| import Mathlib |
| import OAI.MeasureTheory.FalconerCompletion.Measures.PathIndex |
|
|
| namespace OAI |
|
|
| /-! Quantitative estimates for the distance-set construction. -/ |
|
|
| section |
| section |
| namespace CurrentFalconer.TreeND |
| open Set Metric MeasureTheory |
| open scoped ENNReal |
| variable {d : ℕ} |
|
|
| noncomputable def treeCenter (p : CellPath d) : EuclideanSpace ℝ (Fin d) := |
| ambientGridCenter p.length (fun i => (pathIndex p i : ℤ)) |
|
|
| lemma treeCenter_mem (p : CellPath d) : treeCenter p ∈ treeCell p := |
| ambientGridCenter_mem _ _ |
|
|
| noncomputable def occupiedCenterCells (ν : Measure (EuclideanSpace ℝ (Fin d))) |
| (p : ℕ) (x : EuclideanSpace ℝ (Fin d)) (r : ℝ) : Finset (CellPath d) := by |
| classical |
| exact (dyadicPaths p).filter fun q => 0 < ν (treeCell q) ∧ treeCenter q ∈ ball x r |
|
|
| lemma treeCenter_ball_card {g : ℕ} (T : Finset (CellPath d)) (hT : T ⊆ dyadicPaths g) |
| (x : EuclideanSpace ℝ (Fin d)) {r : ℝ} (hr : 0 ≤ r) |
| (hdist : ∀ p ∈ T, dist (treeCenter p) x ≤ r) : |
| (T.card : ℝ) ≤ (2*(2:ℝ)^g*r+5)^d := by |
| classical |
| let f : CellPath d → (Fin d → ℤ) := fun p i => pathIndex p i |
| have hf : Set.InjOn f T := by |
| intro p hp q hq he |
| apply pathIndex_injective (((mem_dyadicPaths _ _).mp (hT hp)).trans |
| ((mem_dyadicPaths _ _).mp (hT hq)).symm) |
| funext i |
| have hei : (pathIndex p i : ℤ) = pathIndex q i := congrFun he i |
| exact_mod_cast hei |
| have hh := ambientGridCenter_ball_card g (T.image f) x hr (fun a ha => by |
| obtain ⟨p,hp,rfl⟩ := Finset.mem_image.mp ha |
| have he : p.length = g := (mem_dyadicPaths _ _).mp (hT hp) |
| simpa only [treeCenter,he] using hdist p hp) |
| rwa [Finset.card_image_of_injOn hf] at hh |
|
|
| end CurrentFalconer.TreeND |
| end |
| end |
|
|
| end OAI |
|
|
