Fractional parts of powers and Sturmian words

Yann Bugeaud; Artūras Dubickas

doi:10.1016/j.crma.2005.06.007

Number Theory

Fractional parts of powers and Sturmian words
[Parties fractionnaires de puissances et mots sturmiens]

Présenté par : Jean-Pierre Kahane

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

Comptes Rendus. Mathématique, Volume 341 (2005) no. 2, pp. 69-74.

Résumé
Abstract

Soit $b ⩾ 2$ 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 ${ξ b^{n}}$ , $n ⩾ 0$ , 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 ${ξ b^{n}}$ , $n ⩾ 0$ .

Let $b ⩾ 2$ be an integer. In terms of combinatorics on words we describe all irrational numbers $ξ > 0$ with the property that the fractional parts ${ξ b^{n}}$ , $n ⩾ 0$ , 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 ${ξ b^{n}}$ , $n ⩾ 0$ , cannot all belong to an interval of length smaller than $1 / b$ .

Reçu le : 2005-05-24
Accepté le : 2005-05-31
Publié le : 2005-06-29

DOI : 10.1016/j.crma.2005.06.007

Affiliations des auteurs :

Yann Bugeaud ¹ ; Artūras Dubickas ²

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     title = {Fractional parts of powers and {Sturmian} words},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {69--74},
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     year = {2005},
     doi = {10.1016/j.crma.2005.06.007},
     language = {en},
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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/

Bibliographie
Cité par

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[2] J. Berstel; J. Karhumäki Combinatorics on words – a tutorial, Bull. EATCS, Volume 79 (2003), pp. 178-228

[3] Y. Bugeaud Linear mod one transformations and the distribution of fractional parts ${ξ {(p / q)}^{n}}$ , Acta Arith., Volume 114 (2004), pp. 301-311

[4] A. Dubickas, Arithmetical properties of powers of algebraic numbers, Bull. London Math. Soc., in press

[5] A. Dubickas, On the distance from a rational power to the nearest integer, submitted for publication

[6] A. Dubickas, Arithmetical properties of linear recurrent sequences, submitted for publication

[7] A. Dubickas, A. Novikas, Integer parts of powers of rational numbers, Math. Z., in press

[8] S. Ferenczi; Ch. Mauduit Transcendence of numbers with a low complexity expansion, J. Number Theory, Volume 67 (1997), pp. 146-161

[9] L. Flatto; J.C. Lagarias; A.D. Pollington On the range of fractional parts ${ξ {(p / q)}^{n}}$ , Acta Arith., Volume 70 (1995), pp. 125-147

[10] M. Lothaire Algebraic Combinatorics on Words, Encyclopedia Math. Appl., vol. 90, Cambridge University Press, Cambridge, 2002

[11] K. Mahler An unsolved problem on the powers of $3 / 2$ , J. Austral. Math. Soc., Volume 8 (1968), pp. 313-321

[12] M. Morse; G.A. Hedlund Symbolic dynamics II: Sturmian sequences, Amer. J. Math., Volume 62 (1940), pp. 1-42

[13] A. Schinzel, On the reduced length of a polynomial with real coefficients, submitted for publication

[14] T. Zaimi, An arithmetical property of powers of Salem numbers, submitted for publication

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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