{
  "abstract": "Let $q$ be an odd prime power and $m\\geq 1$. Let $Q$ be a nondegenerate quadratic form on $\\mathbb F_q^{2m}$. For every sphere $S_j=\\{Q=j\\}$ with $j\\in \\mathbb F_q^\\times$, we prove the sharp extension estimate $R_{S_j}^*(2\\to r)\\lesssim_m1$ for $r\\ge 2(m+1)/m$, uniformly in $q$, $Q$, and $j$. As an application, we show that if $E\\subseteq\\mathbb F_q^{2m}$ satisfies $|E|/q^{m+1/3}\\to\\infty$, then almost every pin $y\\in E$ determines $(1-o(1))q$ values of $Q(x-y)$. The proof combines two arithmetic Hecke operator estimates with an induction in the dimension, a centered sphere--cone estimate, and an orthogonal decomposition.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39876",
  "author_total": 2,
  "authors": [
    {
      "name": "Thang Pham"
    },
    {
      "name": "Boqing Xue"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dmetwnhweqrx2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39876",
  "fulltext_url": null,
  "id": "dmetwnhweqrx2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39876",
  "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.CA"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7215080244",
  "pdf_url": "https://arxiv.org/pdf/2609.39876",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dmetwnhweqrx2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39876",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39876"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sharp spherical extension theorem in $\\mathbb F_q^{2m}$ and applications",
  "title_text": "Sharp spherical extension theorem in 𝔽_q^2m and applications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/dmetwnhweqrx2",
  "venue": "arXiv",
  "versions": []
}