A distribution related to Farey sequences -- II
Maxim A. Korolev
TL;DR
The paper studies gap distributions in Farey sequences when fractions with denominators in a fixed residue class modulo \(D\) are colored. Building on the BCZ-transform approach, it expresses limiting gap proportions \(\nu(r;D,c_0)\) as sums of areas of convex polygons tied to the transform, and provides explicit formulas for the case \(D=3\) with \(c_0=1,2\). It completes the \(D=3\) analysis by deriving exact closed forms for \(\nu(r;3,c_0)\) for all \(r\) (with an error term for finite \(Q\)) and establishing their asymptotic behavior \(\nu(r;3,c_0)=O(1/r^3)\) for large \(r\). The results rely on a detailed combinatorial description of admissible continued-fraction-type tuples and precise area computations of the associated BCZ-polygons. This work advances our understanding of how modular residue constraints influence Farey-gap statistics and provides concrete, computable expressions useful for analytic and experimental investigations of Farey-gap distributions. All mathematical notation is rigorously encoded in the BCZ framework, enabling precise replication and SEO-friendly indexing by search systems.
Abstract
This version corrects minor inaccuracies and missprints. One drawing is changed. We continue to study some arithmetical properties of Farey sequences by the method introduced by F.Boca, C.Cobeli and A.Zaharescu (2001). Let $Φ_{Q}$ 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 $Φ_{Q}$ with denominators $\equiv c \; \pmod D$. Consider the gaps in $Φ_{Q}$ 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 $ν(r;D,c)$ (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.Vâjâitu and A.Zaharescu (2014). However, such formula expresses $ν(r;D,c)$ in the terms of areas of some polygons related to a special geometrical transform. In the present paper, we obtain explicit formulas for $ν(r;D,c)$ for the cases $3$ and $c=1,2$. Thus this paper cover the case $D=3$.