## Distribution of geometric sequences modulo 1.(English)Zbl 1177.11060

The main result of this paper is a lower bound for $$\limsup_{n\to \infty}\| B\alpha^n \|$$ where $$B$$ is a real number and $$\alpha$$ is an algebraic irrational number. The author also proved that for a fixed real number $$C$$ and arbitrary positive numbers $$\delta$$ and $$M$$, the set of $$\alpha >M$$ satisfying $$\limsup_{n\to \infty} \| C\alpha^n \| \leq \frac{1+\delta}{2\alpha}$$ is at least countable and satisfying $$\limsup_{n\to \infty} \| C\alpha^n \| \leq \frac{1+\delta}{\alpha}$$ is at least uncountable.

### MSC:

 11J71 Distribution modulo one 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure 11B85 Automata sequences

### Keywords:

distribution; algebraic numbers; geometric progression
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