{
  "abstract": "We prove a sharp quantitative form of the van der Corput property for squares. For all sufficiently large $N$, we construct a real trigonometric polynomial $P(θ)=a_0+\\sum_{1\\le n\\le\\sqrt N}a_n\\cos(2πn^2θ)$ satisfying \\begin{equation}\\tag{1}\\label{eq:abstract-vdc} P(θ)\\ge0\\quad(θ\\in\\mathbb{R}/\\mathbb{Z}),\\qquad P(0)=1, \\end{equation} with \\[0\\le a_0\\le\\exp(-c\\sqrt{\\log N\\log\\log N}),\\] where $c>0$ is an absolute constant. As a consequence, if $A\\subseteq\\mathbb{N}\\cap[1,N]$ has no nonzero square difference, then \\[|A|\\le N\\exp(-c\\sqrt{\\log N\\log\\log N}),\\] improving upon a recent result of Green and Sawhney. Our construction builds on previous work of Slijepčević, Slijepčević--Ninčević, and Fan--Lott, and the proof of the upper bound is elementary and self-contained. We also show that every real trigonometric polynomial of the above form satisfying \\eqref{eq:abstract-vdc} has \\[a_0\\ge\\exp(-C\\sqrt{\\log N\\log\\log N})\\] for an absolute constant $C>0$. Thus the van der Corput property cannot be used to obtain a power-saving bound for the Furstenberg--Sárközy theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07765",
  "author_total": 2,
  "authors": [
    {
      "name": "Steve Fan"
    },
    {
      "name": "Andrew Lott"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c0e3598v476yc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07765",
  "fulltext_url": null,
  "id": "c0e3598v476yc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07765",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7221050200",
  "pdf_url": "https://arxiv.org/pdf/2610.07765",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c0e3598v476yc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.07765",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07765"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The optimal van der Corput property for squares",
  "title_text": "The optimal van der Corput property for squares",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/c0e3598v476yc",
  "venue": "arXiv",
  "versions": []
}