{
  "abstract": "We study the asymptotically maximal possible number of integer solutions to the linear equation $ax+by+cz = 0$ with a fixed choice of $a, b, c \\in \\mathbb{Z}$ and variables $x, y, z \\in S$ for some finite set $S\\subset \\mathbb{Z}$, as $|S|\\to +\\infty$. Define $γ_{a, b, c}$ to be the largest constant for which there are arbitrary large finite sets $S\\subset \\mathbb{Z}$ such that the number of solutions to $ax+by+cz=0$ with $x,y,z\\in S$ is $γ_{a,b,c}|S|^2-o(|S|^2)$. We prove structural results for general $a, b, c$ and moreover, we show that $5/13\\le γ_{1,1,-3}\\le 1/2-δ$ for some constant $δ>0$. In addition we show that the limit as $a \\rightarrow \\infty$ of $γ_{1,1,-a}$ is equal to precisely $1/5$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06975",
  "author_total": 5,
  "authors": [
    {
      "name": "Žarko Ranđelović"
    },
    {
      "name": "Xuancheng Shao"
    },
    {
      "name": "Max Wenqiang Xu"
    },
    {
      "name": "Shengtong Zhang"
    },
    {
      "name": "Yuan Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5bq94z4xa23km.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06975",
  "fulltext_url": null,
  "id": "5bq94z4xa23km",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06975",
  "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": "W7212056142",
  "pdf_url": "https://arxiv.org/pdf/2609.06975",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5bq94z4xa23km.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06975",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06975"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Maximising the number of solutions to linear equations",
  "title_text": "Maximising the number of solutions to linear equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/5bq94z4xa23km",
  "venue": "arXiv",
  "versions": []
}