{
  "abstract": "For $E\\subset\\mathbb{F}_p^d$, let $Δ(E)=\\{Q_d(x-y):x,y\\in E\\}$, where $Q_d(z)=z_1^2+\\cdots+z_d^2$. We introduce and study the problem of the existence of nondegenerate three-term geometric progressions in the distance set $Δ(E)$ when $d$ is odd. If $M_{GP,d}(p)$ denotes the largest size of a set whose distance set contains no such progression, then, for every prime $p\\geq7$ and every integer $m\\geq0$, \\[ p^mM_{GP,1}(p)\\leq M_{GP,2m+1}(p) \\leq4p^mM_{GP,1}(p). \\] For all sufficiently large primes $p$, we also prove \\[ p^{m+\\frac{7}{26}}\\leq M_{GP,2m+1}(p) \\leq Cp^{m+\\frac{16}{25}}, \\] where $C$ is an absolute positive constant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34758",
  "author_total": 3,
  "authors": [
    {
      "name": "Thang Pham"
    },
    {
      "name": "Dung The Tran"
    },
    {
      "name": "Boqing Xue"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/amst6ptr15hff.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34758",
  "fulltext_url": null,
  "id": "amst6ptr15hff",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34758",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "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": "W7223835023",
  "pdf_url": "https://arxiv.org/pdf/2610.34758",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/amst6ptr15hff.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34758",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34758"
    }
  ],
  "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": "Geometric progressions in distance sets in odd dimensions over prime fields",
  "title_text": "Geometric progressions in distance sets in odd dimensions over prime fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/amst6ptr15hff",
  "venue": "arXiv",
  "versions": []
}