Cubic approximation to Sturmian continued fractions

TL;DR

This paper analyzes the approximation properties of Sturmian continued fractions, providing explicit constants, describing the parametric minima graph, and extending understanding of approximation spectra and exponents in dimension three.

Contribution

It offers a detailed description of approximation constants and the parametric minima graph for Sturmian continued fractions, extending prior two-dimensional results and exploring higher-dimensional exponents.

Findings

Explicit formulas for approximation constants of Sturmian continued fractions.

Description of the combined graph of parametric successive minima functions in dimension three.

New insights into the spectra of approximation exponents and their behavior.

Abstract

We determine the classical approximation constants $w_{3}(\zeta),w_{3}^{\ast}(\zeta),\lambda_{3}(\zeta)$ such as the uniform constants $\widehat{w}_{3}(\zeta),\widehat{w}_{3}^{\ast}(\zeta),\widehat{\lambda}_{3}(\zeta)$ associated to real numbers $\zeta$ whose continued fraction expansions are given by a Sturmian word. We more generally provide a description of the combined graph of the parametric successive minima functions defined by Schmidt and Summerer in dimension three for such Sturmian continued fractions. This both complements similar results due to Bugeaud and Laurent concerning the two-dimensional constants and generalizes a recent result of the author. As a side result we obtain new information on the spectra of certain exponents of approximation. Moreover, we provide some information on the exponents $\lambda_{n}(\zeta)$ for a Sturmian continued fraction $\zeta$ and arbitrary $n$.