  The study of minimal complements in a group or a semigroup was initiated by
Nathanson. The notion of minimal complements and being a minimal complement
leads to the notion of co-minimal pairs which was considered in a prior work of
the authors. In this article, we study which type of subsets in the integers
and free abelian groups of higher rank can be a part of a co-minimal pair. We
show that a majority of lacunary sequences have this property. From the
conditions established, one can show that any infinite subset of any finitely
generated abelian group has uncountably many subsets which is a part of a
co-minimal pair. Further, the uncountable collection of sets can be chosen so
that they satisfy certain algebraic properties.
