Cubic approximation to Sturmian continued fractions

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 $n$.