Exponents of Diophantine approximation in dimension two for a general class of numbers

Abstract

We study the Diophantine properties of a new class of transcendental real numbers which contains, among others, Roy's extremal numbers, Bugeaud-Laurent Sturmian continued fractions, and more generally the class of Sturmian type numbers. We compute, for each real number $\xi$ of this set, several exponents of Diophantine approximation to the pair $(\xi,\xi^2)$, together with $\omega_2^*(\xi)$ and $\widehat{\omega}_2^*(\xi)$, the so-called ordinary and uniform exponent of approximation to $\xi$ by algebraic numbers of degree $\leq 2$. As an application, we get new information on the set of values taken by $\widehat{\omega}_2^*$ at transcendental numbers, and we give a partial answer to a question of Fischler about his exponent $\beta_0$.