Inverse problems for minimal complements and maximal supplements

Given a subset W of an abelian group G, C is called additive complement for if W+C=G; if, moreover, no proper has this property, then we say that minimal W. It natural to ask which subsets can arise as complements some We show in finite every non-empty size |C|≤22/3|G|1/3/((3elog⁡|G|)2/3 As corollary, deduce infinite complement. also derive several analogous results “dual” problems about maximal supplements.