On additive co-minimal pairs

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