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 L S = max z ∈ 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 | + L S implies A ∧ S B = G . We then prove that if | A | + | B | = | G | + L S then | A ∧ S B | ≥ | G | − 2 | S | . We also characterize the triples of sets ( A , B , S ) such that | A | + | B | = | G | + L S and | A ∧ S B | = | G | − 2 | S | . Moreover, in this case, we also provide the structure of the set G ∖ ( A ∧ S B ) .