Multiplicative analogue of Markoff-Lagrange spectrum and Pisot numbers

Abstract

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 setL(α)={lim supn→∞‖ξαn‖|ξ∈R}, where ‖x‖ is the distance from x to the nearest integer. First, we show that L(α) is closed in [0,1/2] for any Pisot number α.Then we consider the case where α is an integer with α≥2, or a quadratic unit with α≥3. We show that L(α) contains a proper interval when α is quadratic but it does not when α 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.