  Given two non-empty subsets $W,W'\subseteq G$ in an arbitrary abelian group
$G$, $W'$ is said to be an additive complement to $W$ if $W + W'=G$ and it is
minimal if no proper subset of $W'$ is a complement to $W$. The notion was
introduced by Nathanson and previous work by him, Chen--Yang,
Kiss--S\`andor--Yang etc. focussed on $G =\mathbb{Z}$. In the higher rank case,
recent work by the authors treated a class of subsets, namely the eventually
periodic sets. However, for infinite subsets, not of the above type, the
question of existence or inexistence of minimal complements is open. In this
article, we study subsets which are not eventually periodic. We introduce the
notion of "spiked subsets" and give necessary and sufficient conditions for the
existence of minimal complements for them. This provides a partial answer to a
problem of Nathanson.
