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d4317d7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 | 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
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