The distribution of rationals in residue classes

TL;DR

This paper investigates the distribution of consecutive reduced fractions within fixed intervals, showing that as denominators grow, the proportion of specific arithmetic progressions approaches a well-defined, interval-independent limit.

Contribution

It provides an explicit description of the limiting distribution of r-tuples of denominators in residue classes for increasing denominators, extending understanding of rational distribution patterns.

Findings

Limit of proportion approaches a fixed value as denominators grow.

Limit is independent of the chosen interval.

Explicit formulas are provided for particular cases.

Abstract

Our purpose is to give an account of the $r$-tuple problem on the increasing sequence of reduced fractions having denominators bounded by a certain size and belonging to a fixed real interval. We show that when the size grows to infinity, the proportion of the $r$-tuples of consecutive denominators with components in certain apriori fixed arithmetic progressions with the same ratio approaches a limit, which is independent on the interval. The limit is given explicitly and it is completely described in a few particular instances.