The extent to which subsets are additively closed

Given a finite abelian group G (written additively), and a subset S of G, the size r (S) of the set {(a, b) : a, b, a + b ∈ S} may range between 0 and | S |², with the extremal values of r (S) corresponding to sum-free subsets and subgroups of G. In this paper, we consider the intermediate values which r (S) may take, particularly in the setting where G is Z / p Z under addition (p prime). We obtain various bounds and results. In the Z / p Z setting, this work may be viewed as a subset generalization of the Cauchy-Davenport Theorem.