| import Mathlib | |
| namespace OAI | |
| namespace DoublingHilbert | |
| abbrev RealL2 := lp (fun _ : ℕ => ℝ) 2 | |
| /-- Intrinsic open-ball doubling with a specified integer bound and centers in the subset. -/ | |
| def DoublingAtMost (S : Set RealL2) (bound : ℕ) : Prop := | |
| ∀ x : S, ∀ r : ℝ, 0 < r → | |
| ∃ centers : Finset S, centers.card ≤ bound ∧ | |
| ∀ y : S, dist y x < r → ∃ c ∈ centers, dist y c < r / 2 | |
| /-- Finite distortion includes an arbitrary positive change of scale. -/ | |
| def AdmitsBiLipschitzEmbedding (S : Set RealL2) (k : ℕ) : Prop := | |
| ∃ f : S → EuclideanSpace ℝ (Fin k), | |
| ∃ a : ℝ, 0 < a ∧ ∃ D : ℝ, 1 ≤ D ∧ | |
| ∀ x y : S, a * dist x y ≤ dist (f x) (f y) ∧ | |
| dist (f x) (f y) ≤ D * a * dist x y | |
| /-- A single fixed subset, independent of dimension and distortion. -/ | |
| theorem main : ∃ S : Set RealL2, | |
| DoublingAtMost S 76800 ∧ | |
| ∀ k : ℕ, 0 < k → ¬ AdmitsBiLipschitzEmbedding S k := by | |
| sorry | |
| end DoublingHilbert | |
| end OAI | |