{
  "abstract": "Let ${\\mathbb P}^1_{\\mathfrak q}$ be the weighted projective line with coprime weights ${\\mathfrak q}=(q_0,q_1)$ and $L=q_0q_1$. The weighted Veronese map $Φ\\colon {\\mathbb P}^1_{\\mathfrak q}\\to {\\mathbb P}^1$ satisfies $h\\circΦ=L\\,{\\mathfrak s}$. Pulling back Roth's theorem gives the optimal exponent $2L$ for the full rational orbit set. For points represented by coprime integral coordinates, counting suggests the smaller exponent $q_0+q_1$. We formulate this as a conjecture and prove an $S$-adic radical form under the $abc$ conjecture. For ${\\mathfrak q}=(2,3)$ the resulting inequality at the target $[1:1]$ is the coprime Hall-type inequality, and Danilov--Elkies examples show that the exponent $5$ is sharp. We also prove the expected exponent unconditionally over characteristic-zero function fields for constant targets and polynomial approximants, and for almost every real target. A weighted $abc$ formulation is introduced; in signature $(3,2)$ finitely many coefficient pairs are equivalent to $abc$, and weighted Belyi maps transfer this input to arbitrary algebraic targets. Applications include a pullback form of Schmidt's subspace theorem, weighted Thue inequalities, and a counterexample to a naive weighted rescaling of Roth's index construction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17589",
  "author_total": 1,
  "authors": [
    {
      "name": "Sajad Salami"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/725fmzjrwc5pr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17589",
  "fulltext_url": null,
  "id": "725fmzjrwc5pr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17589",
  "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.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7221806205",
  "pdf_url": "https://arxiv.org/pdf/2610.17589",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/725fmzjrwc5pr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17589",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17589"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Roth's theorem on weighted projective lines",
  "title_text": "Roth's theorem on weighted projective lines",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/725fmzjrwc5pr",
  "venue": "arXiv",
  "versions": []
}