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 $$\mathcal{L}(\alpha )=\{\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\mathrm{\parallel }\xi {\alpha }^n\mathrm{\parallel }\mid \xi \in \mathbb{R}\},$$ where $\mathrm{\parallel }x\mathrm{\parallel }$ is the distance from $x$ to the nearest integer. First, we show that $\mathcal{L}(\alpha )$ is closed in $[0,1\mathrm{/}2]$ for any Pisot number $\alpha$. Then we consider the case where $\alpha$ is an integer with $\alpha \ge 2$, 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.