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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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