On additive co-minimal pairs

A pair of non-empty subsets (W,W?) in an abelian group G is additive complement if W+W?=G. The set W? said to be minimal W W+(W??{w?})?G,?w??W?. In general, given arbitrary subset a group, the existence complement(s) depends on its structure. dual problem asks that such set, it some subset. Additive complements have been studied context representations integers since time Erd?s, Hanani, Lorentz and others. notion due Nathanson. We study tightness property pairs both are each other. These termed co-minimal we show any finite free belongs pair. also infinite sets forming pairs. At other extreme, motivated by unbounded arithmetic progressions integers, look at which can never part This leads discussion co-minimality, subgroups, approximate subgroups asymptotic G.