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

Jack Anderson, Florin P. Boca, Cristian Cobeli, Alexandru Zaharescu

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 $Q\rightarrow \infty$.

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

Abstract

The sequence of -saturated Farey fractions was defined in our previous work by , where is the multiplicative inverse of in . Here, we prove that the set of -scaled denominators of consecutive fractions in is dense in the region , and provide a formula for their distribution in as .

Paper Structure

This paper contains 5 sections, 11 theorems, 73 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The inclusion scrD subseteq cV
  3. The inclusion cV subseteq scrD
  4. A partition of the region cV
  5. An index identity and the Phi function

Key Result

Theorem 1

The set $\mathscr{D}:=\cup_Q Q^{-1} \mathscr{D}_Q$ is dense in $\mathcal{V}$.

Figures (5)

  • Figure 1: The set of pairs of denominators of consecutive fractions in $\mathscr S_{1200}$. The points are colored differently, depending on the number of intermediate Farey fractions whose insertion is delayed at this level.
  • Figure 2: The polygons defined in \ref{['eq3']} and the partition of the $\mathcal{V}$-shape described in Section \ref{['partition']}. The order in which the polygons are positioned is from right to left and from top to bottom.
  • Figure 3: The highly oscillatory behavior of $\Phi(Q)$.
  • Figure 4: The horizontal and vertical transformations when insertions are made. Thus, a point $(q_1,q_2)\in\mathscr D_{Q}$ that disappears at level $Q$ is replaced in $\mathscr D_{Q+1}$ with two new points $(q_1+q_2,q_2)$ and $(q_1,q_1+q_2)$. Therefore, through insertions, the polygon $ABEF$, in which the points that will disappear at level $Q$ are found, transforms into the polygons $ABCD$ and $GDEF$ that will contain the new points.
  • Figure 5: The points that disappear and the new points created by insertion when passing from level $Q-1$ to level $Q$ for $Q=721,722,723$, and $724$.

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 8 more