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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.

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