–176, 1961/62.
+[148] R. Kaufman. On the approximation of lacunary series by Brownian motion. Acta. Math.
+Acad. Sci. Hung. 35(1-2): 61–66 , 1980.
+[149] M. Kesseb¨ohmer, S. Munday, and B. O. Stratmann. Infinite ergodic theory of numbers. De
+Gruyter, Berlin, 2016.
+[150] A. J. Khinchin. Ein Satz ¨uber Kettenbr¨uche, mit arithmetischen Anwendungen. Mathema-
+tische Zeitschrift 18:289-306, 1923.
+[151] A. J. Khinchin. ¨Uber einen Satz der Wahrscheinlichkeitsrechnung. Fund. Math. 6:9-20, 1924.
+[152] H. Ki and T. Linton. Normal numbers and subsets of N with given densities. Fund. Math.
+144(2):163–179, 1994.
+[153] D. E. Knuth. The art of computer programming. Vol. 2: Seminumerical algorithms. Addison-
+Wesley Publishing Co., 1969.
+[154] J. F. Koksma. Ein mengentheoretischer Satz ¨uber die Gleichverteilung modulo Eins. Com-
+positio Math. 2:250–258, 1935.
+[155] J. F. Koksma. On a certain integral in the theory of uniform distribution. Nederl. Akad.
+Wetensch., Proc. Ser. A. 54 = Indagationes Math., 13:285–287, 1951.
+
+56
+C. AISTLEITNER, I. BERKES AND R. TICHY
+[156] J. F. Koksma. A Diophantine property of summable functions. J. Indian. Math. Soc. 15:87-
+96, 1951.
+[157] J. F. Koksma. Estimations de fonctions `a l’aide d’int´egrales de Lebesgue, Bull. Soc. Math.
+Belg. 6:4–13, 1953.
+[158] A.N. Kolmogorov. ¨Uber das Gesetz des iterierten Logarithmus. Math. Ann. 101:126–135,
+1929.
+[159] J. Koml´os. A generalization of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar. 18:217–
+229, 1967.
+[160] J. Koml´os. Every sequence converging to 0 weakly in L2 contains an unconditional conver-
+gence sequence. Ark. Mat. 12:41–49, 1974.
+[161] L. Kuipers and H. Niederreiter. Uniform distribution of sequences. Wiley-Interscience, 1974.
+[162] J. C. Lagarias. On the normality of arithmetical constants. Experiment. Math. 10(3):355–368,
+2001.
+[163] E. Landau. ¨Uber einen Konvergenzsatz. Nachr. Ges. Wiss. G¨ottingen, Math.-Phys. Kl.
+1907:25–27, 1907.
+[164] C. Lemieux. Monte Carlo and quasi-Monte Carlo sampling. Springer Series in Statistics.
+Springer, New York, 2009.
+[165] G. Leobacher and F. Pillichshammer. Introduction to quasi-Monte Carlo integration and
+applications. Compact Textbooks in Mathematics. Birkh¨auser/Springer, Cham, 2014.
+[166] M. B. Levin. On the discrepancy estimate of normal numbers. Acta Arith. 88(2):99–111,
+1999.
+[167] M. B. Levin. Central limit theorem for Zd
++-actions by toral endomorphisms. Electron. J.
+Probab. 18:no. 35, 42, 2013.
+[168] M. B. Levin. A multiparameter variant of the Salem-Zygmund central limit theorem on
+lacunary trigonometric series. Colloq. Math. 131(1):13–27, 2013.
+[169] M. Lewko and M. Radziwi�l�l. Refinements of G´al’s theorem and applications. Adv. Math.
+305:280–297, 2017.
+[170] B. Li and B. Mance. Number theoretic applications of a class of Cantor series fractal functions,
+II. Int. J. Number Theory 11(2):407–435, 2015.
+[171] M. G. Madritsch. Construction of normal numbers via pseudo-polynomial prime sequences.
+Acta Arith. 166(1):81–100, 2014.
+[172] M. G. Madritsch and B. Mance. Construction of µ-normal sequences. Monatsh. Math.
+179(2):259–280, 2016.
+[173] M. G. Madritsch, A.-M. Scheerer, and R. F. Tichy. Computable absolutely Pisot normal
+numbers. Acta Arith. 184(1):7–29, 2018.
+[174] M. G. Madritsch, J. M. Thuswaldner, and R. F. Tichy. Normality of numbers generated by
+the values of entire functions. J. Number Theory 128(5):1127–1145, 2008.
+[175] B. Mance. Typicality of normal numbers with respect to the Cantor series expansion. New
+York J. Math. 17:601–617, 2011.
+[176] B. Mance. Number theoretic applications of a class of Cantor series fractal functions. I. Acta
+Math. Hungar. 144(2):449–493, 2014.
+[177] J. Marklof. The Berry-Tabor conjecture. In European Congress of Mathematics, Vol. II
+(Barcelona, 2000), volume 202 of Progr. Math., pages 421–427. Birkh¨auser, Basel, 2001.
+[178] J. M. Marstrand. On Khinchin’s conjecture about strong uniform distribution. Proc. London
+Math. Soc. (3), 21:540–556, 1970.
+[179] C. Mauduit and A. S´ark¨ozy. On finite pseudorandom binary sequences. I. Measure of pseu-
+dorandomness, the Legendre symbol. Acta Arith. 82(4):365–377, 1997.
+[180] A. M. Mian, S. Chowla. On the B2 sequences of Sidon. Proc. Nat. Acad. Sci. India Sect. A
+14:3–4, 1944.
+[181] H. L. Montgomery. Ten lectures on the interface between analytic number theory and har-
+monic analysis, volume 84 of CBMS Regional Conference Series in Mathematics. American
+Mathematical Society, Providence, RI, 1994.
+
+LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY
+57
+[182] G. W. Morgenthaler. A central limit theorem for uniformly bounded orthonormal systems.
+Trans. Amer. Math. Soc. 79:281–311, 1955.
+[183] C. M¨ullner and L. Spiegelhofer. Normality of the Thue-Morse sequence along Piatetski-
+Shapiro sequences, II. Israel J. Math. 220(2):691–738, 2017.
+[184] T. Murai. The central limit theorem for trigonometric series. Nagoya Math. J. 87:79–94,
+1982.
+[185] R. Nair. On strong uniform distribution. Acta Arith. 56(3):183–193, 1990.
+[186] Y. Nakai and I. Shiokawa. A class of normal numbers. Japan. J. Math. (N.S.) 16(1):17–29,
+1990.
+[187] Y. Nakai and I. Shiokawa. Normality of numbers generated by the values of polynomials at
+primes. Acta Arith. 81(4):345–356, 1997.
+[188] H. Niederreiter and R. F. Tichy. Solution of a problem of Knuth on complete uniform distri-
+bution of sequences. Mathematika 32(1):26–32, 1985.
+[189] J. S. B. Nielsen and J. G. Simonsen. An experimental investigation of the normality of
+irrational algebraic numbers. Math. Comp. 82(283):1837–1858, 2013.
+[190] E. M. Nikishin. Resonance theorems and superlinear operators. Uspehi Mat. Nauk
+25(6(156)):129–191, 1970.
+[191] E. M. Nikishin. On systems of convergence in measure for ℓ2. Mat. Zametki 13:337–340, 1973.
+[192] E. Novak and H. Wo´zniakowski. Tractability of multivariate problems. Volume II: Standard
+information for functionals, volume 12 of EMS Tracts in Mathematics. European Mathemat-
+ical Society (EMS), Z¨urich, 2010.
+[193] W. Philipp. Limit theorems for lacunary series and uniform distribution mod 1. Acta Arith.
+26(3):241–251, 1974/75.
+[194] W. Philipp. Empirical distribution functions and strong approximation theorems for depen-
+dent random variables. A problem of Baker in probabilistic number theory. Trans. Amer.
+Math. Soc. 345(2):705–727, 1994.
+[195] W. Philipp. Empirical distribution functions and strong approximation theorems for depen-
+dent random variables. A problem of Baker in probabilistic number theory. Trans. Amer.
+Math. Soc. 345(2):705–727, 1994.
+[196] W. Philipp and W. Stout. Almost sure invariance principles for partial sums of weakly de-
+pendent random variables. Mem. Amer. Math. Soc. 2(no. 161):iv+140, 1975.
+[197] A. R´enyi. On stable sequences of events. Sankhy¯a Ser. A 25:293 302, 1963.
+[198] A. R´enyi. Foundations of probability. Holden-Day, 1970.
+[199] Z. Rudnick and A. Zaharescu. The distribution of spacings between fractional parts of lacu-
+nary sequences. Forum Math. 14(5):691–712, 2002.
+[200] I. Z. Ruzsa. An infinite Sidon sequence. J. Number Theory 68(1):63–71, 1998.
+[201] R. Salem and A. Zygmund. On lacunary trigonometric series. Proc. Nat. Acad. Sci. U.S.A.
+33:333–338, 1947.
+[202] R. Salem and A. Zygmund. On lacunary trigonometric series. II. Proc. Nat. Acad. Sci. U.S.A.
+34:54–62, 1948.
+[203] R. Salem and A. Zygmund. La loi du logarithme it´er´e pour les s´eries trigonom´etriques lacu-
+naires. Bull. Sci. Math. (2), 74:209–224, 1950.
+[204] R. Salem and A. Zygmund. Some properties of trigonometric series whose terms have random
+signs. Acta Math. 91:245–301, 1954.
+[205] A.-M. Scheerer. Computable absolutely normal numbers and discrepancies. Math. Comp.
+86(308):2911–2926, 2017.
+[206] W. M. Schmidt. ¨Uber die Normalit¨at von Zahlen zu verschiedenen Basen. Acta Arith. 7:299–
+309, 1961/62.
+[207] W. M. Schmidt. Norm form equations. Ann. of Math. (2), 96:526–551, 1972.
+[208] W. Sierpinski. D´emonstration ´el´ementaire du th´eor`eme de M. Borel sur les nombres ab-
+solument normaux et d´etermination effective d’une tel nombre. Bull. Soc. Math. France
+45:125–132, 1917.
+
+58
+C. AISTLEITNER, I. BERKES AND R. TICHY
+[209] K. Soundararajan. Extreme values of zeta and L-functions. Math. Ann. 342(2):467–486, 2008.
+[210] L. Spiegelhofer. Normality of the Thue-Morse sequence along Piatetski-Shapiro sequences.
+Q. J. Math. 66(4):1127–1138, 2015.
+[211] S. Takahashi. A gap sequence with gaps bigger than the Hadamards. Tohoku Math. J. (2),
+13:105–111, 1961.
+[212] S. Takahashi. An asymptotic property of a gap sequence. Proc. Japan Acad. 38:101–104,
+1962.
+[213] S. Takahashi. The law of the iterated logarithm for a gap sequence with infinite gaps. Tohoku
+Math. J. (2), 15:281–288, 1963.
+[214] S. Takahashi. On the law of the iterated logarithm for lacunary trigonometric series. Tohoku
+Math. J. (2), 24:319–329, 1972.
+[215] N. Technau and A. Zafeiropoulos. The discrepancy of (nkx)∞
+k=1 with respect to certain prob-
+ability measures. Q. J. Math. 71(2):573–597, 2020.
+[216] R. Tijdeman. On integers with many small prime factors. Compositio Math. 26:319–330,
+1973.
+[217] J. Vandehey. On the binary digits of
+√
+2. Integers 18:Paper No. A30, 7, 2018.
+[218] S. Wagon. Is π normal? Math. Intelligencer 7(3):65–67, 1985.
+[219] A. Walfisz. Ein metrischer Satz ¨uber diophantische Approximationen. Fund. Math. 16:361–
+365, 1930.
+[220] M. Weber. On systems of dilated functions. C. R. Math. Acad. Sci. Paris 349:1261–1263,
+2011.
+[221] M. Weiss. The law of the iterated logarithm for lacunary trigonometric series. Trans. Amer.
+Math. Soc. 91:444–469, 1959.
+[222] H. Weyl. ¨Uber die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77(3):313–352, 1916.
+[223] A. Wintner. Diophantine approximations and Hilbert’s space. Amer. J. Math. 66: 564–578,
+1944.
+[224] K. F. Xylogiannopoulos, P. Karampelas, and R. Alhajj. Experimental analysis on the nor-
+mality of π, e, φ,
+√
+2 using advanced data-mining techniques. Exp. Math. 23(2):105–128,
+2014.
+