# On Eventually Periodic Sets as Minimal Additive Complements

### Abstract

We say a subset $C$ of an abelian group $G$ *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 $C'\subset C$ 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 study this further and give, in the form of bounds on the period $m$, some sufficient conditions for an eventually periodic set to arise as 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 (subsets of $\mathbb Z^2$) 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, and we prove some basic properties of these series (e.g. sufficient conditions for inverses to be unique). Through our work we are able to answer a question of Burcroff and Luntzlara (when does $C_1\cup(-C_2)$ arise as a minimal additive complement, where $C_1,C_2$ are eventually periodic sets?) in a large class of cases.