Duke's Theorem and Continued Fractions

For uniformly chosen random $\alpha \in [0,1]$, it is known the probability the $n^{\mathrm{t}\mathrm{h}}$ 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 $d\to \mathrm{\infty }$ 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 $\mathbb{H}$)$.