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 ${\omega }_2^{\ast }(\xi )$ and ${\hat{\omega }}_2^{\ast }(\xi )$, the so-called ordinary and uniform exponent of approximation to $\xi$ by algebraic numbers of degree $\le 2$. As an application, we get new information on the set of values taken by ${\hat{\omega }}_2^{\ast }$ at transcendental numbers, and we give a partial answer to a question of Fischler about his exponent ${\beta }_0$.

28 pages, 2 figures

References in corpus (1)

• Rational approximation to real points on quadratic hypersurfaces