Acta Mathematica Hungarica

, Volume 152, Issue 2, pp 485–504 | Cite as

Finite beta-expansions with negative bases

Article

Abstract

The finiteness property is an important arithmetical property of beta-expansions. We exhibit classes of Pisot numbers β having the negative finiteness property, that is the set of finite (−β)-expansions is equal to \({\mathbb{Z}[\beta^{-1}]}\). For a class of numbers including the Tribonacci number, we compute the maximal length of the fractional parts arising in the addition and subtraction of (−β)-integers. We also give conditions excluding the negative finiteness property.

Mathematics Subject Classification

11A63 11K16 

Key words and phrases

beta-expansion finiteness shift radix system 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. Akiyama, Cubic Pisot units with finite beta expansions, in: Algebraic number theory and Diophantine analysis (Graz, 1998), de Gruyter (Berlin, 2000), pp. 11–26.Google Scholar
  2. 2.
    Akiyama S., Borbély T., Brunotte H., Pethő A., Thuswaldner J.: Generalized radix representations and dynamical systems. I. Acta Math. Hungar., 108, 207–238 (2005)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Akiyama S., Brunotte H., Pethő A., Thuswaldner J.: Generalized radix representations and dynamical systems. II. Acta Arith 121, 21–61 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Akiyama S., Scheicher K.: Symmetric shift radix systems and finite expansions. Math. Pannon. 18, 101–124 (2007)MathSciNetMATHGoogle Scholar
  5. 5.
    Ambrož P., Frougny Ch., Masáková Z., Pelantová E.: Arithmetics on number systems with irrational bases. Bull. Soc. Math. Belg. 10, 641–659 (2003)MathSciNetMATHGoogle Scholar
  6. 6.
    J. Bernat, Computation of \({L_{\oplus}}\) for several cubic Pisot numbers, Discrete Math. Theor. Comput. Sci., 9 (2007), 175–193.Google Scholar
  7. 7.
    H. Brunotte, Symmetric CNS trinomials, Integers, 9 (2009), A19, 201–214.Google Scholar
  8. 8.
    Dombek D., Masáková Z., Vávra T.: Confluent Parry numbers, their spectra, and integers in positive- and negative-base number systems. J. Théor. Nombres Bordeaux, 27, 745–768 (2015)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    K. Dajani and Ch. Kalle, Transformations generating negative β-expansions, Integers, 11B (2011), paper No. A5.Google Scholar
  10. 10.
    Dammak S., Hbaib M.: Number systems with negative bases. Acta Math. Hungar. 142, 475–483 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Frougny Ch.: Confluent linear numeration systems. Theor. Comput. Sci. 106, 183–219 (1992)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Frougny Ch., Lai A. C.: Negative bases and automata. Discrete Math. Theor. Comput. Sci. 13, 75–93 (2011)MathSciNetMATHGoogle Scholar
  13. 13.
    Frougny Ch., Solomyak B.: Finite beta-expansions. Ergodic Theory Dynam. Systems, 12, 713–723 (1992)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Grabner P. J., Pethő A., Tichy R. F., Wöginger G. J.: Associativity of recurrence multiplication. Appl. Math. Letters, 7, 85–90 (1994)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hejda T., Pelantová E.: Spectral properties of cubic complex pisot units. Math. Comp., 85, 401–421 (2016)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    M. Hollander, Linear numeration systems, finite beta expansions, and discrete spectrum of substitution dynamical systems, Ph.D. Thesis, University of Washington (1996).Google Scholar
  17. 17.
    Ito S., Sadahiro T.: Beta-expansions with negative bases. Integers, 9(239–259), 9 239–259 (2009)MathSciNetMATHGoogle Scholar
  18. 18.
    P. Kirschenhofer and J. M. Thuswaldner, Shift radix systems—a survey, in: Numeration and Substitution 2012, RIMS Kôkyûroku Bessatsu, B46, Res. Inst. Math. Sci. (RIMS) (Kyoto, 2014), pp. 1–59.Google Scholar
  19. 19.
    Masáková Z., Pelantová E.: Purely periodic expansions in systems with negative base. Acta Math. Hungar., 139, 208–227 (2013)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Masáková Z., Pelantová E., Vávra T.: Arithmetics in number systems with negative base. Theor. Comp. Sci., 412, 835–845 (2011)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Messaoudi A.: Tribonacci multiplication. Appl. Math. Lett., 15, 981–985 (2002)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Parry W.: On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar., 11, 401–416 (1960)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Rényi A.: Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar., 8, 477–493 (1957)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    P. Surer, ε-shift radix systems and radix representations with shifted digit sets, Publ. Math. Debrecen., 74 (2009), 19–43.Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2017

Authors and Affiliations

  1. 1.Department of Mathematics FNSPECzech Technical University in PraguePraha 2Czech Republic
  2. 2.IRIF, CNRS UMR 8243Université Paris Diderot – Paris 7Paris Cedex 13France

Personalised recommendations