On sequences over a finite abelian group with zero-sum subsequences of forbidden lengths

Weidong Gao ¹   ; Yuanlin Li ²   ; Pingping Zhao ¹   ; Jujuan Zhuang ³

¹ Center for Combinatorics LPMC-TJKLC, Nankai University Tianjin 300071, P.R. China
² Department of Mathematics Brock University St. Catharines, Ontario, Canada L2S 3A1
³ Department of Mathematics Dalian Maritime University Dalian 116024, P.R. China

Colloquium Mathematicum, Tome 144 (2016) no. 1, pp. 31-44

Résumé

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 $G$.

Détail

DOI : 10.4064/cm6488-8-2015

Keywords: additive finite abelian group every positive integer ell mathrm disc ell smallest positive integer each sequence length geq has nonempty zero sum subsequence length equal ell paper determine mathrm disc ell certain finite groups including cyclic groups groups oplus elementary abelian groups following girard define mathrm disc smallest positive integer every sequence geq has nonempty zero sum subsequences distinct lengths shall prove mathrm disc max mathrm disc ell ell geq determine mathrm disc finite abelian p groups where geq rank nbsp