On a problem in additive number theory

Abstract

Let A be a non-empty subset of a finite abelian group G. For x ∈ G, let rA-A(x) = #{(a, a_) ∈ A× A : X = a -a_} the number of representations of x as a difference of two elements from A. Lev [3] proposed the following problem: If rA-A(x) ≥ |A| 2 , x ∈ A - A, is it necessarily true that A - A is either a subgroup or a union of three cosets of a subgroup? By an example, we illustrate that the problem has negative answer for a non cyclic group G. We give an affirmative answer to this problem for a large class of subsets A of a cyclic group G.