Exponents of diophantine approximation in dimension $2$ for numbers of Sturmian type

TL;DR

This paper extends the understanding of Diophantine approximation exponents for Sturmian type numbers in dimension two, generalizing previous Fibonacci-based constructions and analyzing the behavior of successive minima functions.

Contribution

It introduces a new class of Sturmian type numbers, determines their approximation exponents, and describes the structure of their parametric minima functions, extending prior results.

Findings

Explicit formulas for approximation exponents of Sturmian type numbers

Complete description of the parametric minima functions in dimension two

Identification of new families of numbers with well-understood rational approximations

Abstract

We generalize the construction of Roy's Fibonacci type numbers to the case of a Sturmian recurrence and we determine the classical exponents of approximation $\omega_2(\xi)$, $\widehat{\omega}_2(\xi)$, $\lambda_2(\xi)$, $\widehat{\lambda}_2(\xi)$ associated with these real numbers. This also extends similar results established by Bugeaud and Laurent in the case of Sturmian continued fractions. More generally we provide an almost complete description of the combined graph of parametric successive minima functions defined by Schmidt and Summerer in dimension two for such Sturmian type numbers. As a side result we obtain new information on the joint spectra of the above exponents as well as a new family of numbers for which it is possible to construct the sequence of the best rational approximations.