On Eventually Periodic Sets as Minimal Additive Complements

TL;DR

This paper investigates conditions under which eventually periodic sets in abelian groups serve as minimal additive complements, providing bounds, generalizations, and formal frameworks to identify such sets.

Contribution

It establishes sufficient conditions and bounds for eventually periodic sets to be minimal additive complements, extending previous results and introducing a formal power series framework.

Findings

All eventually periodic sets are eventually minimal additive complements.

Provided bounds on the period for such sets to be minimal complements.

Generalized the concept to pattern-based sets and answered related open questions.

Abstract

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.