| import OAI.Geometry.RCD.Metric.DiagonalMeasuredApproximation |
| import OAI.Geometry.RCD.Metric.MeasurablePointedApproximation |
| import OAI.Geometry.RCD.Metric.ApproximationRestriction |
|
|
| namespace OAI |
|
|
| /-! # The published measured-coordinate input on one common sequence of maps -/ |
|
|
| open Set Metric Filter MeasureTheory |
| open scoped Topology |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [ProperSpace X] |
|
|
| theorem LocalMeasuredCoordinateInput.common_approximations {K : ℝ} {n : ℕ} |
| {hX : NoncollapsedRCD K n X} (hinput : LocalMeasuredCoordinateInput hX) (hn : 2 ≤ n) |
| (p : ℕ → X) (r : ℕ → ℝ) (hr : ∀ k, 0 < r k) |
| (hcurv : ∀ δ : ℝ, 0 < δ → ∀ᶠ k in atTop, -δ ≤ K*(r k)^2) |
| (hflat : ∀ S η : ℝ, 0 < S → 0 < η → ∀ᶠ k in atTop, |
| ∃ f : X → Euclidean n, IsPointedApproximation (p k) 0 (r k) S η f) : |
| ∃ (ε : ℕ → ℝ) (V : ℕ → X → Euclidean n), |
| (∀ k, 0 < ε k ∧ ε k ≤ 1) ∧ Tendsto ε atTop (𝓝 0) ∧ |
| ∀ j : ℕ, ∀ᶠ k in atTop, |
| IsMeasuredEuclideanApproximation (hausdorff n X) n (p k) (r k) |
| ((j : ℝ)+1) (ε k) (V k) := by |
| apply diagonal_measured_approximations |
| intro m η hη |
| have hi (i : Fin (m+1)) := hinput hn ((i.val : ℝ)+1) η (by positivity) hη |
| choose δ hδ happly using hi |
| let δ₀ : ℝ := Finset.univ.inf' Finset.univ_nonempty δ |
| have hδ₀ : 0 < δ₀ := (Finset.lt_inf'_iff Finset.univ_nonempty).mpr (fun i _ => hδ i) |
| have hδle (i : Fin (m+1)) : δ₀ ≤ δ i := Finset.inf'_le _ (Finset.mem_univ i) |
| filter_upwards [hcurv δ₀ hδ₀, |
| hflat (2*((m : ℝ)+1)) (δ₀/15) (by positivity) (by positivity)] with k hk hf |
| obtain ⟨f,hf⟩ := hf |
| obtain ⟨V,hVm,hV⟩ := hf.exists_measurable (hr k) (by positivity) (by positivity) |
| refine ⟨V,?_⟩ |
| intro j hj |
| let i : Fin (m+1) := ⟨j,by omega⟩ |
| apply happly i (p k) (r k) (hr k) (le_trans (neg_le_neg (hδle i)) hk) V hVm |
| have hrad : 2*((j : ℝ)+1) ≤ 2*((m : ℝ)+1) := by exact_mod_cast (show 2*(j+1)≤2*(m+1) by omega) |
| have hres := hV.restrict_radius (hr k) (show 0 ≤ 2*((j : ℝ)+1) by positivity) |
| hrad (show 0 ≤ 5*(δ₀/15) by positivity) |
| have herr : 3*(5*(δ₀/15)) ≤ δ i := by linarith [hδle i] |
| exact hres.mono_error herr |
|
|
| end RegularPointCoordinates |
|
|
| end |
|
|
| end OAI |
|
|
