# ANNALES DE L'INSTITUT FOURIER

On the rational approximation to the Thue–Morse–Mahler numbers  [ Sur l’approximation rationnelle des nombres de Thue–Morse–Mahler ]
Annales de l'Institut Fourier, Tome 61 (2011) no. 5, pp. 2065-2076.

Soit ${\left({t}_{k}\right)}_{k\ge 0}$ la suite de Thue–Morse définie sur $\left\{0,1\right\}$ par ${t}_{0}=0$, ${t}_{2k}={t}_{k}$ et ${t}_{2k+1}=1-{t}_{k}$ pour $k\ge 0$. Soit $b\ge 2$ un entier rationnel. Nous démontrons que l’exposant d’irrationalité du nombre de Thue–Morse–Mahler ${\sum }_{k\ge 0}{t}_{k}{b}^{-k}$ est égal à $2$.

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

Reçu le : 2010-04-28
Accepté le : 2010-08-31
DOI : https://doi.org/10.5802/aif.2666
Classification : 11J04,  11J82
Mots clés: mesure d’irrationalité, suite de Thue–Morse, approximant de Padé
@article{AIF_2011__61_5_2065_0,
author = {Bugeaud, Yann},
title = {On the rational approximation to the Thue--Morse--Mahler numbers},
journal = {Annales de l'Institut Fourier},
pages = {2065--2076},
publisher = {Association des Annales de l'institut Fourier},
volume = {61},
number = {5},
year = {2011},
doi = {10.5802/aif.2666},
zbl = {1271.11074},
mrnumber = {2961848},
language = {en},
url = {aif.centre-mersenne.org/item/AIF_2011__61_5_2065_0/}
}
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. https://aif.centre-mersenne.org/item/AIF_2011__61_5_2065_0/

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