Erratum to “ A further study for the upper bound of the cardinality of Farey vertices and applications in discrete geometry ” [ J . Algebra Comb . Discrete Appl . 2 ( 3 ) ( 2015 ) 169-190 ]

The equation (4) on the page 178 of the paper previously published has to be corrected. We had only handled the case of the Farey vertices for which min (⌊ 2m sr′ ⌋ , ⌊ n s′r ⌋) ∈ N∗. In fact we had to distinguish two cases: min (⌊ 2m sr′ ⌋ , ⌊ n s′r ⌋) ∈ N∗ and min (⌊ 2m sr′ ⌋ , ⌊ n s′r ⌋) = 0. However, we highlight the correct results of the original paper and its applications. We underline that in this work, we still brought several contributions. These contributions are: applying the fundamental formulas of Graph Theory to the Farey diagram of order (m,n), finding a good upper bound for the degree of a Farey vertex and the relations between the Farey diagrams and the linear diophantine equations. 2010 MSC: 05A15, 05A16, 05A19, 05C30, 68R01, 68R05, 68R10


Introduction
In [9], 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: m 3 n 3 (m + n) 2 .
In [17], I found that the number of straight Farey lines is asymptotically mn(m + n) ζ(3) 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 [17] with some tools of number theory.
The work which has been done for the case where min 2m sr , n s r ∈ N * , remains correct.But the case where min 2m sr , n s r = 0 has to be handled.
The goal of this study is to understand better how to bound |F V (m, n)| in an optimal way.
Definition 2.1.[17](Farey lines of order (m, n)) A Farey line of order (m, n) is a line whose equation is uα + vβ + w = 0 with (u, v, w) ∈ −m, m × −n, n × Z, 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 sequences of order n) The Farey sequence of order n is the set We mention [14] as a forthcoming modern reference work on the Farey sequences.Several standard variants of the notion of Farey diagram are mentioned there.
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).
Let m and n be two positive integers.We let F m,n denote the set = 0, m − 1 × 0, n − 1 .U m,n denotes the set of all (m, n)-cubes.Furthermore, the proposition 3 of [9] shows that 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 .Definition 2.8.[9]((m, n)-pattern) Let m and n be two positive integers.A (m, n)-pattern is a map w:F m,n −→ Z. m × n is called the size of the (m, n)-pattern w.The set of the (m, n)-patterns will be denoted by M m,n .
Proposition 2.10.(Recall [9]) 1.The (k, l)-th point of the (m, n)-cube at the position (i, j) of the discrete plane P α,β,γ can be computed by the formula :

For all
[ is non-empty, then there exists i, j such that αi 1.
Corollary 2.12.[9] 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 : • for any point p ∈ O in the interior of the segment of vertices p 1 and p 2 , 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 [17], 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 ( [9]).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 and lemmas
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.
Moreover, we remind the Euler's Formula: Theorem 3.3.(Euler's formula for the connex planar graphs) In a connex planar multi-graph, having V vertices, E edges, and F facets, we have: 4. Bound for the degree of a farey vertex where nl(x, m, n) denotes the number of Farey lines of order (m, n) passing through the vertex x.
Proof.In F G(m, n), because a Farey line generates at most 2 edges passing through the Farey vertex P , we have: So, by the handshaking proposition 3.2, We simplify by 2, and we obtain the result.
where C is Euler's constant, and τ the divisor function.
We can apply this theorem and we are able to say in particular: Corollary 4.4.There exists K > 0 such that, ∀n ∈ N \ {0, 1}, we have Proof.There is a classical equality which already exists, where a = b.Here, we generalize it : We multiply by a all the members : Let us define nl max (P, m, n) as following: . Then, we have nl(P, m, n) ≤ nl max (P, m, n) • If p = 0 then we have nl 0, p q ≤ 1 + n q (2m + 1).
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 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 .
Then, it remains to handle the general case: So, we are looking for an optimal bound for the cardinality of (u, v, w) ∈ −m, m × 0, n × Z such that (p ∧ p )(q ∧ q ) urs + vr s (q ∧ q ) 2 ss = −w.
After simplification: If v = 0, then we have: So, in the case where v = 0, We come back to the general equation (with v ≥ 1): In particular, And because of the non-redundancy hypothesis, we have: The diophantine equation becomes: 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): In particular, The determinant of this system in w p ∧ p , k is: 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, Now, we distinguish 2 cases: • If w = 0, by the lemma 4.5, the number of suitable integers k is bounded by min 2 m sr , n s r • w = 0. We can always choose u 0 < 0 and v 0 > 0.
In these conditions, the number of suitable integers k is bounded by: Lemma 4.7.If we consider a Farey vertex V = p q , p q of order (m, n), then q ∨ q ≤ 2mn. Proof.

  
A(m, n, p, q, p , q ) = min 2m sr , n s r B(m, n, p, q, p , q ) = 2 1 d m p q + n p q C(m, n, p, q, p , q ) = A(m, n, p, q, p , q ) × B(m, n, p, q, p , q ) We have: + p q ∈Fm nl p q , 0 + p q ∈Fn nl 0, p q (8) By the corollary 4.1, we have: To conclude, we use the result of the Proposition 2.
4.2.Case of the vertices for which min 2m sr , n s r ∈ N * Proposition 4.9.
Proof.I point out that I choosed n s r , and after 2m sr , in order to obtain a symmetric upper bound.In the following, we use the boundaries for r, r , s, s given by: 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 4.4 (and the same for d s ).

Conclusion of this strategy
By the strategy of the Farey vertices, we obtained some interesting results: • We applied the fundamental formulas of Graph Theory to the Farey diagram of order (m, n).
• We found a good upper bound for the degree of a Farey vertex.
• We made relations between the Farey diagrams and the linear diophantine equations by solving explicit systems of linear diophantine equations.
However, at the moment, this method does not help to improve the known upper bound for the cardinality of the Farey vertices.
We suggest two possible ways of future research for bounding this term D (m, n).
In that case, we could conclude that : • Otherwise we have to search a bound whose order is between 5 and 6.If the optimal order is 6, that would strenghten the importance of our work [17], as it would probably mean that the order of the cardinality of Farey vertices is a homogeneous polynomial of order 6.

Definition 2 . 3 .
(Farey vertex) A Farey vertex of order (m, n) is the intersection of two Farey lines in [0, 1] 2 .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 2.4.(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, denotes the upper 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.ϕ denotes the Euler's totient function.Card(A) or |A| denotes the cardinality of the set A. Definition 2.5.(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.(Farey graph) The Farey graph of order

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 .

it is sufficient to multiply the equation by − 1 .
And we obtain the same line, but (−u, −v, −w) ∈ −m, m × 0, n × Z.
Let us study further B 1 (m, n), then the results for B 2 (m, n) are computed in a similar manner.B 1 (m, n) ≤ m

4. 3 . 6 :
Cases of the vertices for which p = 0 or p = 0 Now, we treat the two simple cases where p = 0 or p = 0 of the proposition 4.Proposition 4.12.

4. 4 .
Case of the vertices for which min 2m It remains to handle the case of the vertices for which min 2m sr , n s r = 0.At this stage of our research, we are not yet able to bound this term D (m, n).
using the definition of the integer part of bx, we have