{
  "abstract": "Let $\\mathcal H=\\ell^2(\\mathbb Z)$ and let $\\mathbb Z$ act by bilateral shifts. We prove that the quotient $\\mathbb S(\\mathcal H)/\\mathbb Z$ of the unit sphere, equipped with its quotient metric, has infinite Hilbert distortion. Consequently, $\\mathcal H/\\mathbb Z$ does not admit a bilipschitz embedding into Hilbert space, answering a question of Cahill, Iverson, and Mixon. The proof first embeds every flat torus $\\mathbb R^n/L$, after rescaling, into the quotient $\\mathbb S\\big(\\ell^2(\\mathbb Z^n)\\big)/\\mathbb Z^n$ with distortion arbitrarily close to $1$, where $\\mathbb Z^n$ acts by translations. We then show that every finite subset of this quotient embeds into $\\mathbb S(\\mathcal H)/\\mathbb Z$ with distortion arbitrarily close to $1$. The lower bounds of Khot and Naor for the Hilbert distortion of flat tori then imply the result.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31468",
  "author_total": 2,
  "authors": [
    {
      "name": "Avik Das"
    },
    {
      "name": "Christian Rosendal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c6qrhgkjpz4c5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31468",
  "fulltext_url": null,
  "id": "c6qrhgkjpz4c5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31468",
  "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.FA"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7223412598",
  "pdf_url": "https://arxiv.org/pdf/2610.31468",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c6qrhgkjpz4c5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.31468",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31468"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A sphere obstruction for the Hilbert shift quotient",
  "title_text": "A sphere obstruction for the Hilbert shift quotient",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/c6qrhgkjpz4c5",
  "venue": "arXiv",
  "versions": []
}