On the law of the iterated logarithm for the discrepancy of lacunary sequences

Abstract

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

A metric discrepancy result with given speed

“…for any ε > 0 (1) and this fails for ε ≤ 0. The discrepancy of exponentially growing sequences has also been investigated extensively.…”

“…See [12,14,15,16,17]. For conditions to have an exact law of the iterated logarithm in (3), see [1,5].…”

“…Since there is a big difference between (1) and (3), it is natural to ask if for intermediate speeds Ψ(N ) between (log N )(log log N ) 1+ε and (N log log N ) 1/2 one can find a sequence {n k } of integers such that the growth speed of D N {n k x} is Ψ(N ) in the above sense. For all γ ∈ (0, 1/2 ], Aistleitner and Larcher [6] constructed an increasing sequence…”

Strong approximation of lacunary series with random gaps

“….. A necessary and sufficient number-theoretic condition for the CLT for f (n k x) under (1.5) was given by Aistleitner and Berkes [4]. For a related sufficient criterion for the law of the iterated logarithm for the discrepancy of {n k x} for almost all x, see Aistleitner [1].…”

On the class of limits of lacunary trigonometric series

“…Aistleitner [4]; the problem to find a sufficient and necessary condition, like in the case of the CLT, is still open). If the number of the solutions of some Diophantine equations of type (6) is "too large", the value of the lim sup in (4) does not have to be equal to f 2 a.e.…”