On the distribution modulo 1 of the sum of powers of a Salem number

Dragan Stankov

doi:10.1016/j.crma.2016.03.012

Number theory

On the distribution modulo 1 of the sum of powers of a Salem number

Presented by: Jean-Pierre Serre

Dragan Stankov ¹

¹ Katedra Matematike RGF-a, University of Belgrade, Beograd, Đušina 7, Serbia

Comptes Rendus. Mathématique, Volume 354 (2016) no. 6, pp. 569-576.

Abstract
Résumé

It is well known that the sequence of powers of a Salem number θ, modulo 1, is dense in the unit interval, but is not uniformly distributed. Generalizing a result of Dupain, we determine, explicitly, the repartition function of the sequence ${(P (θ^{n}) \mod 1)}_{n \geq 1}$ , where P is a polynomial with integer coefficients and θ is quartic. Also, we consider some examples to illustrate the method of determination.

Il est bien connu que la suite des puissances d'un nombre de Salem θ, modulo 1, est dense dans l'intervalle unité, sans être uniformément distribuée. Généralisant un résultat de Dupain, on détermine explicitement la fonction de répartition de la suite ${(P (θ^{n}) \mod 1)}_{n \geq 1}$ , où P est un polynôme à coefficients entiers et θ est quartique. On illustre également la méthode de détermination par quelques exemples.

Received: 2016-02-01
Accepted: 2016-03-17
Published online: 2016-04-14

DOI: 10.1016/j.crma.2016.03.012

Author's affiliations:

Dragan Stankov ¹

¹ Katedra Matematike RGF-a, University of Belgrade, Beograd, Đušina 7, Serbia

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     title = {On the distribution modulo 1 of the sum of powers of a {Salem} number},
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Dragan Stankov. On the distribution modulo 1 of the sum of powers of a Salem number. Comptes Rendus. Mathématique, Volume 354 (2016) no. 6, pp. 569-576. doi : 10.1016/j.crma.2016.03.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.03.012/

References
Cited by

[1] S. Akiyama; Y. Tanigawa Salem numbers and uniform distribution modulo 1, Publ. Math. (Debr.), Volume 64 (2004) no. 3–4, pp. 329-341

[2] M.-J. Bertin; A. Decomps-Guilloux; M. Grandet-Hugot; M. Pathiaux-Delefosse; J.-P. Schreiber Pisot and Salem Numbers, Birkhäuser, Basel, Switzerland, 1992

[3] Y. Bugeaud Distribution Modulo One and Diophantine Approximation, Cambridge Tracts in Mathematics, vol. 193, Cambridge University Press, Cambridge, UK, 2012

[4] C. Doche; M. Mendès France; J.-J. Ruch Equidistribution modulo 1 and Salem numbers, Funct. Approx. Comment. Math., Volume 39 (2008) no. 2, pp. 261-271

[5] Y. Dupain Répartition et discrépance, Université Bordeaux-1, 1978 (PhD thesis)

[6] J.C. Mason; D.C. Handscomb Chebyshev Polynomials, Chapman and Hall/CRC, Boca Raton, London, New York, Washington D.C., 2003

[7] R. Salem Power series with integral coefficients, Duke Math. J., Volume 12 (1945), pp. 153-172

[8] C. Smyth Seventy years of Salem numbers, Bull. Lond. Math. Soc., Volume 47 (2015) no. 3, pp. 379-395

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