On zero-sum subsequences over finite abelian groups of length not exceeding a given number
Abstract
Let $G$ be a finite abelian group and let $k$ be an integer with $k\in [\exp (G), \mathsf{D}(G)- 1]$, where $\exp (G)$ and $\mathsf D(G)$ are the exponent and the Davenport constant of $G$, respectively. Denote by $\mathsf {s}_{\leq k}(G)$ the smallest positive integer $l\in \mathbb {N}\cup \{+\infty \}$ such that each sequence of length $l$ over $G$ has a nontrivial zero-sum subsequence of length at most $k$. We prove that $\mathsf {s}_{\leq \mathsf {D}(G)-2} (G)\leq \mathsf {D}(G)+2$ for all finite non-cyclic abelian groups except for $C_2^3$, $C_2^4$ and $C_2\oplus C_{2n}$. For some classes of $p$-groups $G$, we show that $\mathsf {s}_{\leq k}(G)\leq 2\mathsf {D}(G)-k$ for $k=c_1p^{t+1}\in \bigl [\frac{\mathsf {D}(G)+1}{2},\mathsf {D}(G)\bigr]$ with $c_1\in [0,p-1]$ and $t\geq 0$. In addition, some lower bounds are obtained.
