Gap distribution of Farey fractions under some divisibility constraints

TL;DR

This paper investigates the distribution of gaps between Farey fractions under divisibility constraints, establishing the existence of a limiting gap distribution measure as the order tends to infinity.

Contribution

It proves the existence of a limiting gap distribution for Farey fractions with specific divisibility conditions, extending previous results to new constrained sets.

Findings

Limiting gap distribution measure exists for Farey fractions with divisibility constraints.

Results apply as the order Q tends to infinity.

Extends understanding of Farey sequence distributions under divisibility conditions.

Abstract

For a fixed positive integer d, we show the existence of the limiting gap distribution measure for the sets of Farey fractions a/q of order Q with a not divisible by d, and respectively with q relatively prime with d, as Q tends to infinity.