On the farey fractions with denominators in arithmetic progression

C. Cobeli, A. Zaharescu

Research output: Contribution to journalArticlepeer-review


Let F-fraktur signQ be the set of Farey fractions of order Q. Given the integers ∂ ≥ 2 and 0 ≤ c ≤ ∂ -1, let F-fraktur signQ(c, ∂) be the subset of F-fraktur signQ 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 signQ(c, ∂). This shows that the clusters of points (q0/Q, q1/Q,....,q s/Q) [0, 1]s+1, where q0, q1,..., qs 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.

Original languageEnglish (US)
Pages (from-to)1-26
Number of pages26
JournalJournal of Integer Sequences
Issue number3
StatePublished - Jul 20 2006


  • Arithmetic progressions
  • Congruence constraints
  • Farey fractions

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics


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