About statistics of periods of continued fractions of quadratic irrationalities

Abstract

In this paper we answer certain questions posed by V.I. Arnold, namely, we study periods of continued fractions for solutions of quadratic equations in the form x 2 + px = q with integer p and q, p 2 + q 2 ≤ R 2 . We prove a weak variant of Arnold conjectures about the Gauss-Kuzmin statistics with R → ∞. Statement and discussion of obtained resultsAny value x ∈ R is representable as a continued fraction (CF)where a 0 ∈ Z (if a 0 = 0, then we omit it), a i ∈ N for all i ≥ 1. Let i be a fixed position in the CF expansion and let P i (k) stand for the probability that a i = k with x randomly chosen in the segment [0, 1). Hereinafter we understand the random choice as a realization of the uniform distribution. It is well known that according to the Gauss-Kuzmin theorem,The roots of the quadratic equation x 2 + px = q, where p, q ∈ Z, = p 2 + 4q > 0, are expandable in a periodic CF. Let x + (p, q) = √ /2 − p/2. The fractional part of this value

On the evolution of continued fractions in a fixed quadratic field

“…Although the question of the evolution of the c.f.e along arithmetically defined sequences in a fixed quadratic field is extremely natural, we did not find too many relevant papers to cite. Some earlier works studying the statistics of the period 'in average' (and also not in a fixed field), were initiated by Arnold (see [Arn08], [Arn07], [Ler10] and the references therein). See also [Pol86].…”