{
  "abstract": "Green and Sawhney recently obtained a quasipolynomial bound in the Furstenberg--Sárközy theorem for square differences by proving an ``arithmetic level-d'' inequality, thereby yielding a greatly improved density increment scheme. We apply their method to treat general intersective polynomials $h\\in\\mathbb{Z}[x]$. In particular, let \\[ D(h(\\mathbb{N}),X):= \\max{|A|:\\ A\\subseteq [1,X]\\cap\\mathbb{N} \\text{and}\\ (A-A)\\cap h(\\mathbb{N})\\subseteq\\{0\\}}. \\] We prove that for every $0<μ<1/2$ there are constants $c_0, X_{\\text{min}}>0$ depending on $h$ and $μ$ such that for every $X>X_{\\text{min}}$, \\[D(h(\\mathbb{N}), X)\\leq Xe^{-c_0(\\log X)^μ}.\\] This is the best quantitative upper bound presently known for sets lacking intersective polynomial differences, improving upon the work of Arala. In order to achieve the admissible exponent range $0<μ<1/2$, we use sieve methods to develop novel exponential sum estimates in the style of Rice, and we use the ``random sparsification'' procedure of Green and Sawhney.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.16216",
  "author_total": 9,
  "authors": [
    {
      "name": "Carlo Francisco E. Adajar"
    },
    {
      "name": "Rishika Agrawal"
    },
    {
      "name": "Mukul Rai Choudhuri"
    },
    {
      "name": "Chian Yeong Chuah"
    },
    {
      "name": "Steve Fan"
    },
    {
      "name": "Swaroop Hegde"
    },
    {
      "name": "Andrew Lott"
    },
    {
      "name": "Krishnamohan Nandakumar"
    },
    {
      "name": "Nagendar Reddy Ponagandla"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9g5pk2kxe597k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.16216",
  "fulltext_url": null,
  "id": "9g5pk2kxe597k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.16216",
  "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",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7161564934",
  "pdf_url": "https://arxiv.org/pdf/2605.16216",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9g5pk2kxe597k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2605.16216",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.16216"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Extensions of the Furstenberg-Sárközy theorem via the arithmetic level-$d$ inequality",
  "title_text": "Extensions of the Furstenberg-Sárközy theorem via the arithmetic level-d inequality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/9g5pk2kxe597k",
  "venue": "arXiv",
  "versions": []
}