{
  "abstract": "We prove that any subset $A \\subseteq \\{1, \\dots, N\\}$ lacking non-trivial polynomial progressions $(x, x + y, x + y^2)$ satisfies $\\lvert A \\rvert \\le CN \\exp\\bigl(-c(\\log \\log N)^β\\bigr)$ for effective absolute constants $C, c, β> 0$. The argument develops a density increment framework localized to anisotropic quadratic Bohr sets. To handle mixed linear and quadratic frequencies, we establish a multilinear transference principle coupled with a relative quadratic inverse theorem based on Balog--Szemer{é}di--Gowers additive energy rigidity. Boundary stability across increments is maintained via semi-algebraic model regularization using cylindrical algebraic decomposition and Cauchy--Crofton estimates, while dimension growth is controlled by dissociated basis reduction and multidimensional Erd{ő}s--Tur{á}n--Koksma equidistribution.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13204",
  "author_total": 1,
  "authors": [
    {
      "name": "Yunfan Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b47hmfcccjy5c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13204",
  "fulltext_url": null,
  "id": "b47hmfcccjy5c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13204",
  "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": "W7221554557",
  "pdf_url": "https://arxiv.org/pdf/2610.13204",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b47hmfcccjy5c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.13204",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13204"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the Quantitative Bounds of Nonlinear Roth Theorem",
  "title_text": "On the Quantitative Bounds of Nonlinear Roth Theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/b47hmfcccjy5c",
  "venue": "arXiv",
  "versions": []
}