{
  "abstract": "Let $d \\geq 5$, $0 < γ< d - 2$, and $Ω_N$ be the binomial random subset of $Q_N = [-N,N]^d \\cap \\mathbb Z^d$ with retention probability $p_N = N^{-γ}$. We prove that, with failure probability of optimal exponential order, every subset $B \\subseteq Ω_N$ of fixed positive relative density realizes, at each scale $p_N^{-2/(d - 2)} \\lesssim λ\\lesssim N^2$, a squared distance of the form $q^2 λ$, where $q$ belongs to a fixed finite set depending only on the dimension and the density. The lower scale $p_N^{-2/(d - 2)}$ is sharp. As a consequence,the squared-distance set $D^2(B)$ of $B$ has maximal order $N^2$ and contains affine copies of every fixed finite subset of $\\mathbb Z$. The main new input is a finite multidilate supersaturation theorem for dense subsets of $Q_N$, which, together with boundedness estimates for the associated spherical distance graphs down to the sharp scale, allows us to apply Schacht's transference theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36022",
  "author_total": 2,
  "authors": [
    {
      "name": "Christina Giannitsi"
    },
    {
      "name": "Eyvindur Ari Palsson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2143r6e344beq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36022",
  "fulltext_url": null,
  "id": "2143r6e344beq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36022",
  "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.CA"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7214911548",
  "pdf_url": "https://arxiv.org/pdf/2609.36022",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2143r6e344beq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.36022",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36022"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sharp Thresholds for Distance Patterns in Random Subsets of $\\mathbb Z^d$",
  "title_text": "Sharp Thresholds for Distance Patterns in Random Subsets of ℤ^d",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/2143r6e344beq",
  "venue": "arXiv",
  "versions": []
}