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 ${\widehat{\omega }}_2^{\ast }(\xi )$, the so-called ordinary and uniform exponents 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 ${\widehat{\omega }}_2^{\ast }$ at transcendental numbers, and we give a partial answer to a question of Fischler about his exponent ${\beta }_0$.

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