{
  "abstract": "We establish a uniform product-difference theorem for dense subsets of $\\mathbb Z^2$, which gives an affirmative answer to Problem~2 of Fish and, as consequences, to both parts of his Problem~1. More precisely, we prove that for every $δ>0$ there exists an integer $K(δ)\\geq 1$ such that every set $E\\subseteq\\mathbb Z^2$ with upper Banach density $d^\\star(E)\\geqδ$ satisfies \\[ K(δ)\\mathbb Z \\subseteq \\{ab:(a,b)\\in E-E\\}. \\] As consequences, we obtain affirmative answers to both parts of Fish's Problem~1: for positive-density sets $E_1,E_2\\subseteq\\mathbb Z$ and $E\\subseteq\\mathbb Z^2$, respectively, the sets \\[ (E_1-E_1)^2-(E_2-E_2)^2 \\quad\\text{and}\\quad \\{x^2-y^2:(x,y)\\in E-E\\} \\] contain nontrivial ideals of $\\mathbb Z$, with generators depending only on the corresponding density thresholds. In particular, the latter result also settles a conjecture of Davies concerning differences of the indefinite quadratic form $x^2-y^2$ in dense subsets of $\\mathbb Z^2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06782",
  "author_total": 2,
  "authors": [
    {
      "name": "Sayan Goswami"
    },
    {
      "name": "Chunlin Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4svyjrm15e3jh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06782",
  "fulltext_url": null,
  "id": "4svyjrm15e3jh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06782",
  "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"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7212021338",
  "pdf_url": "https://arxiv.org/pdf/2609.06782",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4svyjrm15e3jh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06782",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06782"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "A Uniform Product-Difference Theorem for Dense Subsets of $\\mathbb Z^2$",
  "title_text": "A Uniform Product-Difference Theorem for Dense Subsets of ℤ^2",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/4svyjrm15e3jh",
  "venue": "arXiv",
  "versions": []
}