Inverse problems for minimal complements and maximal supplements

TL;DR

This paper characterizes which subsets of finite abelian groups can serve as minimal complements for some other subset, revealing that most small subsets are minimal complements and extending results to infinite groups and dual problems.

Contribution

It proves that all sufficiently small subsets in finite abelian groups are minimal complements and extends the concept to infinite groups and dual problems.

Findings

Most small subsets in finite abelian groups are minimal complements.

Every non-empty subset of an infinite abelian group is a minimal complement.

Analogous results are established for maximal supplements.

Abstract

Given a subset $W$ of an abelian group $G$, a subset $C$ is called an additive complement for $W$ if $W+C=G$; if, moreover, no proper subset of $C$ has this property, then we say that $C$ is a minimal complement for $W$. It is natural to ask which subsets $C$ can arise as minimal complements for some $W$. We show that in a finite abelian group $G$, every non-empty subset $C$ of size $|C| \leq 2^{2/3}|G|^{1/3}/((3e \log |G|)^{2/3}$ is a minimal complement for some $W$. As a corollary, we deduce that every finite non-empty subset of an infinite abelian group is a minimal complement. We also derive several analogous results for ``dual'' problems about maximal supplements.