Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T00:59:36.507Z Has data issue: false hasContentIssue false
  • English
  • Français

Beta-expansions of $p$-adic numbers

Published online by Cambridge University Press:  06 November 2014

KLAUS SCHEICHER ,
VÍCTOR F. SIRVENT  and
PAUL SURER
KLAUS SCHEICHER
Affiliation:
Institut für Mathematik, Universität für Bodenkultur, Gregor-Mendel-Straße 33, A-1180 Vienna, Austria email klaus.scheicher@boku.ac.at, me@palovsky.com
VÍCTOR F. SIRVENT
Affiliation:
Departamento de Matemáticas, Universidad Simón Bolívar, Apartado 89000, Caracas 1086-A, Venezuela email vsirvent@usb.ve
PAUL SURER
Affiliation:
Institut für Mathematik, Universität für Bodenkultur, Gregor-Mendel-Straße 33, A-1180 Vienna, Austria email klaus.scheicher@boku.ac.at, me@palovsky.com
Get access Rights & Permissions [Opens in a new window]

Abstract

In the present article, we introduce beta-expansions in the ring $\mathbb{Z}_{p}$ of $p$-adic integers. We characterise the sets of numbers with eventually periodic and finite expansions.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 3 , May 2016 , pp. 924 - 943
Copyright
© Cambridge University Press, 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Akiyama, S.. Pisot numbers and greedy algorithm. Number Theory (Eger, Hungary, 1996). de Gruyter, Berlin, 1998, pp. 9–21.Google Scholar
Akiyama, S., Borbély, T., Brunotte, H., Pethő, A. and Thuswaldner, J. M.. Generalized radix representations and dynamical systems I. Acta Math. Hungar. 108(3) (2005), 207–238.Google Scholar
Akiyama, S., Brunotte, H., Pethő, A. and Thuswaldner, J. M.. Generalized radix representations and dynamical systems II. Acta Arith. 121(1) (2006), 21–61.CrossRefGoogle Scholar
Akiyama, S., Brunotte, H., Pethő, A. and Thuswaldner, J. M.. Generalized radix representations and dynamical systems III. Osaka J. Math. 45(2) (2008), 347–374.Google Scholar
Akiyama, S., Brunotte, H., Pethő, A. and Thuswaldner, J. M.. Generalized radix representations and dynamical systems IV. Indag. Math. (N.S.) 19(3) (2008), 333–348.Google Scholar
Akiyama, S., Rao, H. and Steiner, W.. A certain finiteness property of Pisot number systems. J. Number Theory 107(1) (2004), 135–160.CrossRefGoogle Scholar
Akiyama, S. and Scheicher, K.. Symmetric shift radix systems and finite expansions. Math. Pannon. 18(1) (2007), 101–124.Google Scholar
Bertrand, A.. Développements en base de Pisot et répartition modulo 1. C. R. Acad. Sci. Paris Sér. A–B 285(6) (1977), A419–A421.Google Scholar
Boyd, D. W.. Salem numbers of degree four have periodic expansions. Théorie des nombres (Quebec, PQ, 1987). 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(214) (1996), 861–875.Google Scholar
Brunotte, H., Kirschenhofer, P. and Thuswaldner, J. M.. Contractivity of three-dimensional shift radix systems with finiteness property. J. Difference Equ. Appl. 18(6) (2012), 1077–1099.CrossRefGoogle Scholar
Frougny, C. and Solomyak, B.. Finite beta-expansions. Ergod. Th. & Dynam. Sys. 12(4) (1992), 713–723.CrossRefGoogle Scholar
Hbaib, M. and Mkaouar, M.. Sur le bêta-développement de 1 dans le corps des séries formelles. Int. J. Number Theory 2(3) (2006), 365–378.Google Scholar
Huszti, A., Scheicher, K., Surer, P. and Thuswaldner, J. M.. Three-dimensional symmetric shift radix systems. Acta Arith. 129(2) (2007), 147–166.CrossRefGoogle Scholar
Kirschenhofer, P. and Thuswaldner, J. M.. Shift radix systems—a survey. RIMS Kôkyôroku Bessatsu, to appear; Preprint arXiv:1312.0386.Google Scholar
Meyer, Y.. Algebraic Numbers and Harmonic Analysis (North-Holland Mathematical Library, 2). North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1972.Google Scholar
Neukirch, J.. Algebraic Number Theory (Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 322). Springer, Berlin, 1999.CrossRefGoogle Scholar
Parry, W.. On the 𝛽-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416.Google Scholar
Rényi, A.. Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar 8 (1957), 477–493.Google Scholar
Scheicher, K.. 𝛽-expansions in algebraic function fields over finite fields. Finite Fields Appl. 13(2) (2007), 394–410.CrossRefGoogle Scholar
Scheicher, K., Surer, P., Thuswaldner, J. M. and Van de Woestijne, C. E.. Digit systems over commutative rings. Int. J. Number Theory 10(6) (2014), 1459–1483.CrossRefGoogle Scholar
Scheicher, K. and Thuswaldner, J. M.. Digit systems in polynomial rings over finite fields. Finite Fields Appl. 9(3) (2003), 322–333.Google Scholar
Schmidt, K.. On periodic expansions of Pisot numbers and Salem numbers. Bull. Lond. Math. Soc. 12(4) (1980), 269–278.CrossRefGoogle Scholar
Surer, P.. Characterisation results for shift radix systems. Math. Pannon. 18(2) (2007), 265–297.Google Scholar
Weitzer, M.. Characterization algorithms for shift radix systems with finiteness property. Int. J. Number Theory, to appear, arXiv:401.5259.Google Scholar