| import OAI.MeasureTheory.Falconer.Estimates.DyadicCenters |
| import OAI.MeasureTheory.Falconer.Graphs.GraphSchur |
|
|
| namespace OAI |
|
|
| /-! |
| # Retained graph on the actual dyadic cells |
|
|
| The grid degree estimate is transported to arbitrary finite collections |
| of equal-depth descendants. Their original cube centers define every |
| test. The final Schur bound can therefore be applied directly to the |
| endpoint transforms restricted to those cells. |
| -/ |
|
|
| namespace Falconer |
| attribute [local instance] Classical.propDecidable |
|
|
| def retainedDyadicEdge {d : ℕ} (Q₁ Q₂ : DyadicCube d) (w₁ w₂ : Space d) (τ : ℝ) |
| (row col : List (Fin d → Bool) × List (Fin d → Bool)) : Prop := |
| ‖perpendicularPart w₁ ((Q₁.descendant col.1).center - (Q₁.descendant row.1).center)‖ ≤ τ ∧ |
| ‖perpendicularPart w₂ ((Q₂.descendant col.2).center - (Q₂.descendant row.2).center)‖ ≤ τ ∧ |
| |inner ℝ w₁ ((Q₁.descendant col.1).center - (Q₁.descendant row.1).center) + |
| inner ℝ w₂ ((Q₂.descendant col.2).center - (Q₂.descendant row.2).center)| ≤ τ |
|
|
| theorem retainedDyadicEdge_symm {d : ℕ} (Q₁ Q₂ : DyadicCube d) (w₁ w₂ : Space d) (τ : ℝ) |
| (row col : List (Fin d → Bool) × List (Fin d → Bool)) : |
| retainedDyadicEdge Q₁ Q₂ w₁ w₂ τ row col ↔ retainedDyadicEdge Q₁ Q₂ w₁ w₂ τ col row := by |
| have hn (x y : Space d) : x-y = -(y-x) := by abel |
| simp only [retainedDyadicEdge, hn (Q₁.descendant row.1).center, |
| hn (Q₂.descendant row.2).center, perpendicularPart_neg, norm_neg, |
| inner_neg_right, ← neg_add, abs_neg] |
|
|
| theorem retainedDyadicEdge_iff_grid {d n : ℕ} (Q₁ Q₂ : DyadicCube d) |
| (hside : Q₂.side = Q₁.side) (w₁ w₂ : Space d) (τ : ℝ) |
| (row col : List (Fin d → Bool) × List (Fin d → Bool)) |
| (hr₁ : row.1.length = n) (hr₂ : row.2.length = n) |
| (hc₁ : col.1.length = n) (hc₂ : col.2.length = n) : |
| retainedDyadicEdge Q₁ Q₂ w₁ w₂ τ row col ↔ |
| retainedGridEdge ((2 : ℝ)^n/Q₁.side) w₁ w₂ τ |
| (dyadicCode row.1, dyadicCode row.2) (dyadicCode col.1, dyadicCode col.2) := by |
| unfold retainedDyadicEdge retainedGridEdge |
| rw [Q₁.center_sub_eq_grid_sub col.1 row.1 hc₁ hr₁, |
| Q₂.center_sub_eq_grid_sub col.2 row.2 hc₂ hr₂, hside] |
|
|
| private theorem dyadic_candidate_grid_bound {d n C : ℕ} (Q : DyadicCube d) |
| (A : Finset (List (Fin d → Bool))) (hA : ∀ p ∈ A, p.length = n) |
| (w : Space d) (τ : ℝ) (a : List (Fin d → Bool)) (ha : a.length = n) |
| (hb : (A.filter (fun p => ‖perpendicularPart w |
| ((Q.descendant p).center - (Q.descendant a).center)‖ ≤ τ)).card ≤ C) : |
| ((A.image dyadicCode).filter (fun z => ‖perpendicularPart w |
| (scaledGridPoint ((2 : ℝ)^n/Q.side) z - |
| scaledGridPoint ((2 : ℝ)^n/Q.side) (dyadicCode a))‖ ≤ τ)).card ≤ C := by |
| classical |
| rw [Finset.filter_image] |
| apply Finset.card_image_le.trans |
| have heq : A.filter (fun p => ‖perpendicularPart w |
| (scaledGridPoint ((2 : ℝ)^n/Q.side) (dyadicCode p) - |
| scaledGridPoint ((2 : ℝ)^n/Q.side) (dyadicCode a))‖ ≤ τ) = |
| A.filter (fun p => ‖perpendicularPart w |
| ((Q.descendant p).center - (Q.descendant a).center)‖ ≤ τ) := by |
| apply Finset.filter_congr |
| intro p hp |
| rw [Q.center_sub_eq_grid_sub p a (hA p hp) ha] |
| rwa [heq] |
|
|
| theorem retainedDyadic_degree {d n L C₁ C₂ : ℕ} (Q₁ Q₂ : DyadicCube d) |
| (hQ : 0 < Q₁.side) (hside : Q₂.side = Q₁.side) |
| (A B : Finset (List (Fin d → Bool))) |
| (hA : ∀ p ∈ A, p.length = n) (hB : ∀ p ∈ B, p.length = n) |
| (w₁ w₂ : Space d) (hw₁ : ‖w₁‖ = 1) (hw₂ : ‖w₂‖ = 1) (τ : ℝ) |
| (hL : 4*τ*((2 : ℝ)^n/Q₁.side) ≤ L) |
| (row : List (Fin d → Bool) × List (Fin d → Bool)) (hr : row ∈ A ×ˢ B) |
| (h₁ : (A.filter (fun p => ‖perpendicularPart w₁ |
| ((Q₁.descendant p).center - (Q₁.descendant row.1).center)‖ ≤ τ)).card ≤ C₁) |
| (h₂ : (B.filter (fun p => ‖perpendicularPart w₂ |
| ((Q₂.descendant p).center - (Q₂.descendant row.2).center)‖ ≤ τ)).card ≤ C₂) : |
| ((A ×ˢ B).filter (retainedDyadicEdge Q₁ Q₂ w₁ w₂ τ row)).card ≤ |
| (2*L+1)^d * min C₁ C₂ := by |
| classical |
| let s := (2 : ℝ)^n/Q₁.side |
| let code := fun a : List (Fin d → Bool) × List (Fin d → Bool) => |
| (dyadicCode a.1, dyadicCode a.2) |
| have hrA := (Finset.mem_product.mp hr).1 |
| have hrB := (Finset.mem_product.mp hr).2 |
| have hc₁ := dyadic_candidate_grid_bound Q₁ A hA w₁ τ row.1 (hA _ hrA) h₁ |
| have hc₂ := dyadic_candidate_grid_bound Q₂ B hB w₂ τ row.2 (hB _ hrB) h₂ |
| rw [hside] at hc₂ |
| have hgrid := retainedGrid_degree_min (show 0 < s by dsimp [s]; positivity) hL |
| (A.image dyadicCode) (B.image dyadicCode) w₁ w₂ hw₁ hw₂ (code row) hc₁ hc₂ |
| apply le_trans _ hgrid |
| apply Finset.card_le_card_of_injOn code |
| · intro col hcol |
| obtain ⟨hcol, hedge⟩ := Finset.mem_filter.mp hcol |
| obtain ⟨ha, hb⟩ := Finset.mem_product.mp hcol |
| apply Finset.mem_filter.mpr |
| refine ⟨Finset.mem_product.mpr ⟨Finset.mem_image.mpr ⟨col.1,ha,rfl⟩, |
| Finset.mem_image.mpr ⟨col.2,hb,rfl⟩⟩, ?_⟩ |
| exact (retainedDyadicEdge_iff_grid Q₁ Q₂ hside w₁ w₂ τ row col |
| (hA _ hrA) (hB _ hrB) (hA _ ha) (hB _ hb)).mp hedge |
| · intro col hcol col' hcol' heq |
| obtain ⟨ha,hb⟩ := Finset.mem_product.mp (Finset.mem_filter.mp hcol).1 |
| obtain ⟨ha',hb'⟩ := Finset.mem_product.mp (Finset.mem_filter.mp hcol').1 |
| exact Prod.ext |
| (dyadicCode_injective_at_depth (hA _ ha) (hA _ ha') (congrArg Prod.fst heq)) |
| (dyadicCode_injective_at_depth (hB _ hb) (hB _ hb') (congrArg Prod.snd heq)) |
|
|
| theorem retainedDyadic_schur {d n L C₁ C₂ : ℕ} (Q₁ Q₂ : DyadicCube d) |
| (hQ : 0 < Q₁.side) (hside : Q₂.side = Q₁.side) |
| (A B : Finset (List (Fin d → Bool))) |
| (hA : ∀ p ∈ A, p.length = n) (hB : ∀ p ∈ B, p.length = n) |
| (w₁ w₂ : Space d) (hw₁ : ‖w₁‖ = 1) (hw₂ : ‖w₂‖ = 1) (τ : ℝ) |
| (hL : 4*τ*((2 : ℝ)^n/Q₁.side) ≤ L) |
| (h₁ : ∀ a ∈ A, (A.filter (fun p => ‖perpendicularPart w₁ |
| ((Q₁.descendant p).center - (Q₁.descendant a).center)‖ ≤ τ)).card ≤ C₁) |
| (h₂ : ∀ a ∈ B, (B.filter (fun p => ‖perpendicularPart w₂ |
| ((Q₂.descendant p).center - (Q₂.descendant a).center)‖ ≤ τ)).card ≤ C₂) |
| (F : List (Fin d → Bool) × List (Fin d → Bool) → ℂ) : |
| (∑ row ∈ A ×ˢ B, ∑ col ∈ A ×ˢ B, |
| if retainedDyadicEdge Q₁ Q₂ w₁ w₂ τ row col then ‖F row * star (F col)‖ else 0) ≤ |
| ((2*L+1)^d * min C₁ C₂ : ℕ) * ∑ row ∈ A ×ˢ B, ‖F row‖^2 := by |
| apply finite_graph_schur |
| · intro row hr |
| exact_mod_cast retainedDyadic_degree Q₁ Q₂ hQ hside A B hA hB w₁ w₂ hw₁ hw₂ τ hL |
| row hr (h₁ _ (Finset.mem_product.mp hr).1) (h₂ _ (Finset.mem_product.mp hr).2) |
| · intro col hc |
| have hh := retainedDyadic_degree Q₁ Q₂ hQ hside A B hA hB w₁ w₂ hw₁ hw₂ τ hL |
| col hc (h₁ _ (Finset.mem_product.mp hc).1) (h₂ _ (Finset.mem_product.mp hc).2) |
| have he : (A ×ˢ B).filter (fun row => retainedDyadicEdge Q₁ Q₂ w₁ w₂ τ row col) = |
| (A ×ˢ B).filter (retainedDyadicEdge Q₁ Q₂ w₁ w₂ τ col) := by |
| apply Finset.filter_congr |
| intro row _ |
| exact retainedDyadicEdge_symm Q₁ Q₂ w₁ w₂ τ row col |
| rw [he] |
| exact_mod_cast hh |
|
|
| end Falconer |
|
|
| end OAI |
|
|
