On zero-sum subsequences in a finite abelian group of length not exceeding a given number

2025/06/26 by Kevin Zhao, Zhao, Kevin
Mathematics · Engineering · #Limits and Structures in Graph Theory #Finite Group Theory Research #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2506.21383

Abstract

Let G be an additive finite abelian group and let k∈ [exp(G),D(G)-1] be a positive integer. Denote by s_{≤ k}(G) the smallest positive integer l∈ ℕ∪ \+∞\ such that each sequence of length l over G has a non-empty zero-sum subsequence of length at most k. Let k_G∈ [exp(G),D(G)-1] be the smallest positive integer such that s_{≤ D(G)-d}(G)≤ D(G)+d for D(G)-d≥ k_G. We conjecture that k_G=(D(G)+1)/(2) for finite abelian groups G with r(G)≥ 2 and D(G)=D^*(G). In this paper, we mainly study this conjecture for finite abelian p-groups and get some results to support this conjecture. We also prove that k_G≤ D(G)-2 for all finite abelian groups G with r(G)≥ 2 except C₂³ and C₂⁴. In addition, we also get some lower bounds for the invariant s_{≤ k}(G).

Citations

• On some zero-sum invariants for abelian groups of rank three
• Structure of a sequence with prescribed zero-sum subsequences: Rank Two p-groups
• A Multiplicative Property for Zero-Sums II
• A Multiplicative Property for Zero-Sums I
• On zero-sum subsequences of length exp(G)
• Short Zero-Sum Sequences Over Abelian p-Groups of Large Exponent
• On short zero-sum subsequences of zero-sum sequences
• Zero-sum problems with congruence conditions
• Remarks on a generalization of the Davenport constant

Related

• Zero-sum invariants of finite abelian groups
• On the number of subsequences with a given sum in a finite abelian group
• On short zero-sum subsequences of zero-sum sequences
• On the number of weighted subsequences with zero-sum in a finite abelian group
• The structure of sequences with zero-sum subsequences of the same length on finite abelian groups of rank two