| import OAI.Geometry.DoublingHilbert.Crossings |
| import OAI.Geometry.DoublingHilbert.Doubling |
|
|
| namespace OAI |
|
|
| /-! A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding. -/ |
|
|
| open MeasureTheory Set Filter |
| open scoped BigOperators Topology |
|
|
| namespace DoublingHilbert |
| open Metric |
|
|
|
|
| section Nonembedding |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [CompleteSpace E] [FiniteDimensional ℝ E] |
|
|
| / |
| theorem no_normalized_embedding {f : constructedSet → E} {D : NNReal} |
| (hf : LipschitzWith D f) (hlower : ∀ p q, dist p q ≤ dist (f p) (f q)) : False := by |
| obtain ⟨bound, hbound⟩ := isCompact_exists_packing_bound |
| (isCompact_closedBall (0 : E) (D : ℝ)) (show (0 : ℝ) < 1 by norm_num) |
| obtain ⟨z, hz, hsep⟩ := exists_separated_family hf hlower (bound + 1) (by omega) |
| have hc := hbound z (fun i => by simpa only [mem_closedBall, dist_zero_right] using hz i) hsep |
| omega |
|
|
| / |
| theorem no_scaled_embedding (f : constructedSet → E) {a D : ℝ} |
| (ha : 0 < a) (hD : 1 ≤ D) |
| (hf : ∀ x y, a * dist x y ≤ dist (f x) (f y) ∧ |
| dist (f x) (f y) ≤ D * a * dist x y) : False := by |
| let g : constructedSet → E := fun x => a⁻¹ • f x |
| let D' : NNReal := ⟨D, le_trans zero_le_one hD⟩ |
| have hdist (x y : constructedSet) : dist (g x) (g y) = a⁻¹ * dist (f x) (f y) := by |
| simp only [g, dist_eq_norm, ← smul_sub, norm_smul, Real.norm_eq_abs, abs_inv, abs_of_pos ha] |
| have hg : LipschitzWith D' g := LipschitzWith.of_dist_le_mul fun x y => by |
| rw [hdist] |
| change a⁻¹ * dist (f x) (f y) ≤ D * dist x y |
| calc |
| _ ≤ a⁻¹ * (D * a * dist x y) := |
| mul_le_mul_of_nonneg_left (hf x y).2 (inv_nonneg.mpr ha.le) |
| _ = _ := by field_simp |
| apply no_normalized_embedding hg |
| intro x y |
| rw [hdist] |
| calc |
| dist x y = a⁻¹ * (a * dist x y) := by field_simp |
| _ ≤ _ := mul_le_mul_of_nonneg_left (hf x y).1 (inv_nonneg.mpr ha.le) |
|
|
| end Nonembedding |
| end DoublingHilbert |
|
|
| namespace DoublingHilbert |
|
|
| / |
| theorem main : ∃ S : Set RealL2, |
| DoublingAtMost S 76800 ∧ |
| ∀ k : ℕ, 0 < k → ¬ AdmitsBiLipschitzEmbedding S k := by |
| refine ⟨constructedSet, ?_, ?_⟩ |
| · intro x r hr |
| exact constructedSet_cover x hr |
| · intro k _ hEmbedding |
| obtain ⟨f, a, ha, D, hD, hf⟩ := hEmbedding |
| exact no_scaled_embedding f ha hD hf |
|
|
| end DoublingHilbert |
|
|
| end OAI |
|
|
