Duke's Theorem and Continued Fractions

Details Duke's Theorem and Continued Fractions Duke's Theorem and Continued Fractions

2008-02-20

arxiv.org/abs/0802.2924v1

Mathematics

Number Theory

9 pages

Scientific paper

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, $T^1(\text{SL}_2 \mathbb{Z}\backslash \mathbb{H})$.

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