Comptes Rendus
Number Theory
Fractional parts of powers and Sturmian words
[Parties fractionnaires de puissances et mots sturmiens]
Comptes Rendus. Mathématique, Volume 341 (2005) no. 2, pp. 69-74.

Soit b2 un entier. Au moyen de résultats de la combinatoire des mots, nous caractérisons l'ensemble des nombres réels ξ>0 tels que les parties fractionnaires {ξbn}, n0, appartiennent toutes à un intervalle semi-ouvert ou ouvert de longueur 1/b. La longueur d'un tel intervalle ne peut pas être plus petite, c'est-à-dire, quel que soit le nombre irrationnel ξ, aucun intervalle de longueur strictement inférieure à 1/b ne contient toutes les parties fractionnaires {ξbn}, n0.

Let b2 be an integer. In terms of combinatorics on words we describe all irrational numbers ξ>0 with the property that the fractional parts {ξbn}, n0, all belong to a semi-open or an open interval of length 1/b. The length of such an interval cannot be smaller, that is, for irrational ξ, the fractional parts {ξbn}, n0, cannot all belong to an interval of length smaller than 1/b.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2005.06.007

Yann Bugeaud 1 ; Artūras Dubickas 2

1 Université Louis-Pasteur, UFR de mathématiques, 7, rue René-Descartes, 67084 Strasbourg, France
2 Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225 Vilnius, Lithuania
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Yann Bugeaud; Artūras Dubickas. Fractional parts of powers and Sturmian words. Comptes Rendus. Mathématique, Volume 341 (2005) no. 2, pp. 69-74. doi : 10.1016/j.crma.2005.06.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.06.007/

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Cité par Sources :

The research of the first named author was supported by the Austrian Science Foundation FWF, grant M822-N12. The research of the second named author was partially supported by the Lithuanian State Science and Studies Foundation.

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