Overview
This research investigates the properties of denominators within a sequence of Farey fractions, specifically the $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions denoted as $(\mathscr{S}_Q)_Q$. These fractions were previously defined, and the current work establishes two key findings regarding their denominators: first, the density of a set of $Q$-scaled denominators in a defined region, and second, a formula for their distribution within that region.
Research Context
The sequence $(\mathscr{S}_Q)_Q$ of $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions is foundational to this study. This sequence was introduced in prior work by the same researchers. It is defined as ${\mathscr S}_Q := \{ a/q \in {\mathbb Q} \cap (0,1]: q+a+\bar{a} \le Q\}$, where $a/q$ represents a rational number between 0 and 1 (inclusive of 1). A critical component of this definition is $\bar{a}$, which signifies the multiplicative inverse of $a \pmod q$ within the interval $[1,q)$. The parameter $Q$ is central to the construction of these sequences.
Approach
The research undertakes a mathematical proof to demonstrate the density of specific denominator sets and to derive their distribution formula. The focus is on the $Q$-scaled denominators of consecutive fractions within the defined sequence ${\mathscr S}_Q$. The region of interest for this density and distribution analysis is denoted as ${\mathcal V}$. This region is mathematically defined as ${\mathcal V}:=\{ (x,y)\in [0,1]^2 : \max \{ (1-3x)/2,2x-1\} \le y \le \max \{ x,1-x\} \}$. The approach involves analyzing the behavior of these scaled denominators as the parameter $Q$ tends towards infinity ($Q\rightarrow \infty$).
Findings
- The set of $Q$-scaled denominators corresponding to consecutive fractions within ${\mathscr S}_Q$ is dense in the region ${\mathcal V}$. The region ${\mathcal V}$ is a subset of the unit square $[0,1]^2$, specifically defined by the inequalities $\max \{ (1-3x)/2,2x-1\} \le y \le \max \{ x,1-x\} \$.
- A formula has been provided that describes the distribution of these $Q$-scaled denominators within the region ${\mathcal V}$ as $Q$ approaches infinity.