Asymptotic behaviour 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 $W,W^{\prime }$ in a group $G$, the set $W^{\prime }$ is said to be a complement to $W$ if $W\cdot W^{\prime }=G$ and it is minimal if no proper subset of $W^{\prime }$ 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.