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On weak uniform distribution of sequences of integers. (English) Zbl 0695.10038

A characterization is given of subsets \(A\) of positive integers with the property that there exists a sequence weakly uniformly distributed (mod \(N\)) if and only if \(N\) belongs to \(A\). This solves Problem I in the reviewer’s “Uniform distribution of sequences of integers in residue classes” [Lect. Notes Math. 1087. Berlin: Springer (1984; Zbl 0541.10001), p. 9]. The author utilizes a method used by A. Zame [Colloq. Math. 24, 271–273 (1972; Zbl 0244.10048)] to solve a similar problem concerning uniform distribution. The same result has been obtained independently by another method by I. Z. Ruzsa [Colloq. Math. 57, 183–187 (1989; Zbl 0695.10039)].

MSC:

11K65 Arithmetic functions in probabilistic number theory
11B50 Sequences (mod \(m\))
11N64 Other results on the distribution of values or the characterization of arithmetic functions
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