On the limit points of the fractional parts of powers of Pisot numbers

Arturas Dubickas

Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania, Institute of Mathematics and Informatics, Akademijos 4, LT-08663 Vilnius, Lithuania



E-mail:  arturas.dubickas@maf.vu.lt


We consider the sequence of fractional parts $\{\xi \al^n\}$, $n=1,2,3,\dots$, where $\al>1$ is a Pisot number and $\xi \in {\mathbb Q}(\al)$ is a positive number. We find the set of limit points of this sequence and describe all cases when it has a unique limit point. The case, where $\xi=1$ and the unique limit point is zero, was earlier described by the author and Luca, independently.

AMSclassification. 11J71, 11R06.

Keywords. Pisot numbers, fractional parts, limit points.