The $h$-critical number of finite abelian groups

Abstract

For a finite abelian group $G$ and a positive integer $h$, the unrestricted (resp.~restricted) $h$-critical number $\chi(G,h)$ (resp.~$\chi \hat{\;}(G,h)$) of $G$ is defined to be the minimum value of $m$, if exists, for which the $h$-fold unrestricted (resp.~restricted) sumset of every $m$-subset of $G$ equals $G$ itself. Here we determine $\chi(G,h)$ for all $G$ and $h$; and prove several results for $\chi \hat{\;}(G,h)$, including the cases of any $G$ and $h = 2$, any $G$ and large $h$, and any $h$ for the cyclic group $\mathbb{Z}_n$ of even order. We also provide a lower bound for $\chi \hat{\;}(\mathbb{Z}_n,3)$ that we believe is exact for every $n$---this conjecture is a generalization of the one made by Gallardo, Grekos, et al.~that was proved (for large $n$) by Lev.