Dimension reduction in L_p

OpenAI's theorem (2026) that n points of L_p, 1 < p < infinity, p != 2, embed into l_p with any fixed distortion above 1 using n^o(1) coordinates, while exact embeddings need about n^2/16; through weighted moments, the localization of electrical flows, and Poisson sums.

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

1. Finite subsets of L_p
2. Weighted moments and sampling
3. Electrical flows and a weighted projection
4. Signed Poisson sums and exponents below two
5. Truncation ramps and exponents above two

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Lebesgue's covering theorem and the dimension of cubes · Index theory of elliptic operators (Index theory of elliptic operators — Hermite and Schwartz tools)
• Basic probability (independence, expectations, characteristic functions), finite-dimensional linear algebra and Lebesgue integration.