On additive bases II
Volume 168 / 2015
Abstract
Let $G$ be an additive finite abelian group, and let $S$ be a sequence over $G$. We say that $S$ is regular if for every proper subgroup $H \subseteq G$, $S$ contains at most $|H|-1$ terms from $H$. Let $\mathsf c_0(G)$ be the smallest integer $t$ such that every regular sequence $S$ over $G$ of length $|S|\geq t$ forms an additive basis of $G$, i.e., every element of $G$ can be expressed as the sum over a nonempty subsequence of $S$. The constant $\mathsf c_0(G)$ has been determined previously only for the elementary abelian groups. In this paper, we determine $\mathsf c_0(G)$ for some groups including the cyclic groups, the groups of even order, the groups of rank at least five, and all the $p$-groups except $G=C_p\oplus C_{p^n}$ with $n\geq 2.$
