{
  "abstract": "For $α\\in\\mathbb{R}$ and $k\\in\\mathbb{N}$, let $S_{α,k}(x):=\\#\\{n\\in\\mathbb N:n\\le x,\\ \\gcd(n,\\lfloorαn^k\\rfloor)=1\\}$. Roth conjectured, and Watson proved, that $S_{α,1}(x)\\sim 6x/π^2$ for every irrational $α$. Whether the same asymptotic holds for every fixed $k\\ge2$ is a special case of a question of Bergelson and Richter, and was asked explicitly by Champagne. We answer this for $k=2$, with no Diophantine condition on $α$: for every irrational $α$, $S_{α,2}(x)\\sim x/ζ(2)$ as $x\\to\\infty$. More precisely, for every fixed $A>0$ we obtain an error term $O_A(x/(\\log q(x))^A)$, where $q(x)$ is the largest continued-fraction convergent denominator of $α$ not exceeding $x$. The proof sieves $S_{α,2}(x)$ over divisors, splitting at the parameter $R=q(x)^{1/8}$. Divisors larger than $R$ are handled by reducing the condition $d\\mid\\gcd(n,\\lfloorαn^2\\rfloor)$ to Möbius sums over arithmetic progressions and applying Zhan's uniform estimate for exponential sums twisted by $μ(n)$ over short intervals; divisors at most $R$ are handled instead through the Erdős-Turán inequality together with Weyl-type bounds for quadratic exponential sums of the form $\\sum e(βn^2)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.04345",
  "author_total": 1,
  "authors": [
    {
      "name": "Aahan Chatterjee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b8bc1peb0x96j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.04345",
  "fulltext_url": null,
  "id": "b8bc1peb0x96j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.04345",
  "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.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.04345",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b8bc1peb0x96j.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.04345",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.04345"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On integers $n$ relatively prime to $\\lfloor αn^2\\rfloor$",
  "title_text": "On integers n relatively prime to ⌊ αn^2⌋",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:38Z",
  "url": "https://research.draconitic.com/paper/b8bc1peb0x96j",
  "venue": "arXiv",
  "versions": []
}