Diophantine approximations with Pisot numbers

Let $\alpha$ be a Pisot number. Let $L(\alpha)$ be the largest positive number such that for some $\xi=\xi(\alpha)\in \mathbb R$ the limit points of the sequence of fractional parts $\{\xi \alpha^n\}_{n=1}^{\infty}$ all lie in the interval $[L(\alpha), 1-L(\alpha)]$. In this paper we show that if $\alpha$ is of degree at most 4 or $\alpha\le \frac{\sqrt 5 + 1}{2}$ then $L(\alpha)\ge \frac{3}{17}$. Also we find explicitly the value of $L(\alpha)$ for certain Pisot numbers of degree 3.