no. 5
On the expansions of real numbers in two integer bases
[Sur le développement des nombres réels en deux bases entières]
Bugeaud, Yann1 ; Kim, Dong Han2
1 Université de Strasbourg, CNRS IRMA, UMR 7501 7 rue René Descartes 67084 Strasbourg (France)
2 Dongguk University – Seoul Department of Mathematics Education 30 Pildong-ro 1-gil, Jung-gu Seoul 04620 (Korea)
Annales de l'Institut Fourier, Tome 67 (2017) no. 5, pp. 2225-2235.

Soient r et s deux entiers strictement positifs multiplicativement indépendants. Nous démontrons que les développements en base r et en base s d’un nombre irrationnel, vus comme des mots infinis sur les alphabets {0,1,...,r-1} et {0,1,...,s-1}, respectivement, ne peuvent pas avoir simultanément une trop faible complexité par blocs. En particulier, au plus l’un d’eux est un mot sturmien.

Let r and s be multiplicatively independent positive integers. We establish that the r-ary expansion and the s-ary expansion of an irrational real number, viewed as infinite words on {0,1,...,r-1} and {0,1,...,s-1}, respectively, cannot have simultaneously a low block complexity. In particular, they cannot be both Sturmian words.

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Révisé le :
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DOI : 10.5802/aif.3134
Classification : 11A63, 68R15
Keywords: Combinatorics on words, Sturmian word, complexity, integer base expansion, continued fraction
Mot clés : Combinatoire des mots, mot sturmien, développement en base entière, fraction continue

Bugeaud, Yann 1 ; Kim, Dong Han 2

1 Université de Strasbourg, CNRS IRMA, UMR 7501 7 rue René Descartes 67084 Strasbourg (France)
2 Dongguk University – Seoul Department of Mathematics Education 30 Pildong-ro 1-gil, Jung-gu Seoul 04620 (Korea)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Bugeaud, Yann; Kim, Dong Han. On the expansions of real numbers in two integer bases. Annales de l'Institut Fourier, Tome 67 (2017) no. 5, pp. 2225-2235. doi : 10.5802/aif.3134. https://aif.centre-mersenne.org/articles/10.5802/aif.3134/

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