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 ([14]): ∃C > 0, ∀(m,n) ∈ N, ∣∣∣FV (m,n)∣∣∣ ≤ Cmn(m+ n) ln(mn) Using it, in particular, we show that the number of (m,n)-cubes Um,n verifies: ∃C > 0, ∀(m,n) ∈ N, ∣∣∣Um,n∣∣∣ ≤ Cmn(m+ n) ln(mn) which is an important improvement of the result previously obtained in [6], which was a polynomial of degree 8. This work uses combinatorics, graph theory, and elementary and analytical number theory. 2010 MSC: 05A15, 05A16, 05A19, 05C30, 68R01, 68R05, 68R10


Introduction
Discrete geometry is the meeting point between several domains of mathematics and computer sciences: combinatorics, graph theory and number theory.To understand better the images, one studies very advanced theoretical problems coming from pure mathematics.In 2D, many progresses have been done.In 3D, there is a very active research in understanding the discretization of planes.This article brings progress to the combinatorial studies of 3D-patterns.Paul Erdös obtained many results in the field of combinatorial geometry for some particular configurations (see [9] for example).One particular instance of configurations are the Farey diagrams.These diagrams and Farey sequences have many applications.In particular, Farey sequences and Farey diagrams are directly involved in medicine.For instance, in cardiology to modelize optimized systems for pacemakers, or in research against cancer, in imagery and surgery [28].A very active field of research is the tomography and reconstruction, in which Farey sequences and Farey diagrams also apply [12].Another example of application to vision is given in [20].They can also be used for the detection of pieces of discrete planes in 3D-image.For example in [28].Tomás proves in [27] and [26] that there is an important link between accelerator physics and Farey diagrams.We notice that the Farey diagram of order (n, n) has the same degree as the resonance diagram of order n.The asymptotic behaviour of the two different structures only differs by a factor.There are some similarities between (m, n)-cubes, that we redefine below, and threshold functions on a two-dimensional rectangular grid, for which an asymptotic value for the cardinality of these functions has been derived in [11].And Farey diagrams are used, since long time in computer science: for example, they are also used when we study the preimage of a discrete piece of plane in discrete mathematics, and the Farey diagram for discrete segments were studied by McIlroy in [19].Some problems related to Farey diagrams remain unsolved.We are going to focus on this field to study the Farey diagrams from the point of view of combinatorics and number theory.
In [6], one of the strategies for the enumeration of pieces of discrete planes, was to estimate the number of vertices in a Farey diagram.This work, combined with a basic property of graph theory, yields an upper bound.This upper bound is an homogeneous polynomial of degree 8: In her thesis [7], Debled-Rennesson also studied this problem.Another step forward has been taken by Domenjoud, Jamet, Vergnaud, and Vuillon in [8] where an exact formula (from combinatorial number theory) for the cardinality of the (2, n)-cubes has been derived.In [14], I found that the number of straight Farey lines is asymptotically when m and n go to infinity.
Henceforth, the strategy consisting in focusing on Farey lines to study Farey vertices combinatorics is not sufficient if we want to have a deeper understanding of the combinatorics of the (m, n)-cubes, and we can directly focus on the Farey vertices [14] with some tools of number theory.In the following, I derive an upper bound of degree strictly lower than 6, and not 6, as it was the case in [6].

Preliminaries
Let −m, m denote the set {−m, . . ., −1, 0, 1, . . ., m} of consecutive integers between −m and m.Definition 2.1.[14](Farey lines of order (m, n)) A Farey line of order (m, n) is a line whose equation is uα We mention [10] as a forthcoming modern reference work on the Farey sequences.Several standard variants of the notion of Farey diagram are mentioned there.Definition 2.5.[14](Farey edge) A Farey edge of order (m, n) is an edge of the Farey diagram of order (m, n).We denote the set of Farey edges by F E(m, n).Definition 2.6.[14](Farey graph) The Farey graph of order (m, n) is the graph Definition 2.7.(Farey facet) A Farey facet of order (m, n) is a facet of the Farey graph of order (m, n).We will denote the set of Farey facets of order (m, n) by F F (m, n).
This definition shows that: Now, we recall some results obtained in [6], and some direct consequences of this result.

For all
[ is non-empty, then there exists i, j such that αi 3. By the Proposition 2.10, the set of (m, n)-cubes of the discrete planes P α,β,γ only depends of (α, β) and is denoted by C m,n,α,β .
Corollary 2.12.([6]) Let O be a Farey connected component, then O is a convex polygon and if p 1 , p 2 , p 3 are distinct vertices of the polygon O, then : By this corollary, all the (m, n)-cubes are associated to Farey vertices.And according to the Proposition 2.10, there are at most mn (m, n)-cubes associated to a Farey vertex, therefore We know by [14], that the number of Farey lines, is equivalent to a polynomial of degree 3 in m and n, when m and n go to infinity.According to Lemma 3.1, these lines form a number of vertices, given at most by a polynomial of order 6 ( [6]).But this method is far from giving an optimal upper bound for the cardinality of the Farey vertices.In order to obtain a new and more powerful result of combinatorics on this set of vertices, we are going to study the properties of the Farey lines passing through a Farey vertex.Our idea is to use the theorem: Proposition 3.2.(Reminder of Graph Theory) In a simple graph G = (V, E), we have:

Fundamental properties
where V is the set of vertices, and E is the set of edges, and deg(x) is the degree of the vertex x, that is the number of edges which are adjacent to the vertex x.

Modeling of the problem of Farey vertices of order (m, n)
To have a precise estimate on the number of edges in the Farey graph of order (m, n), it is sufficient to compute the degree of any Farey vertex.Henceforth, below we to study in more detail the mapping defined on F V (m, n): x → deg(x).Let us define r, r , s, s , d and d as follows: (1) • If (p, p ) ∈ N * 2 , then we have • If p = 0 then we have The vertices such that p = 0, are the vertices of the set 0, p q with p q ∈ F n .
• If p = 0, then we have The vertices such that p = 0 are the vertices of the set p q , 0 with p q ∈ F m .
Proof.We can always suppose that in the equation of a Farey line, (of the type: uα And we obtain the same line, but First, we handle the case where p = 0 or p = 0. (because of the preliminary).
There are at most 1 + n q such integers.And there are 2m + 1 integers in the interval −m, m .The vertices such that p = 0, are the vertices of the set There are at most 1 + 2 m q such integers.The vertices such that p = 0, are the vertices of the set p q , 0 with p q ∈ F m .Now, we can handle the general case remaining: In GF (m, n), because a Farey line generates at most 2 edges passing through the Farey vertex P , we have: So, we are looking for an optimal bound for the cardinality of (p ∧ p )(q ∧ q ) urs + vr s (q ∧ q ) 2 ss = −w.
After simplification: (p ∧ p ) urs + vr s (q ∧ q )ss = −w. So, When w is fixed, the consequence of the hypothesis of primality enables to solve this diophantine equation: Let us fix w, where (u 0 , v 0 ) is a particular solution of the diophantine equation in (x, y): The determinant of this system in w p ∧ p , k is: So, one can determine w p ∧ p , k by the Cramer formulas: As in [14], we can always suppose that v ≥ 0 (else we mutiply uα + vβ + w = 0, by −1).Moreover, we have seen that as we have: and as p ∧ p | w, we can deduce that there exists w such that w = w (p ∧ p ). So, As for a Farey line, (u, v) = (0, 0), we deduce that (u , v ) = (0, 0).
First we handle the case where v = 0 we deduce by the Bezout theorem that r ∧ v 0 = 1.In addition to it, we have r ∧ (q ∧ q ) = 1.So, r ∧ (v 0 (q ∧ q )) = 1.And by the Gauss theorem, we have that r | w d .In this case, it is impossible that w = 0, else k = 0 and u = v = w = 0. So, there are at most Farey lines passing through the Farey vertex.
In the second case, v > 0, so we have When w is fixed, Now, we distinguish 2 cases: • If w = 0, the number of suitable integers k is bounded by min 2 m sr , n s r .
• w = 0 Then, the number of suitable w dr is at most 2 1 dr m p q + n p q .
-Else And, of course, the existence of such couples implies that: In particular, which proves the claim.

Bound for |F V (m, n)|
The equations in (1) are: A(m, n, p, q, p , q ) = 2 min 2 m sr , n s r B(m, n, p, q, p , q ) = 4 1 dr m p q + n p q C(m, n, p, q, p , q ) = 4 1 dr m p q + n p q D(m, n, p, q, p , q ) = 2 1 d m p q + n p q E(m, n, p, q, p , q ) = 2A(m, n, r, r , s, s , d ) × D(m, n, r , p, q, p , q ) A(m, n, p, q, p , q ) E(m, n, p, q, p , q ) We have: Proof.By the Proposition 3.2, we have: To conclude,we use the result of the Proposition 4.1, which proves the claim.
Therefore, as we remember that ∀(a, b) We need to recall another theorem: where C is Euler's constant.
We can apply this theorem and we are able to say in particular: Corollary 5.4.There exists K > 0 such that, ∀n ∈ N \ {0, 1}, we have Proposition 5.5. Proof.
A(m, n, p, q, p , q ) I point out that I chose n s r , and after m sr , in order to obtain a symmetric upper bound.In the following, we use the boundaries for r, r , s, s given by: Now, we can continue the calculation of A (m, n).For this purpose, let us permute the sums and let us change the variables by using, as before, In particular, Now, we have to distinguish the case where d s = 1 and the case d s > 1 in the sums in order to use the Corollary 5.4 (and the same for d s ).
Let us study further E 1 (m, n), then the results for E 2 (m, n) are computed in a similar manner.We know [29] that .
Hence, we have: So, we can say that: Hence, in the Farey graph of order (m, n), we have: Moreover, as the Farey graph of order (m, n) is planar, we can apply to it the Euler's Formula: Theorem 5.11.(Recall)(Euler's formula for the connex planar graphs) In a connex planar multi-graph, having V vertices, E edges, and F facets, we have: In particular, this involves that we have V ≤ E + 2.

Summary-conclusion
In order to improve the upper bound for U m,n , an interesting work using combinatorics, graph theory, and number theory, has been to focus on the diophantine aspects of Farey diagrams, combined with some other arguments of graph theory to estimate better the cardinality of Farey vertices.And in this work, we obtained two important results: whereas the previous published result was a polynomial of degree 6 [6].
and which has at least 2 intersection points with the frontier of [0, 1] 2 .(u, v, w) are the coefficients.(α, β) are the variables.Let denote the set of Farey lines of order (m, n) by F L(m, n).Definition 2.2.[14](Farey vertex) A Farey vertex of order (m, n) is the intersection of two Farey lines.We will denote the set of Farey vertices of order (m, n), obtained as intersection points of Farey lines of order (m, n), by F V (m, n).Definition 2.3.[14](Farey diagrams for the pieces of discrete planes of order (m, n) (or (m, n)-cubes)) The Farey diagram for the (m, n)-cubes of order (m, n) is the diagram defined by the passage of Farey lines in [0, 1] 2 .We recall that denotes the integer part, and denotes the fractional part.If a and b are two integers, a ∧ b denotes the greatest common divisor of a and b, and a ∨ b denotes the least common multiple.

Farey
lines of order(3,3) ϕ denotes the Euler's totient function.Card(A) denotes the cardinality of the set A. Definition 2.4.[10](Fareysequences of order n) The Farey sequence of order n is the set

Lemma 3 . 1 .
(Reminder of Graph Theory) Let us consider n straight lines.The number of vertices constructed from these n lines is at most n(n − 1) 2 .

Proposition 4 . 1 .
(Upper bound for the number of Farey lines of order (m, n) passing through a Farey vertex of order (m, n)) Let P = p q , p q be a Farey vertex of order (m, n).