  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.
