{
  "abstract": "Let $p$ be an odd prime, let $m\\geq1$ be an integer, and let $Q$ be a nondegenerate quadratic form on $\\mathbb F_p^{2m}$. For a nonempty set $E\\subseteq\\mathbb F_p^{2m}$, write $Δ_Q(E)=\\{Q(x-y):x,y\\in E\\}$. We prove that, whenever $|E|>p^m$, \\[ |Δ_Q(E)|\\gg \\min\\left\\{\\frac{p}{L_E},\\, \\frac{|E|}{p^{m-1}L_E^2}\\right\\}, \\qquad L_E=\\log\\left(2+\\frac{p^{m+1}}{|E|}\\right), \\] with an absolute implied constant independent of $p$, $m$ and $Q$. This confirms the Erdős--Falconer distance conjecture up to a logarithmic factor. In the anisotropic planar case $m=1$, for small sets, we also prove that \\[ |Δ_Q(E)|\\gg\\frac{|E|}{\\log(2|E|)}, \\qquad (1\\leq |E|\\leq p), \\] which is also optimal up to a logarithmic factor.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01795",
  "author_total": 4,
  "authors": [
    {
      "name": "Thang Pham"
    },
    {
      "name": "Chun-Yen Shen"
    },
    {
      "name": "Dung The Tran"
    },
    {
      "name": "Boqing Xue"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8d57d9br254se.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01795",
  "fulltext_url": null,
  "id": "8d57d9br254se",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01795",
  "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": "W7216372112",
  "pdf_url": "https://arxiv.org/pdf/2610.01795",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8d57d9br254se.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.01795",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01795"
    }
  ],
  "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": "Quadratic distances in even dimensions over prime fields",
  "title_text": "Quadratic distances in even dimensions over prime fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/8d57d9br254se",
  "venue": "arXiv",
  "versions": []
}