On the Farey fractions with denominators in arithmetic progression

Let $F_Q$ be the set of Farey fractions of order $Q$. Given the integers $\text{d}\ge 2$ and $0\le \stackrel{¸}{\le }\text{d}-1$, let $F_Q(c,d)$ be the subset of $F_Q$ of those fractions whose denominators are $\equiv c(\mathrm{m}\mathrm{o}\mathrm{d}d)$, arranged in ascending order. The problem we address here is to show that as $Q\to \mathrm{\infty }$, there exists a limit probability measuring the distribution of $s$-tuples of consecutive denominators of fractions in $F_Q(c,d)$. This shows that the clusters of points $(q_0\mathrm{/}Q,q_1\mathrm{/}Q,\mathrm{.}\mathrm{.}\mathrm{.},q_s\mathrm{/}Q)\in [0,1{]}^{s+1}$, where $q_0,q_1,\mathrm{.}\mathrm{.}\mathrm{.},q_s$ are consecutive denominators of members of $F_Q$ produce a limit set, denoted by $D(c,d)$. The shape and the structure of this set are presented in several particular cases.