A further study for the upper bound of the cardinality of Farey vertices and application in discrete geometry

The aim of the paper is to bring new combinatorial analytical properties of the Farey diagrams of order $(m,n)$, which are associated to the $(m,n)$-cubes. The latter are the pieces of discrete planes occurring in discrete geometry, theoretical computer sciences, and combinatorial number theory. We give a new upper bound for the number of Farey vertices $FV(m,n)$ obtained as intersections points of Farey lines (\cite{khoshnoudiradfarey}): $$\exists C>0, \forall (m,n)\in\mathbb{N}^{*2},\quad \Big|FV(m,n)\Big| \leq C m^2 n^2 (m+n) \ln^2 (mn)$$ Using it, in particular, we show that the number of $(m,n)$-cubes $\mathcal{U}_{m,n}$ verifies: $$\exists C>0, \forall (m,n)\in\mathbb{N}^{*2},\quad \Big|\mathcal{U}_{m,n}\Big| \leq C m^3 n^3 (m+n) \ln^2 (mn)$$ which is an important improvement of the result previously obtained in ~\cite{daurat_tajine_zouaoui_afpdpare}, which was a polynomial of degree 8. This work uses combinatorics, graph theory, and elementary and analytical number theory.