Asymptotic behaviour of minimal complements

TL;DR

This paper explores the properties and inverse problem of minimal complements in groups, analyzing their asymptotic behavior and providing new insights into their structure and possible configurations.

Contribution

It presents new results on the inverse problem for minimal complements and investigates their asymptotic behavior, connecting these concepts in group theory.

Findings

New results on the inverse problem for minimal complements

Insights into the asymptotic behavior of minimal complements

Partial answers to questions about the structure of minimal complements

Abstract

The notion of minimal complements was introduced by Nathanson in 2011 as a natural group-theoretic analogue of the metric concept of nets. Given two non-empty subsets $W,W'$ in a group $G$, the set $W'$ is said to be a complement to $W$ if $W\cdot W'=G$ and it is minimal if no proper subset of $W'$ is a complement to $W$. The inverse problem asks which sets may or not occur as minimal complements. We show some new results on the inverse problem and investigate how the study of the inverse problem naturally gives rise to questions about the asymptotic behaviour of these sets, providing partial answers to some of them.