Acta Mathematica Hungarica

, Volume 153, Issue 2, pp 318–333 | Cite as

Finiteness in real real cubic fields

  • Z. MasákováEmail author
  • M. Tinková


We study finiteness property in numeration systems with cubic Pisot unit base. A base β > 1 is said to satisfy property (F), if the set Fin (β) of numbers with finite β-expansions forms a ring. We show that in every real cubic field which is not totally real, there exists a cubic Pisot unit satisfying (F). On the other hand, there exist totally real cubic fields without such a unit. In such fields, however, one finds a cubic Pisot unit β > 1 satisfying property (−F), i.e., the set Fin (−β) of finite (−β)-expansions forms a ring.

Key words and phrases

beta-expansion finiteness Pisot unit cubic field 

Mathematics Subject Classification

11A63 11R16 


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  1. 1.
    S. Akiyama, Positive finiteness of number systems, in: Number Theory: Tradition and Modernization, ed. by W. Zhang and Y. Tanigawa, Developments in Mathematics, 15, Springer (New York, 2006), pp. 1–10.Google Scholar
  2. 2.
    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
  3. 3.
    M.-J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse and J.-P. Schreiber, Pisot and Salem Numbers, Birkhäuser Verlag (Basel, 1992).Google Scholar
  4. 4.
    A. Bertrand-Mathis, Rauzy’s tilings in the light of uniform distribution; the case + β and the case −β, in: Numeration 2016, Česká technika – nakladatelství ČVUT (Praha, 2016), 26–27.Google Scholar
  5. 5.
    Dammak S., Hbaib M.: Number systems with negative bases. Acta Math. Hungar. 142, 475–483 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Frougny C., Solomyak B.: Finite beta-expansions. Ergodic Theory Dynam. Systems, 12, 713–723 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    M. Hollander, Linear numeration systems, finite beta expansions, and discrete spectrum of substitution dynamical systems, Ph.D. Thesis, University of Washington (1996).Google Scholar
  8. 8.
    S. Ito and T. Sadahiro, Beta-expansions with negative bases, Integers, 9 (2009), A22, 239–259.Google Scholar
  9. 9.
    Krčmáriková Z., Steiner W., Vávra T.: Finite beta-expansions with negative bases. Acta Math. Hungar. 152, 485–504 (2017)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Masáková Z., Pelantová E., Vávra T.: Arithmetics in number systems with negative base. Theoret. Comp. Sci., 412, 835–845 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Rényi A.: Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar., 8, 477–493 (1957)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    P. Ribenboim, Classical Theory of Algebraic Numbers, Springer (New York, 2001).Google Scholar
  13. 13.
    R. Salem, Algebraic Numbers and Fourier Analysis, D. C. Heath and Co. (Boston, Mass., 1963).Google Scholar
  14. 14.
    Schmidt K.: On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc., 12, 269–278 (1980)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    T. Vávra and F. Veneziano, Pisot unit generators in number fields, to appear in J. Symbolic Comput., (2016).Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2017

Authors and Affiliations

  1. 1.Department of Mathematics, FNSPECzech Technical University in PraguePraha 2Czech Republic

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