{
  "abstract": "Fix $0<θ\\leqslant 1$. We prove that if $1\\leqslant p \\leqslant 2/θ$, then the $θ$-snowflake of $\\ell_2^k$, namely, $\\mathbb{R}^k$ equipped with the metric $((x,y)\\in \\mathbb{R}^k\\times \\mathbb{R}^k)\\mapsto \\|x-y\\|_2^θ$, embeds with distortion $O(1)$ into $\\ell_p^m$ for some integer $m\\lesssim_{p,θ}k$, which is optimal as $k\\to \\infty$, as seen by comparing dimensions. However, for $p$ larger than the sharp threshold $2/θ$ the following change in behavior occurs: If a $(1/\\sqrt{k})$-dense subset of the Euclidean sphere $S^{k-1}$ embeds into $\\ell_p^m$ with distortion $O(1)$, then necessarily $m\\gtrsim_{p,θ}( k/\\log k)^{pθ/2}$, which grows super-linearly in $k$ as $pθ/2>1$, and this dimension bound is optimal as $k\\to \\infty$ up to lower order factors. We deduce from this statement that if $2<p<\\infty$, then there exist arbitrarily large $n$-point subsets of $\\ell_p$ with the property that if they embed with distortion $O(1)$ into $\\ell_p^m$, then necessarily $m\\gtrsim_p ((\\log n)/(\\log\\log n)^2)^{p/2}$, thus demonstrating that the statement of the Johnson--Lindenstrauss dimension reduction lemma fails to hold for $\\ell_p$",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01079",
  "author_total": 2,
  "authors": [
    {
      "name": "Assaf Naor"
    },
    {
      "name": "Kevin Ren"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5davem3c8dfz9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01079",
  "fulltext_url": null,
  "id": "5davem3c8dfz9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01079",
  "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.MG"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7206149628",
  "pdf_url": "https://arxiv.org/pdf/2609.01079",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5davem3c8dfz9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.01079",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01079"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A threshold phenomenon for embeddings of Euclidean snowflakes and impossibility of dimension reduction",
  "title_text": "A threshold phenomenon for embeddings of Euclidean snowflakes and impossibility of dimension reduction",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/5davem3c8dfz9",
  "venue": "arXiv",
  "versions": []
}