{
  "abstract": "In this paper, we explore how large a subset of a finite group can be without containing non-trivial solutions to linear equations of the form $aX+bY=cZ+dU$ (where $a+b=c+d$). These equations naturally generalize the classic concept of Sidon sets. Using Tao's slice rank method, we first establish strict upper bounds for the size of such solution-free sets in the vector space $\\mathbb{F}_{p}^{n}$. Next, we turn to cyclic groups $\\mathbb{Z}_{m}$ and construct surprisingly large sets that have no solutions for the asymmetric equation $X+5Y=3U+3Z$. These constructions achieve a logarithmic density of roughly $0.5283$, breaking the expected $0.5$ barrier for any sufficiently large modulus $m$. Finally, we show that this high-density behavior extends to a wider family of equations whose coefficients follow simple rules modulo $8$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09887",
  "author_total": 1,
  "authors": [
    {
      "name": "Katalin Gyarmati"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/03d31qfaargcs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09887",
  "fulltext_url": null,
  "id": "03d31qfaargcs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09887",
  "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.NT"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7221008106",
  "pdf_url": "https://arxiv.org/pdf/2610.09887",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/03d31qfaargcs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.09887",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09887"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Translation-invariant equations in $\\mathbb{F}_p^n$ and groups",
  "title_text": "Translation-invariant equations in 𝔽_p^n and groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/03d31qfaargcs",
  "venue": "arXiv",
  "versions": []
}