On the discrepancy of random subsequences of {nα}

Berkes, István and Borda, Bence (2018) On the discrepancy of random subsequences of {nα}. ACTA ARITHMETICA. pp. 1-30. ISSN 0065-1036

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Abstract

Abstract For irrational α, {nα} 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 {nkα}, with the exception of metric results for exponentially growing (nk). It is therefore natural to consider random (nk), and in this paper we give nearly optimal bounds for the discrepancy of {nkα} in the case when the gaps nk+1−nk 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 nk+1 − nk and the rational approximation properties of α. We also point out an interesting critical phenomenon, i.e. a sudden change of the order of magnitude of the discrepancy of {nkα} as the Diophantine type of α passes through a certain critical value.

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