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Elementary proof of Furstenberg’s diophantine result. (English) Zbl 0815.11036
Let \({\mathcal S} = \{s_ 1, s_ 2, \dots\}\), \(s_ 1 < s_ 2 < s_ 3 < \dots\) be a multiplicative semigroup of positive integers such that (a) there is a pair of multiplicatively independent elements in \({\mathcal S}\), (b) one cannot represent all \(s_ i\) as integral powers of a single element in \({\mathcal S}\), (c) \(\lim_{i \to \infty} (s_{i + 1}/s_ i) = 1\).
The author gives an elementary proof of the fact that for irrational \(\alpha\) the set \(S \alpha\) is dense mod 1. The original, more involved proof is due to H. Furstenberg [Math. Systems Theory 1, 1-49 (1967; Zbl 0146.285)].
Reviewer: R.F.Tichy (Graz)

MSC:
11J71 Distribution modulo one
54H20 Topological dynamics (MSC2010)
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