# Asymptotic complements in the integers

## TL;DR

This paper investigates the existence and non-existence of minimal asymptotic complements in the integers, a concept related to representing integers with sets, building on prior work by Nathanson and others.

## Contribution

It addresses Nathanson's problem by analyzing conditions for the existence or non-existence of minimal asymptotic complements in the integers.

## Key findings

- Characterizes when minimal asymptotic complements exist in 
- Provides examples of sets with and without minimal asymptotic complements
- Extends the understanding of asymptotic complements in additive number theory

## Abstract

Let $W\subseteq \mathbb{Z}$ be a non-empty subset of the integers. A nonempty set $C\subseteq \mathbb{Z}$ is said to be an asymptotic complement to $W$ if $W+C$ contains almost all the integers except a set of finite size. $C$ is said to be a minimal asymptotic complement if $C$ is an asymptotic complement, but $C\setminus \lbrace c\rbrace$ is not an asymptotic complement $\forall c\in C$. Asymptotic complements have been studied in the context of representations of integers since the time of Erd\H{o}s, Hanani, Lorentz and others, while the notion of minimal asymptotic complements is due to Nathanson. In this article, we study minimal asymptotic complements in $\mathbb{Z}$ and deal with a problem of Nathanson on their existence and their inexistence.