On the discrepancy of random subsequences of $\{nα\}$

Abstract: For irrational $\alpha$, ${n\alpha}$ is uniformly distributed mod 1 in the Weyl sense, and the asymptotic behavior of its discrepancy is completely known. In contrast, very few precise results exist for the discrepancy of subsequences ${n_k \alpha}$, with the exception of metric results for exponentially growing $(n_k)$. It is therefore natural to consider random $(n_k)$, and in this paper we give nearly optimal bounds for the discrepancy of ${n_k \alpha}$ in the case when the gaps $n_{k+1}-n_k$ are independent, identically distributed, integer-valued random variables. As we will see, the discrepancy behavior is determined by a delicate interplay between the distribution of the gaps $n_{k+1}-n_k$ and the rational approximation properties of $\alpha$. We also point out an interesting critical phenomenon, a sudden change of the order of magnitude of the discrepancy of ${n_k \alpha}$ as the Diophantine type of $\alpha$ passes through a certain critical value.