Duke’s Theorem and Continued Fractions

For uniformly chosen random α ∈ [0, 1], it is known the probability the nth 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, T (SL2Z\H). 1 Continued Fractions... For any α ∈ [0, 1] we can define the continued fraction expansion in Z by repeating a two step algorithm. First a0 = α and b0 = ⌊α⌋. Now we simply repeat: ak+1 = {1/ak} bk+1 = ⌊ak⌋ (1) The end result is that α can be encoded as a sequence of integers: [b0, b1, b2, . . . ]. If α is rational then we get a finite continued fraction. What if α is the square root of a irrational number? Then we get an eventually repeating sequences of numbers bk. For example, the sequence for √ 7 is [2, 1, 1, 1, 4, 1, 1, 1, 4, . . . ] where the [1, 1, 1, 4] motif repeats forever. How can we get a purely periodic sequence? A theorem by Galois says: