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Local discrepancies in the problem of fractional parts distribution of a linear function

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Abstract

In this paper we consider a problem of distribution of fractional parts of the sequence obtained as multiples of some irrational number with bounded partial quotients of its continued fraction expansion. Local discrepancies are the remainder terms of asymptotic formulas for the number of points of the sequence lying in given intervals. Earlier only intervals with bounded and logarithmic local discrepancies were known. We prove that there exists an infinite set of intervals with arbitrary small growth rate of local discrepancies. The proof is based on the connection of considered problem with some of those from Diophantine approximations.

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Authors and Affiliations

  1. Vladimir State University, pr. Stroitelei 3/7, Vladimir, 600000, Russia

    A. V. Shutov

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  1. A. V. Shutov
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Correspondence to A. V. Shutov.

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Original Russian Text ©A.V. Shutov, 2017, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2017, No. 2, pp. 88–97.

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Shutov, A.V. Local discrepancies in the problem of fractional parts distribution of a linear function. Russ Math. 61, 74–82 (2017). https://doi.org/10.3103/S1066369X17020098

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  • DOI: https://doi.org/10.3103/S1066369X17020098

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