Abstract:

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.