On Eventually Periodic Sets as Minimal Additive Complements

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 is an element of C, w is an element of W} = G and such that there is no proper subset C ' subset of C such that C ' + W = G. In their recent paper, Burcroff and Luntzlara studied, among many other things, the con-ditions under which eventually periodic sets, which are finite unions of infinite (in the positive direction) arithmetic progressions and singletons, arise as minimal ad-ditive complements in 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". More-over, we generalize this to a framework in which "patterns" of points (subsets of Z2) are projected down to 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 C1 boolean OR (-C2) arise as a minimal additive complement, where C1, C2 are eventually periodic sets?) in a large class of cases.