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arXiv · 2508.07951

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

Jack Anderson·Florin P. Boca·Cristian Cobeli·Alexandru Zaharescu
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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 $Q\rightarrow \infty$.

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Jack Anderson, Florin P. Boca, Cristian Cobeli, Alexandru Zaharescu. 2025-08-11. On denominators of consecutive $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions. https://arxiv.org/abs/2508.07951

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