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Actions of sets of integers on irrationals. (English) Zbl 0575.10030
Given a sequence $$(a_ n)$$ of integers, consider the set $$\Delta$$ consisting of all integers of the form $$a_{n_ 1}a_{n_ 2}...a_{n_ k}$$ with $$n_ 1<n_ 2<...<n_ k$$. The paper deals mainly with the following question: Under what conditions on $$(a_ n)$$ does $$\Delta$$ possess the property that the set $$\Delta$$ $$\alpha$$ is dense modulo 1 for every irrational $$\alpha$$ ? Various techniques (elementary, a-adic and topological dyanamics) are employed to get sufficient conditions, and several counter-examples are obtained, too. The results are also applied to investigate global diophantine approximations by small rational groups.

##### MSC:
 11J71 Distribution modulo one 11K06 General theory of distribution modulo $$1$$ 28D99 Measure-theoretic ergodic theory
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