# On minimal additive complements of integers

## TL;DR

This paper investigates the existence of minimal additive complements in the integers, focusing on sets that are eventually periodic or not, thereby addressing a problem posed by Nathanson.

## Contribution

It provides new insights into when minimal additive complements exist for eventually periodic and non-periodic sets, partially resolving Nathanson's problem.

## Key findings

- Minimal complements exist for certain eventually periodic sets.
- Non-periodic sets may lack minimal complements.
- Results extend understanding of additive complement structures.

## 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. Let $X\subseteq \mathbb{N}$. If there exists a positive integer $T$ such that $x+T\in X$ for all sufficiently large integers $x\in X$, then we call $X$ eventually periodic. In this paper, we study the existence of a minimal complement to $W$ when $W$ is eventually periodic or not. This partially answers a problem of Nathanson.