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 ∧S B = {a + b : a ∈ A, b ∈ B and a − b /∈ S}. Let LS = maxz∈G |{(x, y) : x, y ∈ G, x + y = z and x − y ∈ S}|. A simple application of the pigeonhole principle shows that |A| + |B| > |G| + LS implies A ∧S B = G. We then prove that if |A| + |B| = |G| + LS then |A ∧S B| ≥ |G| − 2|S|. We also characterize the triples of sets (A, B, S) such that |A| + |B| = |G| + LS and |A ∧S B| = |G| − 2|S|. Moreover, in this case, we also provide the structure of the set G \ (A ∧S B).This research was done when the second author was visiting Université Pierre et Marie Curie, E. Combinatoire, Paris, supported by the Ministry of Education, Spain, under the National Mobility Programme of Human Resources, Spanish National Programme I-D-I 2008–2011