{
  "abstract": "Let the center of a $d$-dimensional Euclidean ball be uniformly distributed modulo $\\mathbb Z^d$. We study the resulting number of integer lattice points as the dimension and radius tend to infinity. The critical scale is $4π^2R_d^2/(d+2)=\\log d$. In a logarithmic window around this scale, the logarithm of the lattice-point count divided by the volume satisfies a central limit theorem. At a fixed offset from the critical scale this yields a genuine lognormal limit. Above the critical scale, the count-to-volume ratio converges to $1$ in measure; its normalized error has a standard normal limit, and its variance satisfies an asymptotic formula.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30810",
  "author_total": 1,
  "authors": [
    {
      "name": "Xin Hang Ji"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/90efd2v1hfxcc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30810",
  "fulltext_url": null,
  "id": "90efd2v1hfxcc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30810",
  "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"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7204900996",
  "pdf_url": "https://arxiv.org/pdf/2608.30810",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/90efd2v1hfxcc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30810",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30810"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Normal and Lognormal Asymptotics for Lattice-Point Counts in Random Translates of High-Dimensional Balls",
  "title_text": "Normal and Lognormal Asymptotics for Lattice-Point Counts in Random Translates of High-Dimensional Balls",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/90efd2v1hfxcc",
  "venue": "arXiv",
  "versions": []
}