On the evolution of continued fractions in a fixed quadratic field

TL;DR

This paper proves that the continued fraction periods of quadratic irrationals in a fixed quadratic field follow the Gauss-Kuzmin distribution, analyzing their growth and establishing equidistribution results using spectral gap and dynamical methods.

Contribution

It introduces an effective equidistribution theorem for geodesics in the Hecke graph, improving previous results and settling a conjecture on period growth rates.

Findings

Periods of continued fractions follow Gauss-Kuzmin statistics

Growth rate of periods is approximately c k^n

Effective convergence rates are established

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 by-product, the growth rate of the period is analyzed and, for example, it is shown that for a fixed integer $k$ and a quadratic irrational $\al$, the length of the period of the continued fraction expansion of $k^n\al$ equals $c k^n +o(k^{(1-\frac{1}{16})n})$ 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.