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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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##### References:
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