Multiplicative analogue of Markoff-Lagrange spectrum and Pisot numbers

Markoff-Lagrange spectrum uncovers exotic topological properties of Diophantine approximation. We investigate asymptotic properties of geometric progressions modulo one and observe significantly analogous results on the set $${\mathcal L}(\alpha)=\left\{\left.\limsup_{n\to \infty}\|\xi \alpha^n\|\ \right|\ \xi\in {\mathbb R}\right\},$$ where $\|x\|$ is the distance from $x$ to the nearest integer. First, we show that ${\mathcal L}(\alpha)$ is closed in $[0,1/2]$ for any Pisot number $\alpha$. Then we consider the case where $\alpha$ is an integer with $\alpha\geq 2$, or a quadratic unit with $\alpha\ge 3$. We show that ${\mathcal L}(\alpha)$ contains a proper interval when $\alpha$ is quadratic but it does not when $\alpha$ is an integer. We also determine the minimum limit point and all isolated points beneath this point. In the course of the proof, we revisit a property studied by Markoff which characterizes bi-infinite balanced words and sturmian words.