Local Discrepancies in the Problem of the Distribution of the Sequence

The paper deals with local discrepancies in the problem of the distribution of the sequence , i.e., with the remainder terms in asymptotic formulas for the number of points in this sequence lying in prescribed intervals. A construction of intervals for which local discrepancies tend to infinity slow...

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Bibliographic Details
Published inMathematical Notes Vol. 109; no. 3-4; pp. 473 - 482
Main Author Shutov, A. V.
Format Journal Article
LanguageEnglish
Published Moscow Pleiades Publishing 01.03.2021
Springer Nature B.V
Subjects
14/34
639/766/189
639/766/530
639/766/747
Intervals
Mathematics
Mathematics and Statistics
Quotients
local discrepancies
continued fractions
uniform distribution
inhomogeneous Diophantine approximations
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Summary:The paper deals with local discrepancies in the problem of the distribution of the sequence , i.e., with the remainder terms in asymptotic formulas for the number of points in this sequence lying in prescribed intervals. A 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.
ISSN:0001-4346
1067-9073
1573-8876
DOI:10.1134/S0001434621030147