{
  "abstract": "Let $δ\\in [0,1/2]$ and $Q\\geq 1$. Given a compact $C^{\\infty}$-curve $\\mathcal{C}$ in $\\mathbb{R}^n$, denote by $N_{\\mathcal{C}}(δ, Q)$ the number of pairs $(\\mathbf{a}, q)\\in \\mathbb{Z}^n\\times [Q/2,Q]$ such that the rational point $\\mathbf{a}/q$ is $δ/q$-close to $\\mathcal{C}$. We investigate the range of $δ$ in terms of $Q$ which is necessary for the heuristic $N_{\\mathcal{C}}(δ, Q)\\asympδ^{n-1}Q^2$ to be correct for the moment curve. For a nondegenerate curve $\\mathcal{C}\\subset \\mathbb{R}^n$ and for sufficiently large $Q$, Hickman and the second author previously established that $N_{\\mathcal{C}}(δ, Q)\\lesssim_ν δ^{n-1}Q^2$ for all $δ\\in[Q^{α(n+1)+ν},1/2)$ and any $ν>0$. Here $α(n)=-4n^{-2}+O(n^{-3})$ was explicitly computed. We show that this range is surprisingly close to being sharp in the asymptotic sense as $n\\to\\infty$, and can at most be extended to $δ\\in[Q^{α(n)},1/2)$. In particular, the sharp exponent can only be about $O(n^{-3})$ better, even though it was widely believed before that an improvement of $O(n^{-1})$ should be possible. The complementary upper bound in this remaining range has been recently established by Gan--Guo--Oh.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02443",
  "author_total": 3,
  "authors": [
    {
      "name": "Mingfeng Chen"
    },
    {
      "name": "Rajula Srivastava"
    },
    {
      "name": "Niclas Technau"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8vm69t0kxf7z6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02443",
  "fulltext_url": null,
  "id": "8vm69t0kxf7z6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02443",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7220429706",
  "pdf_url": "https://arxiv.org/pdf/2610.02443",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8vm69t0kxf7z6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.02443",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02443"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Counterexamples for Rational Points Near Curves",
  "title_text": "Counterexamples for Rational Points Near Curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/8vm69t0kxf7z6",
  "venue": "arXiv",
  "versions": []
}