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Multidimensional generalization of sums of fractional parts and their number-theoretic applications. (Russian. English summary) Zbl 1434.11148
Summary: In this paper a new multidimensional generalization of the fractional part function is introduced. We obtain a formula which expresses the number of points from the orbit of an irrational shift on the multidimensional torus, lying in a given domain, in terms of sums of multidimensional fractional parts. Also we give some applications of this formula to various number-theoretic problems.

11K31 Special sequences
11K38 Irregularities of distribution, discrepancy
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