Results on doubling, Lipschitz sheets, the energy estimates and the crossings, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Packing in Euclidean space — Lemma 1.1. Points of R^k in a closed ball of radius R with pairwise distances at least s number at most (1 + 2R/s)^k.
• Euclidean subsets are doubling — Proposition 1.2. Every subset of R^k is doubling with constant at most 5^k.
• The doubling bound (OpenAI) — Proposition 3.1. The set S of points over the plane with coloured-strip coordinates at scales 1000^-j is doubling with constant at most 76800.
• Good points — Lemma 1.1. A Lipschitz map of a plane domain into R^k is differentiable at almost every point, with both derivative columns approached in mean square on small squares.
• Line integrals — Lemma 1.2. On almost every horizontal (vertical) line, increments of a Lipschitz map are integrals of its partial derivative, with the Cauchy-Schwarz bound.
• Near the lexicographic maximum — Lemma 3.1. If the mean first coordinate of K-valued pairs tends to its maximum A, the limsup of the mean second coordinate is at most B.
• Mean and variance — Lemma 4.1. The mean square distance of a field from a vector a equals its mean squared length minus |a|^2 minus 2<a, mean - a>.
• Two energy estimates (OpenAI) — Proposition 2.1. Near a near-extremal good point, the row and column selections of the added sheets satisfy n^2 <|G_H^x - F_x|^2> -> 0 and <|G_V^y - F_y|^2> -> 0.
• Lines with small energy — Lemma 1.1. In a period block of small energy, one row line and one column line of each colour carry normalized offsets with oscillation at most N sqrt(2 N eps).
• Crossings — Lemma 2.1. The offsets at the crossings of the chosen lines give N points with pairwise distances at least sqrt 2 - 2 omega.
• The Lang-Plaut problem (OpenAI) — Theorem 3.1. S is a doubling subset of Hilbert space (constant at most 76800) with no bi-Lipschitz embedding into any R^k.
• Finite-dimensional normed targets — Corollary 3.2. S admits no bi-Lipschitz embedding into any finite-dimensional real normed space.

Reading order

Read the lessons in order. The first constructs the set and proves its doubling bound. The second collects the analysis of Lipschitz maps of the plane and defines the lexicographic maximum. The third proves the energy estimates, and the fourth derives the contradiction.