Intervals Between Farey Fractions in the Limit of Infinite Level

TL;DR

This paper investigates the behavior of intervals between new Farey fractions at each level, proving that the total length of these intervals tends to zero as the level approaches infinity, revealing subtle geometric properties.

Contribution

It establishes that the sum of lengths of intervals between new Farey fractions diminishes to zero, and conjectures this limit is zero, highlighting a subtle geometric property.

Findings

The total length of intervals between new Farey fractions tends to zero as level increases.

The sum over inverse squares of new denominators also tends to zero.

Proved bounds for Farey intervals based on parent intervals at lower levels.

Abstract

The modified Farey sequence consists, at each level k, of rational fractions r_k^{(n)}, with n=1, 2, ...,2^k+1. We consider I_k^{(e)}, the total length of (one set of) alternate intervals between Farey fractions that are new (i.e., appear for the first time) at level k, I^{(e)}_k := \sum_{i=1}^{2^{k-2}} (r_k^{(4i)}- r_k^{(4i-2)}) . We prove that \liminf_{k\to \infty} I_k^{(e)}=0, and conjecture that in fact \lim_{k \to \infty}I_k^{(e)}=0. This simple geometrical property of the Farey fractions turns out to be surprisingly subtle, with no apparent simple interpretation. The conjecture is equivalent to $ lim_{k \to \infty}S_{k}=0, where S_{k} is the sum over the inverse squares of the new denominators at level k, S_{k}:=\sum_{n=1}^{2^{k-1}} 1/ (d_k^{(2n)} )^2. Our result makes use of bounds for Farey fraction intervals in terms of their "parent" intervals at lower levels.