Exponents of Diophantine approximation in dimension 2 for a general class of numbers

摘要

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 $hat{omega}_2^*(xi)$, the so-called ordinary and uniform exponents 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 $hat{omega}_2^*$ at transcendental numbers, and we give a partial answer to a question of Fischler about his exponent $beta_0$.