{
  "abstract": "For integers $r\\ge1$ and $n\\ge2$, let $s_r(n)$ be the least $N$ such that every sequence of $N$ vectors in $\\Z^r$, each nonzero modulo $n$, contains $n$ terms whose sum is zero modulo $n$ but nonzero modulo $n^2$. Let $w_r(n)$ be the corresponding least $N$ when any nonempty subsequence is allowed. These constants were introduced as zero-sum problems of new types by Z.-W. Sun. We prove $s_r(n)\\ge(2^{r+1}-2)(n-1)+1$ by constructing an explicit sequence from a lattice polytope, \\emph{disproving} his conjecture $s_2(n)=4n+1$ for $n\\ge4$. We obtain $w_r(2)=2r+1$ by classifying all sequences avoiding such nonempty subsequences modulo $4$. We also \\emph{prove} Sun's conjecture $w_2(n)=4n-3$ for $n=2^k3^\\ell p$, where $p$ is a prime and $k,\\ell\\ge0$ are integers, by classifying the extremal sequences of length $4p-4$ modulo $p^2$ when $n=p$. For every fixed $n\\ge2$, we prove $w_r(n)/s(\\Z_n^r)=O_n(r^2/2^r)\\to0$ as $r\\to\\infty$, where $s(\\Z_n^r)$ is the classical Erdős-Ginzburg-Ziv constant, with an explicit threshold on $r$ for strict inequality. Thus, for every fixed modulus $n\\ge2$, Sun's proposed equality fails for sufficiently large $r$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19965",
  "author_total": 4,
  "authors": [
    {
      "name": "Yang Li"
    },
    {
      "name": "San Ling"
    },
    {
      "name": "Zhonghua Sun"
    },
    {
      "name": "Shixin Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/69d4y8c0jmvt9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19965",
  "fulltext_url": null,
  "id": "69d4y8c0jmvt9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19965",
  "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": "W7222057266",
  "pdf_url": "https://arxiv.org/pdf/2610.19965",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/69d4y8c0jmvt9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19965",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19965"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Proofs and disproofs of some conjectures on zero-sum problems of new types",
  "title_text": "Proofs and disproofs of some conjectures on zero-sum problems of new types",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/69d4y8c0jmvt9",
  "venue": "arXiv",
  "versions": []
}