Duke's Theorem and Continued Fractions

TL;DR

This paper demonstrates that the continued fraction digits of square roots of integers with bounded class number converge to the Gauss-Kuzmin distribution as the integer grows large, extending known results for random reals.

Contribution

It establishes convergence of continued fraction digits of quadratic irrationals to the Gauss-Kuzmin distribution using geodesic flow properties.

Findings

Digits of $ oot{d} $ converge to Gauss-Kuzmin distribution as $d o $

Uses properties of geodesic flow on modular surface

Extends classical results to algebraic irrationals with bounded class number

Abstract

For uniformly chosen random $\alpha \in [0,1]$, it is known the probability the $n^{\rm th}$ digit of the continued-fraction expansion, $[\alpha]_n$ converges to the Gauss-Kuzmin distribution $\mathbb{P}([\alpha]_n = k) \approx \log_2 (1 + 1/ k(k+2))$ as $n \to \infty$. In this paper, we show the continued fraction digits of $\sqrt{d}$, which are eventually periodic, also converge to the Gauss-Kuzmin distribution as $d \to \infty$ with bounded class number, $h(d)$. The proof uses properties of the geodesic flow in the unit tangent bundle of the modular surface, $T^1(\text{SL}_2 \mathbb{Z}\backslash \mathbb{H})$.