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Math-2270 · mathematics · Analysis

Finite-dimensional embeddings of doubling Hilbert subsets

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Suppose a metric space $X$ is doubling, meaning that every ball can be covered by a fixed finite number of balls of half its radius. If $X$ admits a bi-Lipschitz embedding into a Hilbert space, must it admit a bi-Lipschitz embedding into $\mathbb R^m$ for some finite $m$?

Formulation references: [S1], [S2].

Short English statement

Suppose a metric space $X$ is doubling, meaning that every ball can be covered by a fixed finite number of balls of half its radius. If $X$ admits a bi-Lipschitz embedding into a Hilbert space, must it admit a bi-Lipschitz embedding into $\mathbb R^m$ for some finite $m$?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2270-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2270

The displayed question has a negative answer under the manuscript's stated construction. Its counterexample meets the conditions in this problem and defeats the universal assertion, so this is a full disproof claim. The supporting result is manuscript 1 (family 098; 2026-09-25).

Finite-dimensional embeddings of doubling Hilbert subsets

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 098; 2026-09-25), the asserted result is that in each infinite-dimensional real Banach space one can choose a compact subset with a universal doubling constant for which every bi-Lipschitz map into a finite-dimensional normed target is impossible.

Mathematical statement of the claimed result

Manuscript 1.
A single subset $S$ of real $\ell_2$, with the distance inherited from its Hilbert norm, is claimed to satisfy \[ \operatorname{dbl}(S)\le76800,\qquad \forall k\ge1\ \forall D<\infty:\ S\text{ has no bi-Lipschitz embedding into }\mathbb R^k\text{ of distortion }D. \] Here doubling means that every metric ball is covered by at most that many balls of half its radius. The obstruction concerns one fixed doubling space, not a sequence with increasing doubling constants.

Outline of the argument

For manuscript 1 (family 098; 2026-09-25), sheet-like pieces are arranged so that coverings remain uniformly doubling. Near-extremal derivatives of a hypothetical finite-dimensional embedding have small quadratic variation; crossings between separated sheets then force more packed images than the target can accommodate.

Formal verification

For manuscript 1 (family 098; 2026-09-25), OpenAI lists Lean statements for this assertion: in each infinite-dimensional real Banach space one can choose a compact subset with a universal doubling constant for which every bi-Lipschitz map into a finite-dimensional normed target is impossible.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are CompactBanach.lean, DoublingHilbert.lean.

Their named declarations include OAI.CompactBanach.main, OAI.DoublingHilbert.main.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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