Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-24T12:58:39.015Z Has data issue: false hasContentIssue false
  • English
  • Français

On the beta expansion of Salem numbers of degree 8

Part of: Computational number theory Algebraic number theory: global fields Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  01 June 2014

Hachem Hichri
Hachem Hichri*
Affiliation:
Departement de Mathematiques (UR11ES50), Faculté des sciences de Monastir , Université de Monastir, Monastir 5019, Tunisie email [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Boyd showed that the beta expansion of Salem numbers of degree 4 were always eventually periodic. Based on an heuristic argument, Boyd had conjectured that the same is true for Salem numbers of degree 6 but not for Salem numbers of degree 8. This paper examines Salem numbers of degree 8 and collects experimental evidence in support of Boyd’s conjecture.

MSC classification

Secondary: 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. 11R06: PV-numbers and generalizations; other special algebraic numbers; Mahler measure 11Y99: None of the above, but in this section
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 17 , Issue 1 , 2014 , pp. 289 - 301
Copyright
© The Author 2014 

References

Bertrand, A., ‘Développement en base de Pisot et répartition modulo 1’, C. R. Acad. Sci. Paris Sér. I Math. 285 (1977) 419–421.Google Scholar
Boyd, D. W., ‘Salem numbers of degree four have a periodic expansions’, Théories des nombres—Number Theory (Walter de Gruyter, Berlin and New York, 1989) 57–64.Google Scholar
Boyd, D. W., ‘On the beta expansion for Salem numbers of degree 6’, Math. Comp. 65 (1996) 861–875.Google Scholar
L., Flatto, Lagarias, J. C. and Poonen, B., ‘The zeta function of the beta transformation’, Ergodic Theory Dynam. Systems 14 (1994) 237–266.Google Scholar
Frougny, C., ‘Numeration systems’, Algebraic Combinatorics on Words (ed. Lothaire, M.; Cambridge University Press, Cambridge, 1970).Google Scholar
Hare, K. G. and Tweedle, D., ‘Beta-expansions for infinite families of Pisot and Salem numbers’, J. Number Theory 128 (2008) no. 9, 2756–2765.Google Scholar
Kronecker, L., ‘Zwei satze uber gleichungen mit ganzzahligen coefficienten’, J. Reine Angew. Math. 53 (1857) 173–175.Google Scholar
Panju, M., ‘Beta expansion for regular pisot numbers’, J. Integer Seq. 14 (2011) 11–64.Google Scholar
Parry, W., ‘On the beta-expansions of real numbers’, Acta Math. Hungar. 11 (1960) 401–416.Google Scholar
Pisot, C., ‘la répartition modulo 1 et les nombres algébriques’, Ann. Sc. Norm. Super. Pisa 2 (1938) no. 7, 205–248 (Thèse Sc. Math. Paris, 1938).Google Scholar
Rényi, A., ‘Representations for real numbers and their ergodic properties’, Acta Math. Hungar. 8 (1957) 477–493; doi:10.1007/BF02020331.CrossRefGoogle Scholar
Salem, R., ‘Power series with integral coefficients’, Duke Math. J. 12 (1945) 153–172.Google Scholar
Schmidt, K., ‘On periodic expansions of Pisot numbers and Salem numbers’, Bull. Lond. Math. Soc. 12 (1980) 269–278.CrossRefGoogle Scholar
Solomyak, B., ‘Conjugates of beta-numbers and the zero-free domain for a class of analytic functions’, Proc. Lond. Math. Soc. (3) s3-68 (1994) no. 3, 477–498.Google Scholar
Zaimi, T., ‘On numbers having finite beta-expansions’, Ergodic Theory Dynam. Systems 29 (2009) 1659–1668.CrossRefGoogle Scholar
Zaimi, T., ‘On the distribution of certain pisot numbers’, Indag. Math. (N.S.) 23 (2012) 318–326.Google Scholar