On sequences over a finite abelian group with zero-sum subsequences of forbidden lengths
Volume 144 / 2016
Abstract
Let $G$ be an additive finite abelian group. For every positive integer $\ell$, let $\mathrm{disc}_{\ell}(G)$ be the smallest positive integer $t$ such that each sequence $S$ over $G$ of length $|S|\geq t$ has a nonempty zero-sum subsequence of length not equal to $\ell$. In this paper, we determine $\mathrm{disc}_{\ell}(G)$ for certain finite groups, including cyclic groups, the groups $G=C_2\oplus C_{2m}$ and elementary abelian $2$-groups. Following Girard, we define $\mathrm{disc}(G)$ as the smallest positive integer $t$ such that every sequence $S$ over $G$ with $|S|\geq t$ has nonempty zero-sum subsequences of distinct lengths. We shall prove that $\mathrm{disc}(G)=\max \{\mathrm{disc}_{\ell}(G)\,|\, \ell \geq 1 \}$ and determine $\mathrm{disc}(G)$ for finite abelian $p$-groups $G$, where $p\geq r(G)$ and $r(G)$ is the rank of $G$.
