{
  "abstract": "For a positive real lacunary sequence and any probability measure $μ$ with polynomial Fourier decay, we determine, for $μ$-almost every $x$, the asymptotic maximal volume of convex bodies avoiding the first $N$ dilates of $x$ modulo $\\mathbb Z^d$. Among homothets of a fixed convex body, the maximal volume is asymptotic to $(\\log N)/N$. When all convex bodies in the unit cube are allowed, it is asymptotic to $d(\\log N)/N$. For independent uniform points in a fixed convex body $Ω\\subset\\mathbb R^d$, the maximal volume of an empty convex body is asymptotic to $d\\operatorname{vol}(Ω)(\\log N)/N$ almost surely. We also show, by constructing a one-dimensional counterexample, that lacunarity cannot be replaced by sparsity on sublacunary scales.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09790",
  "author_total": 2,
  "authors": [
    {
      "name": "Yuval Peres"
    },
    {
      "name": "Bohan Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8agg8s444jezc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09790",
  "fulltext_url": null,
  "id": "8agg8s444jezc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09790",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
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  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.MG"
    ],
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      "arxiv",
      "math.NT"
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      "arxiv",
      "math.PR"
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      "2604|Point processes and geometric inequalities"
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  "openalex": "W7221003437",
  "pdf_url": "https://arxiv.org/pdf/2610.09790",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8agg8s444jezc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.09790",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09790"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Maximal volume of convex bodies avoiding random points and lacunary dilates",
  "title_text": "Maximal volume of convex bodies avoiding random points and lacunary dilates",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/8agg8s444jezc",
  "venue": "arXiv",
  "versions": []
}