{
  "abstract": "Murphy and Petridis proved that, for subsets $A$ of an arbitrary finite field $\\F_q$, the condition $\\abs A>q^{2/3-δ}$ implies $\\abs{(A-A)(A-A)}>q/2$ whenever $δ<1/13542$ and $q$ is sufficiently large. We improve the explicit saving to $1/1689$. The proof replaces the main physical-space extraction in their large-energy branch by a common Fourier spectrum, followed by a popular-ratio Balog--Szemerédi--Gowers step. The resulting additive energy is treated level by level, retaining difference multiplicities and combining the Reiher--Schoen relation lemma with Mohammadi's difference--ratio estimate. This gives the structural exponent $188$ and hence \\[ \\abs A\\ge Cq^{375/563}=Cq^{2/3-1/1689} \\quad\\Longrightarrow\\quad \\abs{(A-A)(A-A)}>\\frac q2 \\] for an absolute constant $C>0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38244",
  "author_total": 1,
  "authors": [
    {
      "name": "Yao Zhi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2xykjgdzvp1eq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38244",
  "fulltext_url": null,
  "id": "2xykjgdzvp1eq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38244",
  "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"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7215118613",
  "pdf_url": "https://arxiv.org/pdf/2609.38244",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2xykjgdzvp1eq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.38244",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38244"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "An Improved Exponent for Products of Differences over Arbitrary Finite Fields",
  "title_text": "An Improved Exponent for Products of Differences over Arbitrary Finite Fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/2xykjgdzvp1eq",
  "venue": "arXiv",
  "versions": []
}