Duke's Theorem and Continued Fractions

Abstract

For uniformly chosen random $\alpha \in [0,1]$, it is known the probability the $n^{\mathrm{th}}$ 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, $T^1({\text{SL}}_2\mathbb{Z}\backslash \mathbb{H})$.

9 pages

References in corpus (1)

• Sparse equidistribution problems, period bounds, and subconvexity