{
  "abstract": "We extend the simultaneous popular polynomial difference theorem of Conlon, Dong, and the author to polynomial shifts along prescribed directions in $\\mathbb{F}_p^D$. Let $P_1,\\ldots,P_k\\in\\mathbb{Z}[t]$ be linearly independent polynomials with zero constant terms, and let $v_1,\\ldots,v_k\\in\\mathbb{Z}^D\\setminus\\{0\\}$. We show that, for every $\\varepsilon>0$, every sufficiently large prime $p$, and every $A\\subseteq\\mathbb{F}_p^D$ of density $α$, there exists a nonzero $d\\in\\mathbb{F}_p$ such that \\[ \\mathbb{E}_{\\underline{x}\\in\\mathbb{F}_p^D} \\mathbf{1}_A(\\underline{x}) \\prod_{\\substack{1\\leq i\\leq k\\\\ ω_i=1}} \\mathbf{1}_A\\bigl(\\underline{x}+P_i(d)v_i\\bigr) \\geq α^{\\,1+\\sum_{i=1}^kω_i}-\\varepsilon \\] simultaneously for every $ω\\in\\{0,1\\}^k$. The main new ingredient is a directional arithmetic regularity decomposition adapted to mixed directional box norms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30335",
  "author_total": 1,
  "authors": [
    {
      "name": "Guo-Dong Hong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d7awakz86mm1j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30335",
  "fulltext_url": null,
  "id": "d7awakz86mm1j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30335",
  "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": "W7223410541",
  "pdf_url": "https://arxiv.org/pdf/2610.30335",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d7awakz86mm1j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.30335",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30335"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Simultaneous popular polynomial differences in higher dimensions",
  "title_text": "Simultaneous popular polynomial differences in higher dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/d7awakz86mm1j",
  "venue": "arXiv",
  "versions": []
}