Inverse problems for minimal complements and maximal supplements

Given a subset W of an abelian group G, a subset C is called an additive complement for W if W+C=G; if, moreover, no proper subset of C has this property, then we say that C is a minimal complement for W. It is natural to ask which subsets C can arise as minimal complements for some W. We show that in a finite abelian group G, every non-empty subset C of size |C|≤2^2/3|G|^1/3/((3elog⁡|G|)^2/3 is a minimal complement for some W. As a corollary, we deduce that every finite non-empty subset of an infinite abelian group is a minimal complement. We also derive several analogous results for “dual” problems about maximal supplements.