On the law of the iterated logarithm for permuted lacunary sequences

It is known that for any smooth periodic function $f$ the sequence $(f(2^kx))_{k\ge1}$ 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 $(f(2^kx))_{k\ge1}$ can ruin the validity of the law of the iterated logarithm, a very surprising result. In this paper we present an optimal condition on $(n_k)_{k\ge1}$, 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 $(f(n_k x))_{k\geq1}$. A similar result is proved for the discrepancy of the sequence $(\{n_k x\})_{k\geq1}$, where $\{\cdot\}$ denotes the fractional part.