{
  "abstract": "Let $G$ be a finite abelian group with exponent $\\exp(G)$. For positive integer $k$, the generalized Erdős--Ginzburg--Ziv constant, denoted by $\\mathsf s_{k\\exp(G)}(G)$ is the least integer $\\ell$ such that every sequence of length at least $\\ell$ over $G$ has a zero-sum subsequence of length $k\\exp(G)$. In this paper, we prove that, for any $n\\ge 1$, \\[\\mathsf s_{3\\cdot 2^n}(C_{2^n}^3)=6\\cdot 2^n-3 \\quad\\text{and}\\quad \\mathsf s_{4\\cdot 3^n}(C_{3^n}^3)=7\\cdot 3^n-3. \\] This completes all the remaining cases in the study of $\\mathsf s_{kq}(C_q^3)$, where $q$ is a prime power and $k\\ge 3$. As a consequence, for every $k\\ge 3$, we have \\[ \\mathsf s_{kn}(C_n^3)=(k+3)n+O(\\frac{n}{\\log n}). \\] This verifies a conjecture of Kubertin asymptotically. Our approach combines the previous group ring techniques and a recent powerful generalization of the Chevalley-Warning and Ax-Katz theorems introduced by Grynkiewicz",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21577",
  "author_total": 2,
  "authors": [
    {
      "name": "Hanbin Zhang"
    },
    {
      "name": "Shiwen Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/29c4hrny39rvr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21577",
  "fulltext_url": null,
  "id": "29c4hrny39rvr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21577",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7222514757",
  "pdf_url": "https://arxiv.org/pdf/2610.21577",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/29c4hrny39rvr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21577",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21577"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Generalized Erdős--Ginzburg--Ziv Constants for finite abelian groups of rank three",
  "title_text": "Generalized Erdős--Ginzburg--Ziv Constants for finite abelian groups of rank three",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/29c4hrny39rvr",
  "venue": "arXiv",
  "versions": []
}