Journal d'Analyse Mathématique

, Volume 134, Issue 1, pp 335–397 | Cite as

On the evolution of continued fractions in a fixed quadratic field

  • Menny Aka
  • Uri Shapira


We prove that the statistics of the period of the continued fraction expansion of certain sequences of quadratic irrationals from a fixed quadratic field approach the ‘normal’ statistics given by the Gauss-Kuzmin measure. As a byproduct, the growth rate of the period is analyzed and, for example, it is shown that for a fixed integer k and a quadratic irrational α, the length of the period of the continued fraction expansion of k n α equals ck n + o(k15n/16) for some positive constant c. This improves results of Cohn, Lagarias, and Grisel, and settles a conjecture of Hickerson. The results are derived from the main theorem of the paper, which establishes an equidistribution result regarding single periodic geodesics along certain paths in the Hecke graph. The results are effective and give rates of convergence and the main tools are spectral gap (effective decay of matrix coefficients) and dynamical analysis on S-arithmetic homogeneous spaces.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Arn07]
    V. I. Arnol′d, Continued fractions of square roots of rational numbers and their statistics, Uspekhi Mat. Nauk 62 (2007), 3–14.MathSciNetCrossRefGoogle Scholar
  2. [Arn08]
    V. I. Arnol′d, Statistics of the periods of continued fractions for quadratic irrationals, Izv. Ross. Akad. Nauk Ser. Mat. 72, 3–38.Google Scholar
  3. [Art82]
    E. Artin, Collected Papers, Springer-Verlag, New York, 1982.zbMATHGoogle Scholar
  4. [BL05]
    Y. Bugeaud and F. Luca, On the period of the continued fraction expansion of \(\sqrt {{2^{2n + 1}} + 1} \), Indag. Math. (N. S.) 16 (2005), 21–35.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [BO07]
    Y. Benoist and H. Oh, Equidistribution of rational matrices in their conjugacy classes, Geom. Funct. Anal. 17 (2007), 1–32.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [Coh77]
    J. H. E. Cohn, The length of the period of the simple continued fraction of d 1/2, Pacific J. Math. 71 (1977), 21–32.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [CZ04]
    P. Corvaja and U. Zannier, On the rational approximations to the powers of an algebraic number: solution of two problems of Mahler and Mendès France, Acta Math. 193 (2004), 175–191.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [Dir56]
    P. G. L. Dirichlet, Une propriété des formes quadratiques à déterminant positif, J. Math. Pures Appl. (1856), 76–79.Google Scholar
  9. [EW11]
    M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, Springer-Verlag London Ltd., London, 2011.zbMATHGoogle Scholar
  10. [FK10]
    É. Fouvry and J. Klüners, On the negative Pell equation, Ann. of Math. (2) 172 (2010), 2035–2104.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [Gol02]
    E. P. Golubeva, On the class numbers of indefinite binary quadratic forms of discriminant dp 2, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 286 (2002), no. Anal. Teor. Chisel i Teor. Funkts. 18, 40–47, 227–228.Google Scholar
  12. [Gri98]
    G. Grisel, Length of continued fractions in principal quadratic fields, Acta Arith. 85 (1998), 35–49.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [Hic73]
    D. R. Hickerson, Length of period simple continued fraction expansion of √d, Pacific J. Math. 46 (1973), 429–432.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [Kei]
    M. Keith, On the continued fraction expansion of \(\sqrt {{2^{2n + 1}}} \), Unpublished, available on Scholar
  15. [Kim03]
    H. H. Kim, Functoriality for the exterior square of GL 4 and the symmetric fourth of GL 2, J. Amer. Math. Soc. 16 (2003), 139–183.MathSciNetCrossRefGoogle Scholar
  16. [Lag80]
    J. C. Lagarias, On the computational complexity of determining the solvability or unsolvability of the equation X 2DY 2 = −1, Trans. Amer. Math. Soc. 260 (1980), 485–508.MathSciNetzbMATHGoogle Scholar
  17. [Ler10]
    E. Y. Lerner, About statistics of periods of continued fractions of quadratic irrationalities, Funct. Anal. Other Math. 3 (2010), 75–83.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [Mar04]
    G. A. Margulis, On Some Aspects of the Theory of Anosov Systems, Springer-Verlag, Berlin, 2004.CrossRefzbMATHGoogle Scholar
  19. [McM09]
    C. T. McMullen, Uniformly Diophantine numbers in a fixed real quadratic field, Compos. Math. 145 (2009), 827–844.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [MF93]
    M. Mendès-France, Remarks and problems on finite and periodic continued fractions, Enseign. Math. (2) 39 (1993), 249–257.MathSciNetzbMATHGoogle Scholar
  21. [Pol86]
    M. Pollicott, Distribution of closed geodesics on the modular surface and quadratic irrationals, Bull. Soc. Math. France 114 (1986), 431–446.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [PR94]
    V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Academic Press Inc., Boston, MA, 1994.zbMATHGoogle Scholar
  23. [RS92]
    A. M. Rockett and P. Szüsz, Continued Fractions, World Scientific Publishing Co. Inc., River Edge, NJ, 1992.CrossRefzbMATHGoogle Scholar
  24. [Ser85]
    C. Series, The modular surface and continued fractions, J. London Math. Soc. (2) 31 (1985), 69–80.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [Ven10]
    A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity, Ann. of Math. (2) 172 (2010), 989–1094.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2018

Authors and Affiliations

  1. 1.Departement MathematikETH ZurichZurichSwitzerland
  2. 2.Department of MathematicsTechnionHaifaIsrael

Personalised recommendations