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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
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