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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 | import OAI.MeasureTheory.Falconer.Estimates.DistanceFormRemoval
namespace OAI
namespace PlanarFalconer
open Set MeasureTheory
noncomputable def sourceCellwiseDistanceF (P Q : DyadicSquare)
(ρ ν : Measure Plane) (n m : ℕ) (M₁ M₂ : ℝ)
(ψ : CellPath → CellPath → UnitCircle → ℂ)
(b₁ b₂ : Plane → UnitCircle → ℂ) (r : ℝ) : ℂ :=
∑ A ∈ occupiedAnchorPaths P ρ n, ∑ B ∈ occupiedAnchorPaths Q ν m,
sourceDistanceF (ψ A B)
(fun z => sourceAngularTransform ρ (P.descendant A).carrier M₁ b₁ z.1 z.2)
(fun z => sourceAngularTransform ν (Q.descendant B).carrier M₂ b₂ z.1 z.2) r
theorem measurable_sourceCellwiseDistanceF (P Q : DyadicSquare)
(ρ ν : Measure Plane) [IsFiniteMeasure ρ] [IsFiniteMeasure ν]
(n m : ℕ) (M₁ M₂ : ℝ) (ψ : CellPath → CellPath → UnitCircle → ℂ)
(b₁ b₂ : Plane → UnitCircle → ℂ) (hψ : ∀ A B, Measurable (ψ A B))
(h₁ : Measurable (fun z : Plane × UnitCircle => b₁ z.1 z.2))
(h₂ : Measurable (fun z : Plane × UnitCircle => b₂ z.1 z.2)) :
Measurable (sourceCellwiseDistanceF P Q ρ ν n m M₁ M₂ ψ b₁ b₂) := by
unfold sourceCellwiseDistanceF
apply Finset.measurable_sum
intro A _
apply Finset.measurable_sum
intro B _
exact measurable_sourceDistanceF _ _ _ (hψ A B)
(measurable_sourceAngularTransform ρ _ M₁ b₁ h₁)
(measurable_sourceAngularTransform ν _ M₂ b₂ h₂)
theorem sourceCellwiseDistanceF_profile_ratios (P Q : DyadicSquare)
(ρ ν : Measure Plane) [IsFiniteMeasure ρ] [IsFiniteMeasure ν]
(n m : ℕ) (Ma₁ Ma₂ Mp₁ Mp₂ : ℝ) (hMp₁ : Mp₁ ≠ 0) (hMp₂ : Mp₂ ≠ 0)
(ψ : CellPath → CellPath → UnitCircle → ℂ) (b₁ b₂ : Plane → UnitCircle → ℂ)
(hψ : ∀ A B, Measurable (ψ A B))
(h₁ : Measurable (fun z : Plane × UnitCircle => b₁ z.1 z.2))
(h₂ : Measurable (fun z : Plane × UnitCircle => b₂ z.1 z.2))
{W B₁ B₂ : ℝ} (hW : 0 ≤ W) (hB₁ : 0 ≤ B₁) (hB₂ : 0 ≤ B₂)
(hw : ∀ A B dir, ‖ψ A B dir‖ ≤ W) (hb₁ : ∀ x dir, ‖b₁ x dir‖ ≤ B₁)
(hb₂ : ∀ y dir, ‖b₂ y dir‖ ≤ B₂) (r : ℝ) :
sourceCellwiseDistanceF P Q ρ ν n m Ma₁ Ma₂ ψ b₁ b₂ r =
((Mp₁/Ma₁)*(Mp₂/Ma₂)) • sourceCellwiseDistanceF P Q ρ ν n m Mp₁ Mp₂ ψ b₁ b₂ r := by
unfold sourceCellwiseDistanceF
simp_rw [sourceDistanceF_endpoint_integral ρ ν _ _ _ _ _ b₁ b₂ (hψ _ _) h₁ h₂
hW hB₁ hB₂ (hw _ _) hb₁ hb₂]
simp only [← Finset.smul_sum, smul_smul]
congr 1
field_simp
theorem sourceCellwiseDistanceF_constant (P Q : DyadicSquare) (hP : 0 < P.side) (hQ : 0 < Q.side)
(ρ ν : Measure Plane) [IsFiniteMeasure ρ] [IsFiniteMeasure ν] (n m : ℕ)
{M₁ M₂ : ℝ} (hM₁ : 0 < M₁) (hM₂ : 0 < M₂)
(ψ : UnitCircle → ℂ) (b₁ b₂ : Plane → UnitCircle → ℂ) (hψ : Measurable ψ)
(h₁ : Measurable (fun z : Plane × UnitCircle => b₁ z.1 z.2))
(h₂ : Measurable (fun z : Plane × UnitCircle => b₂ z.1 z.2))
{W B₁ B₂ : ℝ} (hW : 0 ≤ W) (hB₁ : 0 ≤ B₁) (hB₂ : 0 ≤ B₂)
(hw : ∀ dir, ‖ψ dir‖ ≤ W) (hb₁ : ∀ x dir, ‖b₁ x dir‖ ≤ B₁)
(hb₂ : ∀ y dir, ‖b₂ y dir‖ ≤ B₂) (r : ℝ) :
sourceCellwiseDistanceF P Q ρ ν n m M₁ M₂ (fun _ _ => ψ) b₁ b₂ r =
sourceDistanceF ψ (fun z => sourceAngularTransform ρ P.carrier M₁ b₁ z.1 z.2)
(fun z => sourceAngularTransform ν Q.carrier M₂ b₂ z.1 z.2) r := by
have hh := sourceDistanceF_split P Q hP hQ ρ ν n m M₁ M₂ M₁ M₂ hM₁.ne' hM₂.ne'
ψ b₁ b₂ hψ h₁ h₂ hW hB₁ hB₂ hw hb₁ hb₂ r
simpa only [sourceCellwiseDistanceF, div_self hM₁.ne', div_self hM₂.ne', one_mul, one_smul] using hh.symm
theorem sourceCellwiseDistanceF_kernel_bound (P Q : DyadicSquare) (hP : 0 < P.side) (hQ : 0 < Q.side)
(ρ ν : Measure Plane) [IsFiniteMeasure ρ] [IsFiniteMeasure ν] (n m : ℕ)
{M₁ M₂ : ℝ} (hM₁ : 0 < M₁) (hM₂ : 0 < M₂)
(ψ : CellPath → CellPath → UnitCircle → ℂ) (b₁ b₂ : Plane → UnitCircle → ℂ)
(hψ : ∀ A B, Measurable (ψ A B))
(h₁ : Measurable (fun z : Plane × UnitCircle => b₁ z.1 z.2))
(h₂ : Measurable (fun z : Plane × UnitCircle => b₂ z.1 z.2))
{W B₁ B₂ ε : ℝ} (hW : 0 ≤ W) (hB₁ : 0 ≤ B₁) (hB₂ : 0 ≤ B₂)
(hw : ∀ A B dir, ‖ψ A B dir‖ ≤ W) (hb₁ : ∀ x dir, ‖b₁ x dir‖ ≤ B₁)
(hb₂ : ∀ y dir, ‖b₂ y dir‖ ≤ B₂) (r : ℝ)
(he : ∀ A ∈ occupiedAnchorPaths P ρ n, ∀ B ∈ occupiedAnchorPaths Q ν m,
∀ x ∈ (P.descendant A).carrier, ∀ y ∈ (Q.descendant B).carrier,
‖sourceDistanceKernel (ψ A B) b₁ b₂ r x y‖ ≤ ε) :
‖sourceCellwiseDistanceF P Q ρ ν n m M₁ M₂ ψ b₁ b₂ r‖ ≤
ε*((ρ P.carrier).toReal/M₁)*((ν Q.carrier).toReal/M₂) := by
unfold sourceCellwiseDistanceF
simp_rw [sourceDistanceF_endpoint_integral ρ ν _ _ _ _ _ b₁ b₂ (hψ _ _) h₁ h₂
hW hB₁ hB₂ (hw _ _) hb₁ hb₂]
exact occupied_pair_error_sum_le P Q hP hQ ρ ν n m hM₁ hM₂ _ he
theorem sourceCellwiseDistanceF_partition (P Q : DyadicSquare) (hP : 0 < P.side) (hQ : 0 < Q.side)
(ρ ν : Measure Plane) [IsFiniteMeasure ρ] [IsFiniteMeasure ν] (n m : ℕ)
{M₁ M₂ : ℝ} (hM₁ : 0 < M₁) (hM₂ : 0 < M₂)
(ψ : UnitCircle → ℂ) (χ : CellPath → CellPath → UnitCircle → ℂ)
(b₁ b₂ : Plane → UnitCircle → ℂ) (hψ : Measurable ψ) (hχ : ∀ A B, Measurable (χ A B))
(h₁ : Measurable (fun z : Plane × UnitCircle => b₁ z.1 z.2))
(h₂ : Measurable (fun z : Plane × UnitCircle => b₂ z.1 z.2))
{W B₁ B₂ : ℝ} (hW : 0 ≤ W) (hB₁ : 0 ≤ B₁) (hB₂ : 0 ≤ B₂)
(hw : ∀ dir, ‖ψ dir‖ ≤ W) (hc : ∀ A B dir, ‖χ A B dir‖ ≤ 1)
(hb₁ : ∀ x dir, ‖b₁ x dir‖ ≤ B₁) (hb₂ : ∀ y dir, ‖b₂ y dir‖ ≤ B₂) (r : ℝ) :
sourceDistanceF ψ (fun z => sourceAngularTransform ρ P.carrier M₁ b₁ z.1 z.2)
(fun z => sourceAngularTransform ν Q.carrier M₂ b₂ z.1 z.2) r =
sourceCellwiseDistanceF P Q ρ ν n m M₁ M₂ (fun A B dir => ψ dir*(1-χ A B dir)) b₁ b₂ r +
sourceCellwiseDistanceF P Q ρ ν n m M₁ M₂ (fun A B dir => ψ dir*χ A B dir) b₁ b₂ r := by
rw [← sourceCellwiseDistanceF_constant P Q hP hQ ρ ν n m hM₁ hM₂ ψ b₁ b₂ hψ h₁ h₂
hW hB₁ hB₂ hw hb₁ hb₂ r]
unfold sourceCellwiseDistanceF
rw [← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro A _
rw [← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro B _
exact sourceDistanceF_retained_add_discarded ψ (χ A B) _ _ r hψ (hχ A B)
((measurable_sourceAngularTransform ρ _ M₁ b₁ h₁).comp (measurable_const.prodMk measurable_id))
((measurable_sourceAngularTransform ν _ M₂ b₂ h₂).comp (measurable_const.prodMk measurable_id))
hW (by positivity) hw (hc A B)
(sourceAngularTransform_norm_le ρ _ M₁ hM₁ b₁ B₁ (fun x _ => hb₁ x) r)
(sourceAngularTransform_norm_le ν _ M₂ hM₂ b₂ B₂ (fun y _ => hb₂ y) r)
end PlanarFalconer
end OAI
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