Exponents of Diophantine approximation in dimension 2 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 ξ of this set, several exponents of Diophantine approximation to the pair ( ξ, ξ 2 ), together with ω ∗ 2 ( ξ ) and b ω ∗ 2 ( ξ ), the so-called ordinary and uniform exponent of approximation to ξ by algebraic numbers of degree ≤ 2. As an application, we get new information on the set of values taken by b ω ∗ 2 at transcendental numbers, and we give a partial answer to a question of Fischler about his exponent β 0 .