Soient et deux entiers strictement positifs multiplicativement indépendants. Nous démontrons que les développements en base et en base d’un nombre irrationnel, vus comme des mots infinis sur les alphabets et , 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 and be multiplicatively independent positive integers. We establish that the -ary expansion and the -ary expansion of an irrational real number, viewed as infinite words on and , respectively, cannot have simultaneously a low block complexity. In particular, they cannot be both Sturmian words.
Révisé le : 2016-09-22
Accepté le : 2016-12-07
Publié le : 2017-11-16
Classification : 11A63, 68R15
Mots clés : Combinatoire des mots, mot sturmien, développement en base entière, fraction continue
@article{AIF_2017__67_5_2225_0, author = {Bugeaud, Yann and Kim, Dong Han}, title = {On the expansions of real numbers in two integer bases}, journal = {Annales de l'Institut Fourier}, pages = {2225--2235}, publisher = {Association des Annales de l'institut Fourier}, volume = {67}, number = {5}, year = {2017}, doi = {10.5802/aif.3134}, language = {en}, url = {archive.numdam.org/item/AIF_2017__67_5_2225_0/} }
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. http://archive.numdam.org/item/AIF_2017__67_5_2225_0/
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