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

Abstract: We consider the ordered 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 ${\mathscr S}_Q \cup { 0}$ defines a unimodular partition of the interval $[0,1]$. We 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 $Q\rightarrow \infty$.