On the farey fractions with denominators in arithmetic progression

Let F-fraktur sign^Q be the set of Farey fractions of order Q. Given the integers ∂ ≥ 2 and 0 ≤ c ≤ ∂ -1, let F-fraktur sign^Q(c, ∂) be the subset of F-fraktur sign^Q of those fractions whose denominators are = c (mod ∂), arranged in ascending order. The problem we address here is to show that as Q → ∞, there exists a limit probability measuring the distribution of s-tuples of consecutive denominators of fractions in F-fraktur sign^Q(c, ∂). This shows that the clusters of points (q₀/Q, q₁/Q,....,q _s/Q) [0, 1]^s+1, where q₀, q₁,..., q_s are consecutive denominators of members of F-fraktur sign ^Q produce a limit set, denoted by D(c, ∂). The shape and the structure of this set are presented in several particular cases.