On non-minimal complements

TL;DR

This paper explores the inverse problem of minimal complements in various groups, extending results beyond abelian groups and demonstrating the existence of many robust non-minimal complements.

Contribution

It provides new insights into the inverse problem for minimal complements in non-abelian and finitely generated abelian groups, including the construction of uncountably many non-minimal complements.

Findings

New results on inverse problems in non-abelian groups

Existence of uncountably many robust non-minimal complements

Extensions of minimal complement theory to broader group classes

Abstract

The notion of minimal complements was introduced by Nathanson in 2011. Since then, the existence or the inexistence of minimal complements of sets have been extensively studied. Recently, the study of inverse problems, i.e., which sets can or cannot occur as minimal complements has gained traction. For example, the works of Kwon, Alon--Kravitz--Larson, Burcroff--Luntzlara and also that of the authors, shed light on some of the questions in this direction. These works have focussed mainly on the group of integers, or on abelian groups. In this work, our motivation is two-fold: (i) to show some new results on the inverse problem, (ii) to concentrate on the inverse problem in not necessarily abelian groups. As a by-product, we obtain new results on non-minimal complements in the group of integers and more generally, in any finitely generated abelian group of positive rank and in any free abelian group of positive rank. Moreover, we show the existence of uncountably many subsets in such groups which are "robust" non-minimal complements.