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On the limit points of the fractional parts of powers of Pisot numbers. (English) Zbl 1164.11026

Let \(\alpha >1\) be an algebraic integer and \(\xi >0\) be a real number. Under the assumption that \(\xi \notin \mathbb Q(\alpha )\) if \(\alpha \) is a Pisot number or a Salem number, the author gave a lower bound (in terms of \(\alpha \)) for the distance between the largest and the smallest limit points of the sequence of fractional parts \(\{\xi \alpha ^n\}\) (see [Bull.Lond.Math.Soc.38, 70-80 (2006; Zbl 1164.11025)]).
The paper under review is devoted to the case when \(\alpha >1\) is a Pisot number and a positive \(\xi \in \mathbb Q(\alpha )\). The set of all limit points of \(\{\xi \alpha ^n\}\) is described and a simple criterion of determining whether \(\{\xi \alpha ^n\}\) has just one limit point is given.
The author proves that for an algebraic \(\alpha >1\) there exists a real number \(\xi >0\) such that \(\{\xi \alpha ^n\}\) tends to a limit if and only if \(\alpha \) is either a strong Pisot number, or \(\alpha =2\), or \(\alpha \) is a Pisot number whose minimal polynomial \(P\) satisfies \(P(1)\leq -2\).

MSC:

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

Citations:

Zbl 1164.11025
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