A Note on Minimal Additive Complements

TL;DR

This paper investigates the structure of minimal additive complements in the integers, providing partial solutions to a specific class of sets and establishing foundational results on their existence.

Contribution

It offers a partial characterization of minimal additive complements for certain structured sets and proves an existence theorem analogous to Nathanson's initial results.

Findings

Partial answer to the structure of minimal additive complements for specific sets

Introduction of a dual problem characterizing sets as minimal additive complements

Proved an existence theorem for minimal additive complements in the integers

Abstract

Let $C, W \subseteq \mathbb{Z}$. If $C + W = \mathbb{Z}$, then the set $C$ is called an additive complement to $W$ in $\mathbb{Z}$. If no proper subset of $C$ is an additive complement to $W$, then $C$ is called a minimal additive complement. We provide a partial answer to a question posed by Kiss, S\'andor, and Yang regarding the minimal additive complement of sets of the form $W = (n \mathbb{N} + A) \cup F \cup G$, where $|F|<\infty, (F \mod{n}) \subseteq (A \mod{n})$ and $(G \mod{n}) \cap (A \mod{n}) = \emptyset$. We also introduce the dual problem of characterizing sets that arise as the minimal additive complements of some set of integers, proving the analog of Nathanson's initial result on existence of minimal additive complements.