{
  "abstract": "Given an integer $k \\geq 2$ and a prime $p$ large enough depending on $k$, we show that, if $A \\subseteq \\mathbb F^n_p$ does not contain a restricted polynomial progression of the form \\begin{align*} x, x+\\begin{pmatrix} y_1 \\\\ y_2 \\\\ \\vdots \\\\ y_n \\end{pmatrix}, x+\\begin{pmatrix} y_1^2 \\\\ y_2^2 \\\\ \\vdots \\\\ y_n^2 \\end{pmatrix}, \\ldots, x+\\begin{pmatrix} y_1^{k-1} \\\\ y_2^{k-1} \\\\ \\vdots \\\\ y_n^{k-1} \\end{pmatrix}, \\end{align*} where $x \\in \\mathbb F^n_p$ and $y \\in \\{0,2, 3, \\ldots, M_k\\}^n \\backslash \\{\\vec{0} \\}$, for some fixed $M_k$, then $A$ must have density bounded above by $\\frac{C}{(\\log^{(k^{O(k)})} n)^c}$, where $C, c>0$ only depend on $k$ and $p$. For $k\\geq 4$, the best previously known bound came from the density Hales--Jewett theorem and was of inverse Ackermann type. The proof uses a recent inverse theorem for noise stability of Bhangale, Khot, Liu and Minzer, which arose in the study of constraint satisfaction problems. The main new ingredient is a proof that the underlying distribution counting restricted polynomial progressions does not admit a nontrivial abelian embedding.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12831",
  "author_total": 5,
  "authors": [
    {
      "name": "Alex Burgin"
    },
    {
      "name": "Anastasios Fragkos"
    },
    {
      "name": "David Kruzel"
    },
    {
      "name": "Michael Lacey"
    },
    {
      "name": "Amelia Stokolosa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/423bppqp142x4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12831",
  "fulltext_url": null,
  "id": "423bppqp142x4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12831",
  "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.CA"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7221561980",
  "pdf_url": "https://arxiv.org/pdf/2610.12831",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/423bppqp142x4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.12831",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12831"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Reasonable bounds for Restricted Polynomial Progressions in $\\mathbb F^n_p$",
  "title_text": "Reasonable bounds for Restricted Polynomial Progressions in 𝔽^n_p",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/423bppqp142x4",
  "venue": "arXiv",
  "versions": []
}