{
  "abstract": "We prove that for any set $A$ with positive multiplicative upper density, there exist $x$, $y$, $z$ and $w$ in $A$ such that $x^2 + y^2w^2 = z^2$, generalizing a result of Frantzikinakis, Klurman and Moreira. Our techniques apply more broadly to a wide class of systems of equations, allowing us to solve these equations simultaneously and furthermore to restrict the w variable in an entirely different multiplicatively dense set. To do this, we prove a novel multiple recurrence result for multiplicative dynamical systems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04078",
  "author_total": 3,
  "authors": [
    {
      "name": "James Leng"
    },
    {
      "name": "Redmond McNamara"
    },
    {
      "name": "Andreas Mountakis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dp43febxhv5qp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04078",
  "fulltext_url": null,
  "id": "dp43febxhv5qp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04078",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7220701449",
  "pdf_url": "https://arxiv.org/pdf/2610.04078",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dp43febxhv5qp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04078",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04078"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Monochromatic Almost-Pythagorean Quadruples",
  "title_text": "Monochromatic Almost-Pythagorean Quadruples",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/dp43febxhv5qp",
  "venue": "arXiv",
  "versions": []
}