Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.4

On the Farey Fractions with Denominators in Arithmetic Progression

C. Cobeli and A. Zaharescu
Institute of Mathematics of the Romanian Academy
P. O. Box 1-764
Bucharest 70700


Let $ {\mathfrak{F}^{Q}}$ be the set of Farey fractions of order $ Q$. Given the integers $ \mathfrak{d}\ge 2$ and $ 0\le \mathfrak{c}\le \mathfrak{d}-1$, let $ {\mathfrak{F}^{Q}}(\mathfrak{c},\mathfrak{d})$ be the subset of $ {\mathfrak{F}^{Q}}$ of those fractions whose denominators are $ \equiv \mathfrak{c}$ (mod $ \mathfrak{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 $ {\mathfrak{F}^{Q}}(\mathfrak{c},\mathfrak{d})$. This shows that the clusters of points $ (q_0/Q,q_1/Q,\dots,q_s/Q)\in[0,1]^{s+1}$, where $ q_0,q_1,\dots,q_s$ are consecutive denominators of members of $ {\mathfrak{F}^{Q}}$ produce a limit set, denoted by $ \mathcal{D}(\mathfrak{c},\mathfrak{d})$. The shape and the structure of this set are presented in several particular cases.

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Received September 15 2004; revised version received May 20 2005; July 20 2006. Published in Journal of Integer Sequences July 20 2006.

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