On the expansions of real numbers in two integer bases
Annales de l'Institut Fourier, Volume 67 (2017) no. 5, p. 2225-2235

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.

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.

Received : 2016-01-04
Revised : 2016-09-23
Accepted : 2016-12-08
Published online : 2017-11-17
DOI : https://doi.org/10.5802/aif.3134
Classification:  11A63,  68R15
Keywords: Combinatorics on words, Sturmian word, complexity, integer base expansion, continued fraction 
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {5},
     year = {2017},
     pages = {2225-2235},
     doi = {10.5802/aif.3134},
     language = {en},
     url = {http://www.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, Volume 67 (2017) no. 5, pp. 2225-2235. doi : 10.5802/aif.3134. http://www.numdam.org/item/AIF_2017__67_5_2225_0/

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