On the law of the iterated logarithm for permuted lacunary sequences

Abstract

It is known that for any smooth periodic function f the sequence (f(2kx))k≥1 behaves like a sequence of i.i.d. random variables; for example, it satisfies the central limit theorem and the law of the iterated logarithm. Recently Fukuyama showed that permuting (f(2kx))k≥1 can ruin the validity of the law of the iterated logarithm, a very surprising result. In this paper we present an optimal condition on (nk)k≥1, formulated in terms of the number of solutions of certain Diophantine equations, which ensures the validity of the law of the iterated logarithm for any permutation of the sequence (f(nkx))k≥1. A similar result is proved for the discrepancy of the sequence ({nkx})k≥1, where {·} denotes the fractional part.