Duke's Theorem and Continued Fractions
| dc.creator | Mangual, John | |
| dc.date | 2008-02-20 | |
| dc.date.accessioned | 2026-07-07T09:22:07Z | |
| dc.date.available | 2026-07-07T09:22:07Z | |
| dc.description | For uniformly chosen random $α\in [0,1]$, it is known the probability the $n^{\rm th}$ digit of the continued-fraction expansion, $[α]_n$ converges to the Gauss-Kuzmin distribution $\mathbb{P}([α]_n = k) \approx \log_2 (1 + 1/ k(k+2))$ 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, $T^1(\text{SL}_2 \mathbb{Z}\backslash \mathbb{H})$. | |
| dc.description | 9 pages | |
| dc.identifier | https://arxiv.org/abs/0802.2924 | |
| dc.identifier | http://arxiv.org/abs/0802.2924 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/155264 | |
| dc.subject | Number Theory | |
| dc.subject | Dynamical Systems | |
| dc.subject | 11J70 | |
| dc.title | Duke's Theorem and Continued Fractions | |
| dc.type | text |