{
  "abstract": "For integers $m\\geq3$ and $n\\geq1$, an $m$-gonal form is called tight $\\mathcal{T}(n)$-universal if it represents exactly the positive integers $\\mathcal{T}(n)=\\{ n, n+1, n+2, \\ldots \\}$. In this paper, we study the minimal criterion set $\\mathrm{CS}(m,n)$ for tight $\\mathcal{T}(n)$-universality. Our main result determines $\\mathrm{CS}(m,n)$ for all $n \\geq8$, $3 \\leq m \\leq \\left\\lfloor \\frac{3n+1}{2} \\right\\rfloor$. More precisely, \\[ \\mathrm{CS}(m,n)= \\begin{cases} \\{ n, n+1, \\ldots, 2n-1 \\}, & m=5,\\newline \\{ n, n+1, \\ldots, 2n \\}, & m\\neq5. \\end{cases} \\] We also establish the corresponding tight $\\mathcal{T}(n)$-universality results and show that the upper bound on $m$ is optimal.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12779",
  "author_total": 2,
  "authors": [
    {
      "name": "Byeong Moon Kim"
    },
    {
      "name": "Ji Young Kim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4z218ppwcp8ec.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12779",
  "fulltext_url": null,
  "id": "4z218ppwcp8ec",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12779",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7213152338",
  "pdf_url": "https://arxiv.org/pdf/2609.12779",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4z218ppwcp8ec.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2609.12779",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12779"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Tight universality of $m$-gonal forms with minimal criterion sets",
  "title_text": "Tight universality of m-gonal forms with minimal criterion sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/4z218ppwcp8ec",
  "venue": "arXiv",
  "versions": []
}