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

TL;DR

This paper studies the distribution of scaled denominators of consecutive Farey fractions within a specific set, proving their density in a region and deriving a formula for their distribution as the parameter grows large.

Contribution

It establishes the density and distribution formula of scaled denominators of consecutive Farey fractions in a defined region, extending understanding of their asymptotic behavior.

Findings

The set of scaled denominators is dense in a specific region.

A formula for the distribution of these denominators as Q approaches infinity is provided.

The results describe the asymptotic distribution of Farey fractions in the defined region.

Abstract

The sequence $({\mathscr S}_Q)_Q$ of $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions was defined in our previous work by ${\mathscr S}_Q := \{ a/q \in {\mathbb Q} \cap (0,1]: q+a+\bar{a} \le Q\}$, where $\bar{a}$ is the multiplicative inverse of $a\pmod{q}$ in $[1,q)$. Here, we prove that the set of $Q$-scaled denominators of consecutive fractions in ${\mathscr S}_Q$ is dense in the region ${\mathcal V}:=\{ (x,y)\in [0,1]^2 : \max \{ (1-3x)/2,2x-1\} \le y \le \max \{ x,1-x\} \}$, and provide a formula for their distribution in ${\mathcal V}$ as $Q\rightarrow \infty$.