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On distribution of sequences of integers. (English) Zbl 0803.11038
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 $$1$$
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##### References:
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