Duke's Theorem and Continued Fractions

2008/02/20 by Mangual, John
#11J70 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.0802.2924

Abstract

For uniformly chosen random α∈ [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 ℙ([α]_n = k) ≈ log₂ (1 + 1/ k(k+2)) as n → ∞. In this paper, we show the continued fraction digits of √(d), which are eventually periodic, also converge to the Gauss-Kuzmin distribution as d → ∞ with bounded class number, h(d). The proof uses properties of the geodesic flow in the unit tangent bundle of the modular surface, T¹(SL₂ ℤ\backslash ℍ).

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