Multiplicative analogue of Markoff-Lagrange spectrum and Pisot numbers

TL;DR

This paper explores the properties of a multiplicative analogue of the Markoff-Lagrange spectrum related to Pisot numbers and quadratic units, revealing topological characteristics and the structure of limit points in these spectra.

Contribution

It introduces a multiplicative spectrum analogous to the Markoff-Lagrange spectrum, analyzing its topological properties and structure for Pisot numbers and quadratic units.

Findings

${ m extbf{L}}(oldsymbol{eta})$ is closed for any Pisot number $eta$.

${ m extbf{L}}(oldsymbol{eta})$ contains a proper interval for quadratic units $eta$.

Identifies the minimum limit point and all isolated points below it.

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 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.