More on the $h$-critical numbers of finite abelian groups

Abstract

For a finite abelian group $G$, a nonempty subset $A$ of $G$, and a positive integer $h$, we let $hA$ denote the $h$-fold sumset of $A$; that is, $hA$ is the collection of sums of $h$ not-necessarily-distinct elements of $A$. Furthermore, for a positive integer $s$, we set $[0,s] A=\cup_{h=0}^s h A$. We say that $A$ is a generating set of $G$ if there is a positive integer $s$ for which $[0,s] A=G$. The $h$-critical number $\chi (G,h)$ of $G$ is defined as the smallest positive integer $m$ for which $hA=G$ holds for every $m$-subset $A$ of $G$; similarly, $\chi (G,[0,s])$ is the smallest positive integer $m$ for which $[0,s]A=G$ holds for every $m$-subset $A$ of $G$. We define $\widehat{\chi} (G, h)$ as the smallest positive integer $m$ for which $hA=G$ holds for every generating $m$-subset $A$ of $G$; $\widehat{\chi} (G, [0,s])$ is defined similarly. The value of $\chi (G,h)$ has been determined by this author for all $G$ and $h$, and $\widehat{\chi} (G, [0,s])$ was introduced and resolved for some special cases by Klopsch and Lev. Here we determine the remaining two quantities in all cases.