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On the evolution of continued fractions in a fixed quadratic field

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Abstract

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 knα equals ckn + 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.

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Author information

Authors and Affiliations

  1. Departement Mathematik, ETH Zurich, Ramistrasse 101, 8092, Zurich, Switzerland

    Menny Aka

  2. Department of Mathematics, Technion, Haifa, Israel

    Uri Shapira

Authors
  1. Menny Aka
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  2. Uri Shapira
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Corresponding author

Correspondence to Menny Aka.

Additional information

M. A. acknowledges the support of the Advanced research Grant 226135 from the European Research Council, the ISEF foundation, the Ilan and Asaf Ramon memorial foundation, and the Hoffman Leadership and Responsibility fellowship program at the Hebrew University of Jerusalem.

U. S. acknowledges the support of the Advanced research Grant 228304 from the European Research Council.

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Aka, M., Shapira, U. On the evolution of continued fractions in a fixed quadratic field. JAMA 134, 335–397 (2018). https://doi.org/10.1007/s11854-018-0012-4

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  • DOI: https://doi.org/10.1007/s11854-018-0012-4

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