Letters in Mathematical Physics

, Volume 87, Issue 1–2, pp 181–195 | Cite as

Repetitions in Beta-Integers

  • L’ubomíra Balková
  • Karel Klouda
  • Edita Pelantová


Classical crystals are solid materials containing arbitrarily long periodic repetitions of a single motif. In this Letter, we study the maximal possible repetition of the same motif occurring in β-integers—one dimensional models of quasicrystals. We are interested in β-integers realizing only a finite number of distinct distances between neighboring elements. In such a case, the problem may be reformulated in terms of combinatorics on words as a study of the index of infinite words coding β-integers. We will solve a particular case for β being a quadratic non-simple Parry number.

Mathematics Subject Classification (2000)

11R06 52C23 11B05 68R15 


repetitions in words powers of factors index of infinite words beta-integers quasicrystals aperiodic structures 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Balková L’., Gazeau J.-P., Pelantová E.: Asymptotic behavior of beta-integers. Lett. Math. Phys. 84, 179–198 (2008)CrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Frougny Ch., Gazeau J.-P., Krejcar R.: Additive and multiplicative properties of point-sets based on beta-integers. Theor. Comput. Sci. 303, 491–516 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Berstel, J.: On the index of Sturmian words. In: Jewels are Forever, pp. 287–294. Springer, Heidelberg (1999)Google Scholar
  4. 4.
    Justin J., Pirillo G.: Fractional powers in Sturmian words. Theor. Comput. Sci. 255, 363–376 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Mignosi F., Pirillo G.: Repetitions in the Fibonacci infinite word. RAIRO Inf. Theor. Appl. 26, 199–204 (1992)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Masáková, Z., Pelantová, E.: Relation between powers of factors and recurrence function characterizing Sturmian words (submitted). arXiv 0809.0603v2[math.CO]Google Scholar
  7. 7.
    Carpi A., de Luca A.: Special factors, periodicity, and an apllication to Sturmian words. Acta Inf. 36, 983–1006 (2000)zbMATHCrossRefGoogle Scholar
  8. 8.
    Damanik D., Lenz D.: The index of Sturmian sequences. Eur. J. Combin. 23, 23–29 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Damanik D.: Singular continuous spectrum for a class of substitution Hamiltonians II. Lett. Math. Phys. 54, 25–31 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hof A., Knill O., Simon B.: Singular continuous spectrum for palindromic Schrödinger operators. Commun. Math. Phys. 1, 149–159 (1995)CrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Balková L’., Masáková Z.: Palindromic complexity of infinite words associated with non-simple Parry numbers. RAIRO Theor. Inf. Appl. 43, 145–163 (2009)zbMATHCrossRefGoogle Scholar
  12. 12.
    Balková L’., Pelantová E., Turek O.: Combinatorial and arithmetical properties of infinite words associated with quadratic non-simple Parry numbers. RAIRO Theor. Inf. Appl. 41, 307–328 (2007)zbMATHCrossRefGoogle Scholar
  13. 13.
    Hedlund G.A., Morse M.: Symbolic dynamics. Am. J. Math. 60, 815–866 (1938)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Berstel, J.: Sturmian and episturmian words (a survey of some recent results). In: Bozapalidis, S., Rahonis, G. (eds.) Conference on Algebraic Informatics, Thessaloniki. Lecture Notes Comput. Sci., vol. 4728, pp. 23–47 (2007)Google Scholar
  15. 15.
    Lothaire M.: Algebraic combinatorics on words. Encyclopedia of Mathematics and its Applications, vol. 90. Cambridge University Press, London (2002)Google Scholar
  16. 16.
    Minc H.: Nonnegative Matrices. Wiley, New York (1988)zbMATHGoogle Scholar
  17. 17.
    Queffélec M.: Substitution dynamical systems—spectral analysis. Lecture Notes in Mathematics. Springer, Heidelberg (1987)Google Scholar
  18. 18.
    Fabre S.: Substitutions et β-systèmes de numération. Theor. Comput. Sci. 137, 219–236 (1995)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer 2009

Authors and Affiliations

  • L’ubomíra Balková
    • 1
  • Karel Klouda
    • 1
  • Edita Pelantová
    • 1
  1. 1.Department of Mathematics, Doppler Institute, FNSPECzech Technical University in PraguePraha 2Czech Republic

Personalised recommendations