On permutations of Hardy-Littlewood-Pólya sequences

Abstract:

Let ${\mathcal H}=(q_1, \ldots , q_r)$ be a finite set of coprime integers and let $n_1, n_2, \ldots$ denote the multiplicative semigroup generated by $\mathcal H$ and arranged in increasing order. The distribution of such sequences has been studied intensively in number theory, and they have remarkable probabilistic and ergodic properties. In particular, the asymptotic properties of the sequence $\{n_kx\}$ are similar to those of independent, identically distributed random variables; here $\{\cdot \}$ denotes fractional part. In this paper we prove that under mild assumptions on the periodic function $f$, the sequence $f(n_kx)$ obeys the central limit theorem and the law of the iterated logarithm after any permutation of its terms. Note that the permutational invariance of the CLT and LIL generally fails for lacunary sequences $f(m_kx)$ even if $(m_k)$ has Hadamard gaps. Our proof depends on recent deep results of Amoroso and Viada on Diophantine equations. We will also show that $\{n_kx\}$ satisfies a strong independence property (“interlaced mixing”), enabling one to determine the precise asymptotic behavior of permuted sums $S_N (\sigma )= \sum _{k=1}^N f(n_{\sigma (k)} x)$.