# On additive co-minimal pairs

## TL;DR

This paper investigates the structure and properties of co-minimal pairs in abelian groups, establishing existence results for finite sets, exploring infinite cases, and analyzing related concepts like subgroups and approximate subgroups.

## Contribution

It introduces the concept of co-minimal pairs, proves that all non-empty finite sets in free abelian groups are part of such pairs, and examines infinite sets and related subgroup structures.

## Key findings

- Finite non-empty sets in free abelian groups belong to co-minimal pairs
- Infinite sets forming co-minimal pairs are characterized
- Sets that cannot be part of any minimal pair are identified

## Abstract

A pair of non-empty subsets $(W,W')$ in an abelian group $G$ is an additive complement pair if $W+W'=G$. $W'$ is said to be minimal to $W$ if $W+(W'\setminus \{w'\}) \neq G, \forall \,w'\in W'$. In general, given an arbitrary subset in a group, the existence of minimal complement(s) depends on its structure. The dual problem asks that given such a set, if it is a minimal complement to some subset. Additive complements have been studied in the context of representations of integers since the time of Erd\H{o}s, Hanani, Lorentz and others. The notion of minimal complements is due to Nathanson. We study tightness property of complement pairs $(W,W')$ such that both $W$ and $W'$ are minimal to each other. These are termed co-minimal pairs and we show that any non-empty finite set in an arbitrary free abelian group belongs to some co-minimal pair. We also study infinite sets forming co-minimal pairs. At the other extreme, motivated by unbounded arithmetic progressions in the integers, we look at sets which can never be a part of any minimal pair. This leads to a discussion on co-minimality, subgroups, approximate subgroups and asymptotic approximate subgroups of $G$.