A distribution related to Farey sequences -- I

TL;DR

This paper investigates the distribution of gaps in Farey sequences with specific modular constraints, providing explicit formulas for certain cases and extending previous geometric-based results.

Contribution

It derives explicit formulas for the proportions of gaps with a fixed number of fractions for Farey sequences with denominators constrained by modular conditions, specifically for D=2,3 and c=0.

Findings

Explicit formulas for gap proportions when D=2,3 and c=0.

Extension of previous geometric methods to derive these formulas.

Provides a clearer understanding of the distribution of Farey sequence gaps under modular constraints.

Abstract

Minor corrections to previous version. We study some arithmetical properties of Farey sequences by the method introduced by F.Boca, C.Cobeli and A.Zaharescu (2001). Let $\Phi_{Q}$ be the classical Farey sequence of order $Q$. Having the fixed integers $D\geqslant 2$ and $0\leqslant c\leqslant D-1$, we colour to the red the fractions in $\Phi_{Q}$ with denominators $\equiv c \pmod D$. Consider the gaps in $\Phi_{Q}$ with coloured endpoints, that do not contain the fractions $a/q$ with $q\equiv c \pmod D$ inside. The question is to find the limit proportions $\nu(r;D,c)$ (as $Q\to +\infty$) of such gaps with precisely $r$ fractions inside in the whole set of the gaps under considering ($r = 0,1,2,3,\ldots$). In fact, the expression for this proportion can be derived from the general result obtained by C.Cobeli, M.V\^{a}j\^{a}itu and A.Zaharescu (2014). However, such formula expresses $\nu(r;D,c)$ in the terms of areas of some polygons related to a special geometrical transform. In the present paper, we obtain an explicit formulas for $\nu(r;D,c)$ for the cases $D = 2, 3$ and $c=0$.