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

Bugeaud, Yann

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

Bugeaud, Yann

Annales de l'Institut Fourier, Tome 61 (2011), p. 2065-2076 / Harvested from Numdam

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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 $2$ .

Publié le : 2011-01-01

MR 2961848 | Zbl 1271.11074

DOI : https://doi.org/10.5802/aif.2666
Classification: 11J04, 11J82
Mots clés: mesure d’irrationalité, suite de Thue–Morse, approximant de Padé

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     author = {Bugeaud, Yann},
     title = {On the rational approximation to the Thue--Morse--Mahler numbers},
     journal = {Annales de l'Institut Fourier},
     volume = {61},
     year = {2011},
     pages = {2065-2076},
     doi = {10.5802/aif.2666},
     zbl = {1271.11074},
     mrnumber = {2961848},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2011__61_5_2065_0}
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Bugeaud, Yann. On the rational approximation to the Thue–Morse–Mahler numbers. Annales de l'Institut Fourier, Tome 61 (2011) pp. 2065-2076. doi : 10.5802/aif.2666. http://gdmltest.u-ga.fr/item/AIF_2011__61_5_2065_0/

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