On the rational approximation to the Thue–Morse–Mahler numbers

Bugeaud, Yann

doi:10.5802/aif.2666

On the rational approximation to the Thue–Morse–Mahler numbers
[Sur l’approximation rationnelle des nombres de Thue–Morse–Mahler]

Bugeaud, Yann ¹

¹ Université de Strasbourg Département de Mathématiques 7, rue René Descartes 67084 STRASBOURG (France)

Annales de l'Institut Fourier, Tome 61 (2011) no. 5, pp. 2065-2076.

Résumé
Abstract

Soit ${(t_{k})}_{k \geq 0}$ la suite de Thue–Morse définie sur ${0, 1}$ par $t_{0} = 0$ , $t_{2 k} = t_{k}$ et $t_{2 k + 1} = 1 - t_{k}$ pour $k \geq 0$ . Soit $b \geq 2$ un entier rationnel. Nous démontrons que l’exposant d’irrationalité du nombre de Thue–Morse–Mahler $\sum_{k \geq 0} t_{k} b^{- k}$ est égal à $2$ .

Let ${(t_{k})}_{k \geq 0}$ be the Thue–Morse sequence on ${0, 1}$ defined by $t_{0} = 0$ , $t_{2 k} = t_{k}$ and $t_{2 k + 1} = 1 - t_{k}$ for $k \geq 0$ . Let $b \geq 2$ be an integer. We establish that the irrationality exponent of the Thue–Morse–Mahler number $\sum_{k \geq 0} t_{k} b^{- k}$ is equal to $2$ .

MR Zbl | 2 citations dans Numdam

DOI : 10.5802/aif.2666

Classification : 11J04, 11J82
Keywords: Irrationality measure, Thue–Morse sequence, Padé approximant
Mot clés : mesure d’irrationalité, suite de Thue–Morse, approximant de Padé

Affiliations des auteurs :

Bugeaud, Yann ¹

¹ Université de Strasbourg Département de Mathématiques 7, rue René Descartes 67084 STRASBOURG (France)

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     title = {On the rational approximation to the {Thue{\textendash}Morse{\textendash}Mahler} numbers},
     journal = {Annales de l'Institut Fourier},
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     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {61},
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Bugeaud, Yann. On the rational approximation to the Thue–Morse–Mahler numbers. Annales de l'Institut Fourier, Tome 61 (2011) no. 5, pp. 2065-2076. doi : 10.5802/aif.2666. http://www.numdam.org/articles/10.5802/aif.2666/

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