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

We consider the set ${\mathcal{S}}_Q$ of Farey fractions $d\mathrm{/}b$ of order $Q$ with the property that there exists a matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \mathrm{SL}(2,\mathbb{Z})$ of trace at most $Q$, with positive entries and $a\ge \max \{b,c\}$. For every $Q\ge 3$, the set ${\mathcal{S}}_Q\cup \{0\}$ is shown to define a unimodular partition of the interval $[0,1]$. We also prove that the elements of ${\mathcal{S}}_Q$ are asymptotically distributed with respect to the probability measure with density $(1\mathrm{/}(1+x)-1\mathrm{/}(2+x))\mathrm{/}\log (4\mathrm{/}3)$ and that the sequence of sets $({\mathcal{S}}_Q{)}_Q$ has a limiting gap distribution as $Q\to \mathrm{\infty }$.