{
  "abstract": "Let $P=\\{0,a,b\\}$, where $0<a<b$ and $\\gcd(a,b)=1$. For a finite set $A\\subset\\mathbb Z$, let $M_P^+(A)$ count the copies $x,x+ad,x+bd\\in A$ with $d>0$, and let $M_P(A)$ count the copies with any $d\\ne0$. We prove that every such three-point pattern other than the arithmetic progression $\\{0,1,2\\}$ satisfies \\[ M_P^+(A)\\le \\frac{99}{400}|A|^2+O(|A|), \\qquad M_P(A)\\le \\frac{13}{28}|A|^2+O_P(|A|). \\] For the particular pattern $P=\\{0,1,3\\}$ -- the subject of a question raised by Ganguly and recorded as Problem 24 in Green's list of open problems -- we sharpen the bound allowing both signs of the dilation to \\[ M_{\\{0,1,3\\}}(A)\\le \\frac{47}{122}|A|^2+O(|A|). \\]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02308",
  "author_total": 1,
  "authors": [
    {
      "name": "Samuel Korsky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0aan4yenmkp13.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02308",
  "fulltext_url": null,
  "id": "0aan4yenmkp13",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02308",
  "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.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7207693726",
  "pdf_url": "https://arxiv.org/pdf/2609.02308",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0aan4yenmkp13.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02308",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02308"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Affine Copies of Three-Point Patterns in Sets of Integers",
  "title_text": "Affine Copies of Three-Point Patterns in Sets of Integers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/0aan4yenmkp13",
  "venue": "arXiv",
  "versions": []
}