A sphere obstruction for the Hilbert shift quotient

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 $\mathbb{S}({\ell }^2({\mathbb{Z}}^n))/{\mathbb{Z}}^n$ 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.

arXiv commentThe proof of the main theorem was found using OpenAI's ChatGPT 5.5