On permutations of Hardy-Littlewood-P\'olya sequences

Let $\mathcal{H}=(q_1,\dots q_r)$ be a finite set of coprime integers and let $n_1,n_2,\dots$ 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. For example, the asymptotic properties of the sequence $\{n_{k}x\}$ are very similar to those of independent, identically distributed random variables; here $\{\cdot \}$ denotes fractional part. However, the behavior of this sequence depends sensitively on the generating elements of $(n_k)$ and the combination of probabilistic and number-theoretic effects results in a unique, highly interesting asymptotic behavior. In particular, the properties of $\{n_{k}x\}$ are not permutation invariant, in contrast to i.i.d. behavior. The purpose of this paper is to show that $\{n_{k}x\}$ satisfies a strong independence property (“interlaced mixing”), enabling one to determine the precise asymptotic behavior of permuted sums $S_N (\sigma)= \sum_{k=1}^N f(n_{\sigma(k)} x)$\mathrm{.}Aswewillsee,thebehaviorof$S_N(\sigma)$ still follows that of sums of independent random variables, but its growth speed (depending on $\sigma$) is given by the classical G'al function of Diophantine approximation theory. Some examples describing the class of possible growth functions are given.