{
  "abstract": "Fix $k,s\\in\\mathbb{N}$. We prove that there exists a subadditive density on $\\mathbb{N}$ such that, for every polynomial $P\\in\\mathbb{Z}[y]$ of degree $k$ and with $P(0)=0$, every set of positive density contains configurations $\\{x,x+P(y),xy^s\\}$ for arbitrarily large $x>y\\geq 2$. This provides a density strengthening of Moreira's result on partition regularity for $\\{x,x+y,xy\\}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30410",
  "author_total": 2,
  "authors": [
    {
      "name": "Florian K. Richter"
    },
    {
      "name": "Konstantinos Tsinas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/97pwr32ynn9zv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30410",
  "fulltext_url": null,
  "id": "97pwr32ynn9zv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30410",
  "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"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7214654194",
  "pdf_url": "https://arxiv.org/pdf/2609.30410",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/97pwr32ynn9zv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2609.30410",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30410"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Density regularity of $\\{x,x+y,xy\\}$ in the integers",
  "title_text": "Density regularity of {x,x+y,xy} in the integers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/97pwr32ynn9zv",
  "venue": "arXiv",
  "versions": []
}