On the distribution of $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions

TL;DR

This paper studies the distribution and gap properties of a special subset of Farey fractions linked to SL(2, N) matrices, revealing their asymptotic distribution and partitioning of the interval [0,1].

Contribution

It introduces a new class of Farey fractions associated with SL(2, N) matrices and analyzes their distribution, partitioning, and gap structure as the order Q increases.

Findings

The set ${\\mathscr S}_Q$ partitions [0,1] with 0 included.

Elements of ${\mathscr S}_Q$ follow a specific asymptotic distribution.

The sequence ${\mathscr S}_Q$ has a limiting gap distribution.

Abstract

We consider the set ${\mathscr S}_Q$ of Farey fractions $d/b$ of order $Q$ with the property that there exists a matrix $\left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \in \operatorname{SL}(2,{\mathbb Z})$ of trace at most $Q$, with positive entries and $a\ge \max\{ b,c\}$. For every $Q\ge 3$, the set ${\mathscr S}_Q \cup \{ 0\}$ is shown to define a unimodular partition of the interval $[0,1]$. We also prove that the elements of ${\mathscr S}_Q$ are asymptotically distributed with respect to the probability measure with density $(1/(1+x) -1/(2+x) )/\log (4/3) $ and that the sequence of sets $({\mathscr S}_Q)_Q$ has a limiting gap distribution as $Q\rightarrow \infty$.