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

Bugeaud, Yann

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 $\{0,1\}$ 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 $\{0,1\}$ 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 $2$.

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

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