On the discrepancy of random subsequences of
Volume 191 / 2019
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.