{
  "abstract": "Ferraguti and Micheli conjectured that non-trivial $F$-sets of every prescribed width exist over each finite field. We prove the finite-width part of their conjecture over $\\mathbb{F}_q$ for every $q\\neq2,3$. Fix a suitable prime $\\ell$. For a degree bound $D$, we consider the saturated family of all irreducible polynomials $g(X^{\\ell^j})$ whose core $g$ has degree at most $D$. A factor-descent lemma shows that every irreducible factor arising from a shifted difference belongs to the same family and has strictly smaller core degree. The saturated family is therefore an $F$-set of finite width. Dirichlet's theorem for $\\mathbb{F}_q[X]$ produces core ladders of arbitrary length, and Kummer lifting reproduces each ladder at infinitely many scales. These parallel ladders force the width to be sufficiently large; an appropriate tail of the nullity filtration then has the prescribed width.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05305",
  "author_total": 1,
  "authors": [
    {
      "name": "Alessandro Giannoni"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/079wbj0mbxfkc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05305",
  "fulltext_url": null,
  "id": "079wbj0mbxfkc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05305",
  "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",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7211914214",
  "pdf_url": "https://arxiv.org/pdf/2609.05305",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/079wbj0mbxfkc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.05305",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05305"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "F-sets of arbitrary finite width",
  "title_text": "F-sets of arbitrary finite width",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/079wbj0mbxfkc",
  "venue": "arXiv",
  "versions": []
}