Acta Mathematica Hungarica

, Volume 139, Issue 3, pp 208–227 | Cite as

Purely periodic expansions in systems with negative base

Article
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Abstract

We study the question of pure periodicity of expansions in the negative base numeration system. In analogy of Akiyama’s result for positive Pisot unit base β, we find a sufficient condition so that there exist an interval J containing the origin such that the (−β)-expansion of every rational number from J is purely periodic. We focus on the case of quadratic bases and demonstrate the following difference between the negative and positive bases: It is known that the finiteness property (\(\mathop {\mathrm {Fin}}\nolimits \, (\beta)= {\mathbb {Z}}[\beta]\)) is not only sufficient, but also necessary in the case of positive quadratic and cubic bases. We show that \(\mathop {\mathrm {Fin}}\nolimits \, (-\beta)= {\mathbb {Z}}[\beta]\) is not necessary in the case of negative bases.

Key words and phrases

negative base periodic expansions Pisot numbers 

Mathematics Subject Classification

11K16 11A63 
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References

  1. [1]
    B. Adamczewski, Ch. Frougny, A. Siegel and W. Steiner, Rational numbers with purely periodic beta-expansion, Bull. Lond. Math. Soc., 42 (2010), 538–552. MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    S. Akiyama, Cubic Pisot units with finite beta expansions, in: F. Halter-Koch and R. F. Tichy (Eds.), Algebraic Number Theory and Diophantine Analysis, de Gruyter (2000), pp. 11–26. Google Scholar
  3. [3]
    S. Akiyama, Pisot numbers and greedy algorithm, in: Number Theory (Eger, 1996), de Gruyter (Berlin, 1998), pp. 9–21. Google Scholar
  4. [4]
    S. Akiyama, G. Barat, V. Berthé and A. Siegel, Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions, Monatsh. Math., 155 (2008), 377–419. MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    P. Ambrož, D. Dombek, Z. Masáková and E. Pelantová, Numbers with integer expansion in the numeration system with negative base, to appear in Funct. Approx. (2012), http://arxiv.org/abs/0912.4597.
  6. [6]
    D. Dombek, Substitutions over infinite alphabet generating (−β)-integers, in: Proceedings WORDS 2011, EPTCS 63, (2011), pp. 115–121. Google Scholar
  7. [7]
    M. Hama and T. Imahashi, Periodic β-expansions for certain classes of Pisot numbers, Comment. Math. Univ. St. Paul., 46 (1997), 103–116. MathSciNetMATHGoogle Scholar
  8. [8]
    S. Ito and T. Sadahiro, (−β)-expansions of real numbers, Integers, 9 (2009), 239–259. MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Ch. Frougny and A. C. Lai, On negative bases, in: Proceedings of DLT 09, Lectures Notes in Computer Science (2009), p. 5583. Google Scholar
  10. [10]
    L. Liao and W. Steiner, Dynamical properties of the negative beta transformation, to appear in Ergod. Theor. Dyn. Syst. (2012). Google Scholar
  11. [11]
    Z. Masáková and E. Pelantová, Ito-Sadahiro numbers vs. Parry numbers, Acta Polytechnica, 51 (2011), 59–64. Google Scholar
  12. [12]
    Z. Masáková, E. Pelantová and T. Vávra, Arithmetics in number systems with negative base, Theor. Comp. Sci., 412 (2011), 835–845. MATHCrossRefGoogle Scholar
  13. [13]
    Z. Masáková and T. Vávra, Arithmetics in number systems with negative quadratic base, Kybernetika, 47 (2011), 74–92. MathSciNetMATHGoogle Scholar
  14. [14]
    W. Parry, On the β-expansions of real numbers, Acta Math. Acad. Sci. Hung., 11 (1960), 401–416. MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung., 8 (1957), 477–493. MATHCrossRefGoogle Scholar
  16. [16]
    K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers, Bull. Lond. Math. Soc., 12 (1980), 269–278. MATHCrossRefGoogle Scholar
  17. [17]
    W. Steiner, On the Delone property of (−β)-integers, in: Proceedings WORDS 2011, EPTCS 63 (2011), pp. 247–256. Google Scholar
  18. [18]
    W. Steiner, On the structure of (−β)-integers, RAIRO – Theor. Inf. Appl., 46 (2012), 181–200. MATHCrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2012

Authors and Affiliations

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

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