Statistics of incomplete quotients of continued fractions of quadratic irrationalities

TL;DR

This paper proves that the statistical distribution of incomplete quotients in continued fractions of quadratic irrationals aligns with the Gauss--Kuz'min distribution, supported by theoretical bounds and probabilistic analysis.

Contribution

It provides a rigorous proof of Arnold's experimental observations for roots of quadratic equations with fixed coefficients and random constant term, extending understanding of continued fraction statistics.

Findings

Limit of incomplete quotient statistics matches Gauss--Kuz'min distribution.

Average incomplete quotients are logarithmically small before taking the limit.

Upper bounds on the proportion of 'red' numbers among sums of two squares.

Abstract

V.I. Arnold has experimentally established that the limit of the statistics of incomplete quotients of partial continued fractions of quadratic irrationalities coincides with the Gauss--Kuz'min statistics. Below we briefly prove this fact for roots of the equation $r x^2+p x=q$ with fixed $p$ and $r$ ($r>0$), and with random $q$, $q\le R$, $R\to \infty$. In Section 3 we estimate the sum of incomplete quotients of the period. According to the obtained bound, prior to the passage to the limit, incomplete quotients in average are logarithmically small. We also upper estimate the proportion of the "red" numbers among those representable as a sum of two squares.