A note on Farey fractions with denominators in arithmetic progressions

TL;DR

This paper investigates the distribution of numerators in differences of consecutive Farey fractions when denominators are restricted to an arithmetic progression, providing asymptotic frequency results.

Contribution

It extends previous work by deriving asymptotic frequencies for Farey fractions with denominators in arithmetic progressions, focusing on differences of consecutive fractions.

Findings

Computed asymptotic frequencies for numerator values

Extended understanding of Farey fractions with restricted denominators

Builds on prior investigations into Farey sequence properties

Abstract

As the conclusion of a line of investigation undertaken in two previous papers, we compute asymptotic frequencies for the values taken by numerators of differences of consecutive Farey fractions with denominators restricted to lie in arithmetic progression.