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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.

11J71 Distribution modulo one
11K06 General theory of distribution modulo \(1\)
28D99 Measure-theoretic ergodic theory
Full Text: DOI EuDML