{
  "abstract": "For fixed integers $d,r\\ge1$ with $dr\\ge2$, the $r$-free points of $\\mathbb{Z}^d$ are those whose coordinate gcd is not divisible by the $r$th power of any prime. Fix $1\\le q\\le \\infty$. A lattice point is $R$-deep if every lattice point in the closed $\\ell_q$-ball of radius $R$ centred there is non-$r$-free. We mark each finite nearest-neighbour component of non-$r$-free points containing an $R$-deep point by its lexicographically least such point. Uniformly for $1\\le q\\le\\infty$, the densities of $R$-deep points and of these representatives both have the asymptotic form $\\exp\\left\\{-\\frac{v_{d,q}R^d}{ζ(dr)}\\left[d(dr-1)\\log R+dr\\log\\log R-Υ_{d,r,q}+r\\frac{\\log\\log R}{\\log R}-\\frac{Λ_{d,r,q}}{\\log R}+O_{d,r}\\left(\\frac{(\\log\\log R)^2}{(\\log R)^2}\\right)\\right]\\right\\}$ as $R\\to\\infty$, where $v_{d,q}$ is the volume of the unit $\\ell_q$-ball and $Υ_{d,r,q},Λ_{d,r,q}$ are explicit constants. For deep points, this sharpens the positive-density hole constructions of Baake, Moody, and Pleasants and of Pleasants and Huck, and the latter authors' upper bounds for sparse-pattern frequencies. For $d=1$, consider the densities of $r$-free integers followed by at least $g-1$ consecutive non-$r$-free integers, or by exactly $g-1$ such integers and then another $r$-free integer. Both have the asymptotics $\\exp\\left\\{-\\frac{g}{ζ(r)}\\left[(r-1)\\log g+r\\log\\log g-\\widehatΥ_r+r\\frac{\\log\\log g}{\\log g}-\\frac{\\widehatΛ_r}{\\log g}+O_r\\left(\\frac{(\\log\\log g)^2}{(\\log g)^2}\\right)\\right]\\right\\}$ as $g\\to\\infty$, where $\\widehatΥ_r=Υ_{1,r,1}+(r-1)\\log2$ and $\\widehatΛ_r=Λ_{1,r,1}+r\\log2$. The exact-gap expansion improves Grimmett's leading asymptotic and refines the formula in Jiang's recent preprint. For fixed $r$, we also obtain estimates uniform in growing dimensions $d=O_r((\\log R)^r)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25457",
  "author_total": 1,
  "authors": [
    {
      "name": "Francesco Cellarosi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/96x51txhn0rwq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25457",
  "fulltext_url": null,
  "id": "96x51txhn0rwq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25457",
  "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": "W7214246044",
  "pdf_url": "https://arxiv.org/pdf/2609.25457",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/96x51txhn0rwq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2609.25457",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25457"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Density of large holes among power-free lattice points",
  "title_text": "Density of large holes among power-free lattice points",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/96x51txhn0rwq",
  "venue": "arXiv",
  "versions": []
}