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On the beta-expansions of 1 and algebraic numbers for a Salem number beta

Published online by Cambridge University Press:  26 February 2014

HAJIME KANEKO
HAJIME KANEKO*
Affiliation:
Department of Mathematics, College of Science and Technology, Nihon University, 1-8-14 Kanda-Surugadai, Chiyoda-ku, Tokyo 101-8308, Japan email [email protected]
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Abstract

We study the digits of $\beta $-expansions in the case where $\beta $ is a Salem number. We introduce new upper bounds for the numbers of occurrences of consecutive 0s in the expansion of 1. We also give lower bounds for the numbers of non-zero digits in the $\beta $-expansions of algebraic numbers. As applications, we give criteria for transcendence of the values of power series at certain algebraic points.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 4 , June 2015 , pp. 1243 - 1262
Copyright
© Cambridge University Press, 2014 

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References

Adamczewski, B.. Transcendance ‘à la Liouville’ de certains nombres réels. C. R. Acad. Sci. Paris 338 (2004), 511–514.CrossRefGoogle Scholar
Adamczewski, B. and Bugeaud, Y.. Dynamics for beta-shifts and Diophantine approximation. Ergod. Th. & Dynam. Syst. 27 (2007), 1695–1711.CrossRefGoogle Scholar
Adamczewski, B. and Bugeaud, Y.. On the complexity of algebraic numbers I. Expansions in integer bases. Ann. of Math. (2) 165 (2007), 547–565.CrossRefGoogle Scholar
Adamczewski, B. and Faverjon, C.. Chiffres non nuls dans le développement en base entière des nombres algébriques irrationnels. C. R. Acad. Sci. Paris 350 (2012), 1–4.CrossRefGoogle Scholar
Bailey, D. H., Borwein, J. M., Crandall, R. E. and Pomerance, C.. On the binary expansions of algebraic numbers. J. Théor. Nombres Bordeaux 16 (2004), 487–518.CrossRefGoogle Scholar
Bertrand, A.. Développements en base de Pisot et répartition modulo 1. C. R. Acad. Sci. Paris A–B 285 (1977), A419–A421.Google Scholar
Blanchard, F.. $\beta $-expansions and symbolic dynamics. Theoret. Comput. Sci. 65 (1989), 131–141.CrossRefGoogle Scholar
Borel, É.. Sur les chiffres décimaux de $\sqrt{2}$ et divers problèmes de probabilités en chaîne. C. R. Acad. Sci. Paris 230 (1950), 591–593.Google Scholar
Boyd, D. W.. Salem numbers of degree four have periodic expansions. Number Theory. Walter de Gruyter, Berlin, 1989, pp. 57–64.Google Scholar
Boyd, D. W.. On the beta expansion for Salem numbers of degree 6. Math. Comp. 65 (1996), 861–875.CrossRefGoogle Scholar
Bugeaud, Y.. Distribution modulo one and diophantine approximation (Cambridge Tracts in Mathematics, 193). Cambridge University Press, Cambridge, 2012.CrossRefGoogle Scholar
Bugeaud, Y.. On the $b$-ary expansion of an algebraic number. Rend. Semin. Mat. Univ. Padova 118 (2007), 217–233.Google Scholar
Bugeaud, Y.. On the $\beta $-expansion of an algebraic number in an algebraic base $\beta $. Integers 9 (2009), 215–226.CrossRefGoogle Scholar
Corvaja, P. and Zannier, U.. Some new applications of the subspace theorem. Compositio Math. 131 (2002), 319–340.CrossRefGoogle Scholar
Dubickas, A.. On $\beta $-expansions of unity for rational and transcendental numbers $\beta $. Math. Slovaca 61 (2011), 705–716.CrossRefGoogle Scholar
Kaneko, H.. On the binary digits of algebraic numbers. J. Aust. Math. Soc. 89 (2010), 233–244.CrossRefGoogle Scholar
Kaneko, H.. On the number of digit changes in base-$b$ expansions of algebraic numbers. Unif. Distrib. Theory 7 (2012), 141–168.Google Scholar
Liouville, J.. Remarques relatives $1^{\circ }$ à des classes très-étendues de quantités dont la valeur n’est ni rationnelle ni même réducible à des irrationnelles algébriques; $2^{\circ }$ à un passage du livre des Principes où Newton calcule l’action exercée par une sphère sur un point extérieur. C. R. Acad. Sci. Paris 18 (1844), 883–885.Google Scholar
Liouville, J.. Nouvelle démonstration d’un théorème sur les irrationnelles algébriques. C. R. Acad. Sci. Paris 18 (1844), 910–911.Google Scholar
Mahler, K.. Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen. Math. Ann. 101 (1929), 342–366.CrossRefGoogle Scholar
Nishioka, K.. Mahler Functions and Transcendence (Lecture Notes in Mathematics, 1631). Springer, Berlin, 1996.CrossRefGoogle Scholar
Parry, W.. On the $\beta $-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416.CrossRefGoogle Scholar
Rényi, A.. Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hung. 8 (1957), 477–493.CrossRefGoogle Scholar
Rivoal, T.. On the bits counting function of real numbers. J. Aust. Math. Soc. 85 (2008), 95–111.CrossRefGoogle Scholar
Schmidt, K.. On periodic expansions of Pisot and Salem numbers. Bull. Lond. Math. Soc. 12 (1980), 269–278.CrossRefGoogle Scholar
Shidlovskii, A. B.. Transcendental Numbers (Walter de Gruyter Studies in Mathematics, 12). Walter de Gruyter, Berlin, 1989.CrossRefGoogle Scholar
Verger-Gaugry, J. L.. On gaps in Rényi $\beta $-expansions of unity for $\beta >1$ an algebraic number. Ann. Inst. Fourier (Grenoble) 56 (2006), 2565–2579.CrossRefGoogle Scholar