Comptes Rendus
Number Theory
Fractional parts of powers and Sturmian words
Comptes Rendus. Mathématique, Volume 341 (2005) no. 2, pp. 69-74.

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.

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.

Received:
Accepted:
Published online:
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/

[1] S. Akiyama, Ch. Frougny, J. Sakarovitch, On number representation in a rational base, submitted for publication

[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

Cited by 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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