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Sequences of algebraic numbers and density modulo 1. (English) Zbl 1164.11027
H. Furstenberg [Math. Syst. Theory 1, 1–49 (1967; Zbl 0146.28502)] showed that the set $$\{a\xi\mid a\in A\}$$ is dense modulo $$1$$ for any irrational $$\xi$$ if $$A$$ is a non-lacunary sub-semigroup of the natural numbers. D. Berend [J. Number Theory 26, 246–256 (1987; Zbl 0623.10038)] and B. Kra [Proc. Am. Math. Soc. 127, 1951–1956 (1999; Zbl 0921.11034)] found related results for other sets, involving both non-integer powers and translates. Here aspects of both these results are extended: the sets $$\{\mu^m\lambda^n\xi+r_m\mid m,n\geq 1\}$$ and $$\{\mu^m\lambda^n\xi+r^{m+n}\beta\mid m,n\geq1\}$$ with $$\mu,\lambda$$ rationally independent real algebraic numbers with some additional technical assumptions, and $$r,\beta,r_m$$ real, are shown to be dense modulo one.

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