On the Farey fractions with denominators in arithmetic progression

Abstract

Let $F_Q$ be the set of Farey fractions of order $Q$. Given the integers $\d\ge 2$ and $0\le \c \le \d-1$, let $F_Q(c,d)$ be the subset of $F_Q$ of those fractions whose denominators are $\equiv c \pmod d$, arranged in ascending order. The problem we address here is to show that as $Q\to\infty$, 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 $(q_0/Q,q_1/Q,...,q_s/Q)\in[0,1]^{s+1}$, where $q_0,q_1,...,q_s$ 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.