A distribution related to Farey sequences -- I

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 $c=0$.