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 may range between 0 and 2|S|, 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/pZ under addition (p prime). We obtain various bounds and results. In the Z/pZ setting, this work may be viewed as a subset generalization of the Cauchy–Davenport Theorem.