Minimun overlap problem on finite groups

Let A, B be disjoint sets such that A∪B=[1,2n]⊂Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A\cup B= [1,2n]\subset {\mathbb {Z}}$$\end{document} and |A|=|B|=n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert A\vert = \vert B\vert = n$$\end{document}. Let us call m(A,B)=maxt∈Z|(t+B)∩A|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m (A, B)=\max _{t \in {\mathbb {Z}}}\vert (t+B)\cap A \vert $$\end{document} and consider M(n):=min(A,B)m(A,B)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(n):=\min \limits _{(A,B)} m(A,B)$$\end{document}(over all partitions with A∪B=1,2n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \cup B=\left[ 1,2n\right] $$\end{document}). There are well-known upper and lower bounds of M(n). In this paper we studied a variation of this problem, i.e. we considered a finite abelian group G with |G|=k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert G \vert =k$$\end{document}, we define M(G) which is analogous to M(n) and we obtained upper and lower bounds for M(G).


Introduction
In this paper, Z, Z + and F q denote the set of integers, positive integers and the finite field with q elements respectively.For n ∈ Z + , let [1, 2n] := {1, 2, 3, . . ., 2n} and U be a set, then P U denotes the partitions of U .
Let A, B be disjoint sets such that A ∪ B = [1, 2n] ⊂ Z and |A| = |B| = n.We write A = {a 1 , a 2 , . . .a n }, B = {b 1 , b 2 , . . ., b n } and we denote the number of solutions of the equation a i − b j = x, 1 ≤ i, j ≤ n by R A−B (x) = |(x + B) ∩ A|, where x is an integer.Let us call m(A, B) the maximum number of representations of x as difference of elements of A and B, i.e. m(A, B) := max{R A−B (x) : x ∈ Z} and consider M(n) as follows: The problem of finding accurate bounds of M(n) was proposed by Erdös [1] and is described by Guy [2].Furthermore, exact values are known for M(n) up to n = 15 by [2].As few exact values are known for M(n), it is important to find upper or lower bounds, in this sense P. Erdös proved in [1] that: Over the years, researchers have found improvements to this lower bound showing that: (n) by Haugland [5].
Asymptotically by [3] it is known that the limit lim n→∞ M(n) n exists and it is less than 0.38201.
An important problem is to regard the minimal overlap problem for other groups.Generalizations of this problem are studied in [3,[8][9][10].
Let A, B be subsets of an arbitrary finite abelian group G such that A ∪ B = G and A ∩ B = ∅.M(G) is defined as follows: In this paper we considered G a finite abelian group with |G| = k, and we obtained upper and lower bounds for M(G) that are expressed as follows.

Theorem 1.1 Let G be an abelian finite group with |G| = k and A, B ⊂ G. If
On the other hand, note that if k is an even number, with On the other hand, we obtained an upper bound for M(G).This upper bound can be improved if G = (Z/ pZ) n with p an odd prime..

Proofs of the theorems
In this section we present the proofs of Theorems 1.1 and 1.2.First we prove a result that allows us to find a pair of sets that generate a partition of a finite field with odd size and that later we can extend to other groups.
where Q = {a 2 : a ∈ F} and U = F\Q.
Proof Write ψ : F * −→ F * where ψ(a) = a 2 , and notice that For each d ∈ ψ(F * ), the polynomial x 2 − d has exactly two roots so , and then (1) gives For each c ∈ F, set Since Q and U are complementary in F, we get that T c and S c are complementary in For each d ∈ Q, there is one and only one e ∈ F such that d − e = c so and hence (3) imples that Notice that for each d ∈ F * , both entries of ϕ c (d) are in Q and Next we will show that for each (a, b) ∈ ϕ c (F * ), and multiplying this equality by (2de) 2 , we get The nonzero polynomial (cx + d 2 x) 2 − (cd + dx 2 ) 2 has a degree of at most four so it cannot have more than 4 roots.From ( 7), the elements of ϕ −1 c (a, b) are roots of this polynomial giving (6).Then (by ( 2)) which gives and this implies plainly the claim of the lemma since c is arbitrary.
Note that the above lemma implies that M(Z/ pZ) ≤ p+3 4 .Now we are going to prove Theorem 1.1.

Proof. Theorem 1.1 Let G be an abelian group, |G| = k, A, B ⊂ G and |B|
Let H be fixed with this property, then By Lemma 2.1 we know that there are (A , B ) ∈ P Z/ pZ such that m A ,B ≤ p+3 4 , i.e.A = {a 2 : a ∈ Z/ pZ} and B = Z/ pZ \ A .
Let us now take the projection π :  Taking the maximun of these values, we get Finally, the Lemma 2.1 implies that implies the claim.

Conclusion
In this article we studied the minimum overlap problem for finite abelian groups obtaining upper and lower bounds of M(G) for big family of groups.These results generalize those obtained by Haugland [5].It would be interesting to consider other cases such as if |G| = 2 k with k ∈ Z + .A future we can work on the following items: where |G| = pm with p an odd prime.

Theorem 1 . 2
Let G be an abelian finite group with |G| = k, k = pm and p an odd prime an isomorphism of groups, we have that for all (a, b) ∈ G × G and c ∈ G we get that a − b = c if and only if ϕ(a) − ϕ(b) = ϕ(c).This gives for any partition A ∪ B = G with || A | − | B ||≤ 1 that ϕ(A) ∪ ϕ(B) is a partition of F with || ϕ(A) | − | ϕ(B) ||≤ 1 and m A,B = m ϕ(A),ϕ(B) .