On the zero-sum subsequences of modular restricted lengths

Abstract

Let G be an additive finite abelian group and let ℓ be a positive integer. Denote by discℓ(G) the smallest positive integer t such that every sequence S over G of length |S|≥t has a nonempty zero-sum subsequence T with length |T|≠ℓ. Let disc(G) denote the smallest positive integer t such that every sequence S over G of length |S|≥t has two nonempty zero-sum subsequences of distinct lengths. Gao et al. [8] proved that disc(G)=max⁡{discℓ(G):ℓ∈N0}. In this paper, we continue to investigate this invariant discℓ(G) by introducing a new invariant Ek,u(G). Let k and u be positive integers with k≥2 and u∈[1,k]. Denote by Ek,u(G) the smallest positive integer t such that every sequence S over G of length |S|≥t has a nonempty zero-sum subsequence T with length |T|≢u (mod k). In addition to consider the exact value and inverse problem of discℓ(G), we also study the relationship between Ek,u(G) and disc(G), discℓ(G) for various types of abelian groups. In particular, we determine the exact value of Ek,u(G) for G=Cn with u∈[1,k−1] and that of Ek,1(G) for elementary abelian 2-groups G=C2r, respectively.