Numerators of differences of nonconsecutive Farey fractions

TL;DR

This paper investigates the numerators of differences between nonconsecutive Farey fractions, establishing algebraic identities and calculating their average values using measure-preserving transformations of the Farey triangle.

Contribution

It introduces new algebraic identities for differences of nonconsecutive Farey fractions and computes their average values through a novel measure-theoretic approach.

Findings

Derived algebraic identities for nonconsecutive Farey fraction differences

Calculated average values of these differences using measure-preserving transformations

Extended understanding of Farey fractions beyond consecutive cases

Abstract

An elementary but useful fact is that the numerator of the difference of two consecutive Farey fractions is equal to one. For triples of consecutive fractions the numerators of the differences are well understood and have applications to several interesting problems. In this paper we investigate numerators of differences of fractions which are farther apart. We establish algebraic identities between such differences which then allow us to calculate their average values by using properties of a measure preserving transformation of the Farey triangle.