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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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