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

*Arturas** Dubickas*

**Address****.**

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

** **

**Abstract****.**

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.