  We say a subset $C$ of an abelian group $G$ \textit{arises as a minimal
additive complement} if there is some other subset $W$ of $G$ such that
$C+W=\{c+w:c\in C,\ w\in W\}=G$ and such that there is no proper subset
$C'\supset C$ such that $C'+W=G$. In their recent paper, Burcroff and Luntzlara
studied, among many other things, the conditions under which "eventually
periodic sets", which are finite unions of infinite (in the positive direction)
arithmetic progressions and singletons, arise as minimal additive complements
in $\mathbb Z$. In the present paper we shall study this question further. We
give, in the form of bounds on the period $m$, some sufficient conditions for
an eventually periodic set to be a minimal additive complement; in particular
we show that "all eventually periodic sets are eventually minimal additive
complements". Moreover, we generalize this to a framework in which "patterns"
of points are projected down to $\mathbb Z$, and we show that all sets which
arise this way are eventually minimal additive complements. We also introduce a
formalism of formal power series, which serves purely as a bookkeeper in
writing down proofs. Through our work we are able to answer a question of
Burcroff and Luntzlara in a large class of cases.
