On the discrepancy of random subsequences of $\{n\alpha\}$

TL;DR

This paper investigates the discrepancy of random subsequences of the form {n_k lpha}, revealing how the distribution of gaps and the Diophantine properties of lpha influence their uniformity, with critical phenomena observed.

Contribution

It provides nearly optimal bounds for the discrepancy of random subsequences with i.i.d. integer gaps, highlighting the interplay with lpha's Diophantine approximation.

Findings

Discrepancy bounds depend on gap distribution and lpha's approximation properties.

Identifies a critical transition in discrepancy behavior related to Diophantine type.

Shows a sudden change in discrepancy magnitude at a specific Diophantine threshold.

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.