Bias in the number of steps in the Euclidean algorithm and a conjecture of Ito on Dedekind sums

TL;DR

This paper analyzes the average number of steps in variants of the Euclidean algorithm over Farey fractions, revealing different behaviors on subintervals and proving a conjecture of Ito related to Dedekind sums.

Contribution

It provides asymptotic formulas for Euclidean algorithm steps over Farey fractions and proves Ito's conjecture on Dedekind sums distribution.

Findings

Different asymptotic behaviors on (0,1/2) and (1/2,1) intervals.

Established links between Euclidean algorithm steps and continued fraction distributions.

Proved Ito's conjecture on Dedekind sums.

Abstract

We investigate the number of steps taken by three variants of the Euclidean algorithm on average over Farey fractions. We show asymptotic formulae for these averages restricted to the interval $(0,1/2)$, establishing that they behave differently on $(0,1/2)$ than they do on $(1/2,1)$. These results are tightly linked with the distribution of lengths of certain continued fraction expansions as well as the distribution of the involved partial quotients. As an application, we prove a conjecture of Ito on the distribution of values of Dedekind sums. The main argument is based on earlier work of Zhabitskaya, Ustinov, Bykovski\u{i} and others, ultimately dating back to Heilbronn, relating the quantities in question to counting solutions to a certain system of Diophantine inequalities. The above restriction to only half of the Farey fractions introduces additional complications.