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.