On distribution of sequences of integers. (English) Zbl 0803.11038

Paštéka, Milan; Porubský, Štefan

Math. Slovaca 43, No. 5, 521-539 (1993).

Let \(\Gamma= \{h(j,m)\): \(0\leq j<m\), \(m\in \mathbb{N}\}\) be a system of non- negative real numbers. A sequence \(\omega= (w_ n)\) of positive integers is called \(\Gamma\)-distributed if and only if \[ \lim_{N\to\infty} {{\#\{n\leq N:\;w_ n\equiv j\pmod m\}} \over N} = h(j,m), \] for every \(j,m\in \mathbb{N}\) and \(0\leq j<m\). Applying a general metric result for uniform distribution in compact spaces, the authors show the existence of a \(\Gamma\)-distributed sequence \(\omega\) provided that \(\omega\) satisfies \(h(0,1) =1\) and \(h(j,m)= \sum_{r=0}^{k-1} h(j+ rm, km)\) for every \(k,m\in \mathbb{N}\), \(0\leq j<m\). The authors make use of an interesting connection to Novoselov’s theory of polyadic numbers. In a final chapter transformations of \(\Gamma\)-distributed sequences are studied.

Reviewer: R.F.Tichy (Graz)

MSC:

11K06

General theory of distribution modulo \(1\)

Keywords:

existence of a \(\Gamma\)-distributed sequence; Novoselov’s theory of polyadic numbers; transformations

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