Comments on the fractional parts of Pisot numbers

summary:Let $L(\theta ,\lambda )$ be the set of limit points of the fractional parts $\lbrace \lambda \theta ^{n}\rbrace $, $n=0,1,2, \dots $, where $\theta $ is a Pisot number and $\lambda \in \mathbb{Q}(\theta )$. Using a description of $L(\theta ,\lambda )$, due to Dubickas, we show that there is a sequence $(\lambda _{n})_{n\ge 0}$ of elements of $\mathbb{Q}(\theta )$ such that $\operatorname{Card}\,(L(\theta ,\lambda _{n}))< \operatorname{Card}\,(L(\theta ,\lambda _{n+1}))$, $\forall $ $n\ge 0$. Also, we prove that the fractional parts of Pisot numbers, with a fixed degree greater than 1, are dense in the unit interval