Large restricted sumsets in general Abelian groups

Let A, B and S be subsets of a finite Abelian group G. The restricted sumset of A and B with respect to S is defined as A^∧SB = {a + b : a ∈ A, b ∈ Banda - b ∉ S} Let _{L S} = _{max z ∈G}| { (x, y) : x, y ∈ G, x + y = zandx - y ∈ S} |. A simple application of the pigeonhole principle shows that |A| + |B| > |G| + _{L S} implies A^∧SB = G. We then prove that if |A| + |B| = |G| + _{L S} then |A^∧SB| ≥ |G| - 2|S|. We also characterize the triples of sets (A, B, S) such that |A| + |B| = |G| + _{L S} and |A^∧SB| = |G| - 2|S|. Moreover, in this case, we also provide the structure of the set G {set minus} (A^∧SB)