Beta-expansions of p-adic numbers

About the paper

coauthors Klaus Scheicher
Víctor F. Sirvent
language English
published in Ergodic Theory and Dynamical Systems 38, No. 3 (2016)
pages 924 to 943
DOI 10.1017/etds.2014.84

Abstract

In the present article, we introduce beta-expansions in the ring Zp of p-adic integers. We characterise the sets of numbers with eventually periodic and finite expansions.

References

  1. S. Akiyama, Pisot numbers and greedy algorithm., Number theory (Eger, 1996), 9—21, de Gruyter, Berlin, 1998.
  2. S. Akiyama, T. Borbély, H. Brunotte, A. Pethő, J. M. Thuswaldner, Generalized radix representations and dynamical systems I, Acta Math. Hungar. 108 (2005), 207—238.
  3. S. Akiyama, H. Brunotte, A. Pethő, J. M. Thuswaldner, Generalized radix representations and dynamical systems II, Acta Arith. 121 (2006), 21—61.
  4. S. Akiyama, H. Brunotte, A. Pethő, J. M. Thuswaldner, Generalized radix representations and dynamical systems III, Osaka J. Math. 45 (2008), 347—374.
  5. S. Akiyama, H. Brunotte, A. Pethő, J. M. Thuswaldner, Generalized radix representations and dynamical systems IV, Indag. Math. (N.S.) 19 (2008), 333—348.
  6. S. Akiyama, H. Rao, W. Steiner, A certain finiteness property of Pisot number systems, J. Number Theory 107 (2004) 135—160.
  7. S. Akiyama, K. Scheicher, Symmetric shift radix systems and finite expansions, Math. Pannon. 18 (2007), 101—124.
  8. A. Bertrand, Développements en base de Pisot et répartition modulo 1, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), A419—A421.
  9. D. W. Boyd, Salem numbers of degree four have periodic expansions, Théorie des nombres (Quebec, PQ, 1987), 57—64, de Gruyter, Berlin, 1989
  10. D. W. Boyd, On the beta expansion for Salem numbers of degree 6, Math. Comp. 65 (1996), 861—875, S29—S31.
  11. H. Brunotte, P. Kirschenhofer, J. M. Thuswaldner, Contractivity of three-dimensional shift radix systems with finiteness property, J. Difference Equ. Appl. 18 (2012), 1077—1099.
  12. C. Frougny, B. Solomyak, Finite beta-expansions, Ergod. Th. and Dynam. Sys. 12 (1992), 713—723.
  13. M. Hbaib, M. Mkaouar, Sur le bêta-développement de 1 dans le corps des séries formelles, Int. J. Number Theory 2 (2006), 365—378.
  14. A. Huszti, K. Scheicher, P. Surer, J. M. Thuswaldner, Three-dimensional symmetric shift radix systems, Acta Arith., 129 (2007), 147—166.
  15. P. Kirschenhofer, J. M. Thuswaldner, Shift radix systems - a survey, RIMS Kôkyûroku Bessatsu, to appear.
  16. J. Neukirch, Algebraic number theory, volume 322 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999.
  17. W. Parry, On the β-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401—416.
  18. A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477—493.
  19. K. Scheicher, β-expansions in algebraic function fields over finite fields, Finite Fields Appl. 13 (2007), 394—410.
  20. K. Scheicher, P. Surer, J. M. Thuswaldner, C. van de Woestijne, Digit systems over commutative rings, Int. J. Number Theory 10 (2014), 1459—1483.
  21. K. Scheicher, J. M. Thuswaldner, Digit systems in polynomial rings over finite fields, Finite Fields Appl. 9 (2003), 322—333.
  22. K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc., 12 (1980), 269—278.
  23. P. Surer, Characterisation results for shift radix systems, Math. Pannon., 18 (2007), 265—297.
  24. M. Weitzer, Characterization algorithms for shift ratix systems with finiteness property, submitted.

Links

Ergodic Theory and Dynamical Systems
Preprint on arXiv.org