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

Yann Bugeaud[1]

• [1] Université de Strasbourg Département de Mathématiques 7, rue René Descartes 67084 STRASBOURG (France)
• Volume: 61, Issue: 5, page 2065-2076
• ISSN: 0373-0956

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## Abstract

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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 $2$.

## How to cite

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Bugeaud, Yann. "On the rational approximation to the Thue–Morse–Mahler numbers." Annales de l’institut Fourier 61.5 (2011): 2065-2076. <http://eudml.org/doc/219765>.

@article{Bugeaud2011,
abstract = {Let $(t_k)_\{k \ge 0\}$ be the Thue–Morse sequence on $\lbrace 0, 1\rbrace$ 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 $2$.},
affiliation = {Université de Strasbourg Département de Mathématiques 7, rue René Descartes 67084 STRASBOURG (France)},
author = {Bugeaud, Yann},
journal = {Annales de l’institut Fourier},
keywords = {Irrationality measure; Thue–Morse sequence; Padé approximant; irrationality measure; Thue-Morse sequence},
language = {eng},
number = {5},
pages = {2065-2076},
publisher = {Association des Annales de l’institut Fourier},
title = {On the rational approximation to the Thue–Morse–Mahler numbers},
url = {http://eudml.org/doc/219765},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Bugeaud, Yann
TI - On the rational approximation to the Thue–Morse–Mahler numbers
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 5
SP - 2065
EP - 2076
AB - Let $(t_k)_{k \ge 0}$ be the Thue–Morse sequence on $\lbrace 0, 1\rbrace$ 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 $2$.
LA - eng
KW - Irrationality measure; Thue–Morse sequence; Padé approximant; irrationality measure; Thue-Morse sequence
UR - http://eudml.org/doc/219765
ER -

## References

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