On the Farey fractions with denominators in arithmetic progression

Cobeli, C. ; Zaharescu, A.

Journal of integer sequences, Tome 9 (2006) no. 3.

Résumé

Summary: 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.

Détail

Classification : 11B57
Keywords: Farey fractions, arithmetic progressions, congruence constraints