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.