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A sphere obstruction for the Hilbert shift quotient

  • Avik Das
  • Christian Rosendal
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Abstract

Let ℋ=ℓ2(ℤ) and let ℤ act by bilateral shifts. We prove that the quotient 𝕊(ℋ)/ℤ of the unit sphere, equipped with its quotient metric, has infinite Hilbert distortion. Consequently, ℋ/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 ℝn/L, after rescaling, into the quotient 𝕊(ℓ2(ℤn))/ℤn with distortion arbitrarily close to $1$, where ℤn acts by translations. We then show that every finite subset of this quotient embeds into 𝕊(ℋ)/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.

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Cite

BibTeXRIS
BibTeX
@misc{das2026sphere,
  title = {{A sphere obstruction for the Hilbert shift quotient}},
  author = {Avik Das and Christian Rosendal},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31468},
  eprint = {2610.31468},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31468},
}
RIS
TY  - UNPB
TI  - A sphere obstruction for the Hilbert shift quotient
AU  - Avik Das
AU  - Christian Rosendal
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.31468
UR  - https://arxiv.org/abs/2610.31468
AB  - Let ℋ=ℓ^2(ℤ) and let ℤ act by bilateral shifts. We prove that the quotient 𝕊(ℋ)/ℤ of the unit sphere, equipped with its quotient metric, has infinite Hilbert distortion. Consequently, ℋ/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 ℝ^n/L, after rescaling, into the quotient 𝕊(ℓ^2(ℤ^n))/ℤ^n with distortion arbitrarily close to $1$, where ℤ^n acts by translations. We then show that every finite subset of this quotient embeds into 𝕊(ℋ)/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.
ER  -