Exponents of Diophantine Approximation and Sturmian Continued Fractions

Bugeaud, Yann; Laurent, Michel

Annales de l'Institut Fourier, Tome 55 (2005) no. 3, pp. 773-804.

Soient $\xi$ un nombre réel et $n$ un entier strictement positif. Nous définissons quatre exposants d’approximation diophantienne, qui viennent compléter les exposants $w_n\left(\xi \right)$ et $w_n^{*}\left(\xi \right)$ définis par Mahler et Koksma. Nous calculons leurs six valeurs lorsque $n=2$ et $\xi$ est un nombre réel dont le développement en fraction continue est, aux premiers termes près, une suite sturmienne d’entiers positifs. En particulier, nous obtenons l’exposant exact d’approximation d’une telle fraction continue $\xi$ par des nombres quadratiques

Let $\xi$ be a real number and let $n$ be a positive integer. We define four exponents of Diophantine approximation, which complement the exponents $w_n\left(\xi \right)$ and $w_n^{*}\left(\xi \right)$ defined by Mahler and Koksma. We calculate their six values when $n=2$ and $\xi$ is a real number whose continued fraction expansion coincides with some Sturmian sequence of positive integers, up to the initial terms. In particular, we obtain the exact exponent of approximation to such a continued fraction $\xi$ by quadratic surds.

Classification : 11J13, 11J82
Keywords: Diophantine approximation, Sturmian sequence, simultaneous approximation, transcendence measure
Mot clés : approximation diophantienne, suite sturmienne, approximation simultanée, mesure de transcendance

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