# Minimal additive complements in finitely generated abelian groups

## TL;DR

This paper investigates the existence of minimal additive complements in finitely generated abelian groups, introducing 'spiked subsets' and establishing conditions for their minimal complements, extending previous work beyond periodic sets.

## Contribution

It introduces the concept of 'spiked subsets' in finitely generated abelian groups and characterizes when they have minimal additive complements, advancing understanding beyond periodic sets.

## Key findings

- Necessary and sufficient conditions for minimal complements of spiked subsets
- Extension of additive complement theory to non-periodic subsets
- Partial resolution of Nathanson's open problem

## Abstract

Given two non-empty subsets $W,W'\subseteq G$ in an arbitrary abelian group $G$, $W'$ is said to be an additive complement to $W$ if $W + W'=G$ and it is minimal if no proper subset of $W'$ is a complement to $W$. The notion was introduced by Nathanson and previous work by him, Chen--Yang, Kiss--S\`andor--Yang etc. focussed on $G =\mathbb{Z}$. In the higher rank case, recent work by the authors treated a class of subsets, namely the eventually periodic sets. However, for infinite subsets, not of the above type, the question of existence or inexistence of minimal complements is open. In this article, we study subsets which are not eventually periodic. We introduce the notion of "spiked subsets" and give necessary and sufficient conditions for the existence of minimal complements for them. This provides a partial answer to a problem of Nathanson.