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The Lang-Plaut problem: does every doubling subset of Hilbert space bi-Lipschitz embed in some Euclidean space?

Claims a negative answer: a fixed doubling subset of $\ell_2$ (doubling constant at most 76800) with no bi-Lipschitz embedding into any $\mathbb R^k$ ; more generally every infinite-dimensional real Banach space contains a compact set of universally bounded doubling constant with no bi-Lipschitz embedding into any finite-dimensional normed space, and the same metric embeds isometrically in every $L_p$ , $1\le p<\infty$ . It does NOT bear on snowflaked embeddings (Assouad). Priority note: Schioppa announced a negative answer in 2017 (arXiv:1703.10265) but withdrew it.

Exact FrontierDelta

Prior state unknown→disproved

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Occurred: Sep 25, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Lang-Plaut problem: does every doubling subset of Hilbert space bi-Lipschitz embed in some Euclidean space? · Lang-Plaut problem

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Registry verification: lean checked · announcement · candidate

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VibeMathed record: The Lang-Plaut problem: does every doubling subset of Hilbert space bi-Lipschitz embed in some Euclidean space?

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No independent mathematician has checked this yet. Theorem 1.1 and the abstract's Banach-space statement were read against Question 2.4 of Lang-Plaut: a fixed subset of real $\ell_2$ with doubling constant at most 76800 admits no bi-Lipschitz embedding into any $\mathbb R^k$ at any distortion. Both are in formalization.yaml. ComparatorChallenges/DoublingHilbert.lean (OAI.DoublingHilbert.main, OAI/Geometry/DoublingHilbert/Main.lean) states exactly Theorem 1.1, with doubling defined by open balls centered in the subset and embeddings allowing any scale and distortion. ComparatorChallenges/CompactBanach.lean (OAI.CompactBanach.main) states the stronger form: one constant $\Lambda$ such that every infinite-dimensional real Banach space contains a compact $\Lambda$ -doubling set with no bi-Lipschitz embedding into any finite-dimensional normed space. Permitted axioms propext, Quot.sound, Classical.choice. Not rebuilt here.