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 $\{\xi {\alpha }^n{\}}_{n=1}^{\mathrm{\infty }}$ 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 $L(\alpha)\ge \frac{3}{17}$\mathrm{.}Alsowefindexplicitlythevalueof$L(\alpha)$ for certain Pisot numbers of degree 3.