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Matematicheskie Zametki, 2021, Volume 109, Issue 3, Pages 452–463
DOI: https://doi.org/10.4213/mzm11326 (Mi mzm11326) 		 
Local Discrepancies in the Problem of the Distribution of the Sequence $\{k\alpha\}$

A. V. Shutov

Vladimir State University
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DOI: https://doi.org/10.4213/mzm11326
Abstract: The paper deals with local discrepancies in the problem of the distribution of the sequence $\{k\alpha\}$, i.e., the remainder terms in asymptotic formulas for the number of points in this sequence lying in prescribed intervals. The construction of intervals for which local discrepancies tend to infinity slower than any given function is presented. Moreover, it is shown that there exists an uncountable set of such intervals. Previously, similar results were obtained only for irrationalities with bounded partial quotients of their continued fraction expansions.
Keywords: uniform distribution, local discrepancies, inhomogeneous Diophantine approximations, continued fractions.
Received: 28.07.2016
Revised: 06.10.2020
English version:
Mathematical Notes, 2021, Volume 109, Issue 3, Pages 473–482
DOI: https://doi.org/10.1134/S0001434621030147
Bibliographic databases:
Document Type: Article
UDC: 511.43
Language: Russian
Citation: A. V. Shutov, “Local Discrepancies in the Problem of the Distribution of the Sequence $\{k\alpha\}$”, Mat. Zametki, 109:3 (2021), 452–463; Math. Notes, 109:3 (2021), 473–482
Citation in format AMSBIB
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