Abstract

Let 𝔉^{Q} be the set of Farey fractions of order Q. Given the integers 𝔡 ≥ 2 and 0 ≤ 𝔠 ≤ 𝔡 − 1, let 𝔉^{Q}(𝔠, 𝔡) be the subset of 𝔉^{Q} of those fractions whose denominators are ≡ 𝔠 (mod 𝔡), 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^{Q}(𝔠, 𝔡). This shows that the clusters of points (q_{0}/Q, q_{1}/Q, …, q_{s}/Q) ∈ [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.