The Distribution of Special Subsets of the Farey Sequence

TL;DR

This paper extends previous work on Farey fractions by analyzing subsets with denominators not divisible by a prime and studies the joint distribution of numerators of differences among consecutive fractions.

Contribution

It introduces new results on the distribution of Farey fractions with restricted denominators and examines the joint distribution of numerator differences in these sets.

Findings

Distribution results for Farey fractions with non-divisible denominators

Joint distribution of numerator differences in these subsets

Generalization of previous Farey sequence results

Abstract

In this paper we generalize some of our results from, `A note on Farey fractions with odd denominators' to subsets of Farey fractions consisting of fractions with denominators not divisible by a given prime. We also investigate the joint distribution of numerators of differences of h-tuples of consecutive fractions in these sets.