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Description and exact maximum and minimum values of the remainder in the problem of the distribution of fractional parts. (English. Russian original) Zbl 1303.11078
Math. Notes 89, No. 1, 59-67 (2011); translation from Mat. Zametki 89, No. 1, 43-52 (2011).
The paper deals with the distribution of the function \(f(x)=\langle \alpha x\rangle\) where \(\langle x\rangle\) is the fractional part of the \(x.\) The aim of the paper is to investigate how much can be the deviation of this distribution from the uniform one. Let \[ r(\alpha,n,I)=\#\{i: 0\leq i< n, \langle i\alpha\rangle \in I\}-n|I| \] \[ r^+(\alpha)=\sup_{n}r(\alpha,n,I),\, r^-(\alpha)=\inf_{n}r(\alpha,n,I). \] The authors consider the intervals of the form \(I=[\delta;\delta+\langle m\alpha\rangle]\) and such that \(|r^{\pm}(\alpha)|<\infty.\) Using Hecke’s formula for \(r(\alpha,n,I)\) and investigating different properties of piecewise linear functions, the authors obtain new formula for \(r^{\pm}(\alpha)\). As a corollary they prove that \(r^{\pm}(\alpha)\) can be computed in \(O(m)\) operations.

MSC:
11J71 Distribution modulo one
11K38 Irregularities of distribution, discrepancy
11K31 Special sequences
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