The Lang–Plaut problem

OpenAI's negative answer (2026) to the Lang–Plaut problem: a doubling subset of Hilbert space that admits no bi-Lipschitz embedding into any finite-dimensional space. The course builds the set from coloured strips at rapidly decreasing scales and proves its doubling bound, differentiates the Lipschitz sheets of a supposed embedding near a lexicographically maximal derivative, proves two energy estimates, and turns crossings of rows and columns into a packing contradiction in Euclidean space.

Four lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Doubling spaces and coloured strips
2. Lipschitz sheets and a lexicographic maximum
3. Two energy estimates
4. Crossings and the Lang–Plaut problem

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Rademacher's theorem, absolute continuity and the fundamental theorem of calculus for Lipschitz functions, Lebesgue's differentiation theorem, Fubini's theorem and the scaling of Lebesgue measure (core course Measure and Integration: Fremlin, Measure Theory, Volume 2, 262Q, 262B, 225E, 261E, 251N, 252B, 252D and 263A; Volume 1, 134A); compactness of the unit sphere of R^k (core course Real Analysis II).