On the Farey Fractions with Denominators in Arithmetic Progression

Let ${\mathfrak{F}^{Q}}$ be the set of Farey fractions of order

. 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

-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.

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

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 Romania

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