Superexpanders and Separation of L-p from Hilbert Embeddability

Definitions and mathematical statement

For a metric space $(X,d)$, write \[ B_X(x,R)=\{y\in X:d(x,y)\le R\}. \] The cardinality convention for bounded geometry is \[ \forall R>0\quad \sup_{x\in X}|B_X(x,R)|<\infty. \] Some conventions additionally require uniform discreteness, meaning that there exists $\delta>0$ such that $d(x,y)\ge\delta$ whenever $x\ne y$. The supplied record does not specify whether this additional condition is intended.

For $2<p<\infty$, let $\ell_p$ be the space of scalar sequences $a=(a_n)_{n\ge1}$ satisfying $\sum_{n=1}^{\infty}|a_n|^p<\infty$, equipped with the metric induced by \[ \|a\|_p=\left(\sum_{n=1}^{\infty}|a_n|^p\right)^{1/p}. \] The scalar field may be ${\mathbb{R}}$ or ${\mathbb{C}}$; the source does not specify one. A coarse embedding of a metric space $(X,d_X)$ into a metric space $(Y,d_Y)$ is a map $f:X\to Y$ for which there exist nondecreasing functions $\rho_-,\rho_+:[0,\infty)\to[0,\infty)$ such that \[ \lim_{t\to\infty}\rho_-(t)=\infty, \qquad \rho_-(d_X(x,y))\le d_Y(f(x),f(y))\le\rho_+(d_X(x,y)) \quad(x,y\in X). \]

Question under the coarse-embedding interpretation. Do there exist a real number $p$ with $2<p<\infty$ and a metric space $X$ of bounded geometry such that $X$ admits a coarse embedding into $\ell_p$, while for every Hilbert space $H$, equipped with its norm metric, there is no coarse embedding $X\to H$?

The word "embeds" is unqualified in the supplied statement. The coarse-embedding interpretation is suggested by the report topic and supplied literature links, but is not explicitly confirmed by the record. Consequently, the displayed question is a conditional formalization, pending confirmation of both the embedding notion and the bounded-geometry convention. The corrected source statement retains only this separation question; the original additional question about whether all expander families are super-expanders was removed as a duplicate of OWR-16160-001.

Short English statement

Under the suggested coarse-embedding interpretation, the question asks whether some bounded-geometry metric space coarsely embeds into $\ell_p$ for at least one $2<p<\infty$, yet coarsely embeds into no Hilbert space. The embedding notion and any uniform-discreteness requirement need confirmation. The supplied record labels the problem open; this draft does not independently verify that status.

Sources

• Supplied local registry record, data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30003787$, exact source code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied. The proposed year is recorded as 2018.
• Supplied source citation: Mini-Workshop: Superexpanders and Their Coarse Geometry (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/19, pp. 1117--1160. https://doi.org/10.4171/owr/2018/19. The supplied title year is 2019, whereas the proposed-year field and DOI contain 2018; this discrepancy is unresolved.
• Supplied literature source, "workshop report": https://ems.press/content/serial-article-files/46739.
• Supplied literature source, "non-embedding of $\ell_p$": https://arxiv.org/abs/math/0410427.
• Supplied literature source, "coarse embeddings into $\ell_q$": https://arxiv.org/abs/1210.0588.
• Literature assessment checked at 2026-08-21: supplied dated curation, not independently verified. The record labels the problem open and reports that the known non-Hilbert example $\ell_p$, for $p>2$, is not itself a bounded-geometry discrete space, and that no general separating example was found by that cutoff. The record describes this as literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The supplied statement-status label is corrected_verified. Its accompanying explanation says that the first clause of the original statement duplicated OWR-16160-001 and was removed, retaining the distinct Banach-space embedding question. These are supplied editorial qualifications, not independent verification by this draft.