Finding integral diagonal pairs in a two dimensional N–set

According to [1] an n-dimensional N–set is a compact subset A of Rn such that for every x ∈ Rn there is y ∈ A with y − x ∈ Zn. We prove that every two dimensional N–set A must contain distinct points x, y such that x − y is in Z2 and x − y is neither horizontal nor vertical. This answers a question of P. Hegarty and M. Nathanson. For any sets A,B in an abelian group A±B denotes the set {a±b : a ∈ A and b ∈ B}. During one of the problem sessions in CANT (Combinatorial and Additive Number Theory Workshop) 2009 M. Nathanson asked the following question which was originally raised by P. Hegarty: Question 1 Can we find an N–set A ⊆ R, i.e., a compact set A ⊆ R with the property that R = A+ Z, such that (A− A) ∩ Z ⊆ (Z× {0}) ∪ ({0} × Z)? Question 1 is motivated by the study of a general inverse problem in order to determine which set E ⊆ Z can be represented by the form of (A−A)∩Z for some N –set A. Notice that (A − A) ∩ Z contains the origin and is symmetric about the origin. This inverse problem is completely solved in one dimensional case. It is shown in [1] that a finite set E of positive integers is relatively prime if and only if there is an N –set A ⊆ R such that E = (A − A) ∩ N. By the fundamental observation of geometric group theory (see [1]) if A is an n-dimensional N –set, then (A − A) ∩ Z is a finite set of generators of the group Z. Clearly, for a one dimensional N –set A ⊆ R, (A − A) ∩ Z is a set of generators if and only if (A − A) ∩ N is relatively prime. Hence the next logical step is to ask whether a symmetric set of generators of Z together with the origin (0, 0) can be represented by the form of (A−A) ∩ Z for some two dimensional N –set A. For example, it is interesting to ask whether the set E = {(0, 0),±(0, 1),±(1, 0)} can be represented by (A− A) ∩ Z for some N –set A. The main theorem in this paper will show that the answer is “no”. The following is the main theorem. Theorem 2 Every N–set A ⊆ R contains x, y such that x− y ∈ (Z r {0}). 2000 Mathematics Subject Classification. Primary 11B75, 11H06, 11P21.