On the law of the iterated logarithm for permuted lacunary sequences

Abstract

It is known that for any smooth periodic function $f$ the sequence $(f(2^{k}x){)}_{k\ge 1}$ 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 $(f(2^{k}x){)}_{k\ge 1}$ can ruin the validity of the law of the iterated logarithm, a very surprising result. In this paper we present an optimal condition on $(n_k{)}_{k\ge 1}$, 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 $(f(n_{k}x){)}_{k\ge 1}$. A similar result is proved for the discrepancy of the sequence $(\{n_{k}x\}{)}_{k\ge 1}$, where $\{\cdot \}$ denotes fractional part.

arXiv commentDedicated to the memory of Professor Anatolii Alexeevitch Karatsuba