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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 | import OAI.MeasureTheory.Falconer.Recursion.DescriptorMasks
namespace OAI
namespace PlanarFalconer
open Set Metric MeasureTheory
open scoped ENNReal
theorem descriptor_mask_failure_bound (Q : DyadicSquare) (hQ : 0 < Q.side) (ρ : Measure Plane)
(m : ℝ → ℝ) (N : ℕ) (lam sig : ℝ) (J : ℕ) (ha : SourceAngularBounds Q ρ m N lam sig)
{b : PolicyDescriptor} (hb : ValidDescriptor N b) {x : Plane}
(hx : ∃ P ∈ occupiedAnchorPaths Q ρ (descriptorAnchor b), x ∈ (Q.descendant P).carrier) :
sourceArcLength {dir | descriptorSpatialMask Q ρ m N lam sig J 0 b x dir ≠ 1} ≤
ENNReal.ofReal (((2:ℝ)^N)^(-(lam+sig))) := by
obtain ⟨P,hP,hx⟩ := hx
simp only [descriptorSpatialMask,spatialAnchorMask_eq Q hQ ρ _ _ _ _ hP hx]
apply (circleTestMask_failure_le _ (measurable_descriptorGoodSet _ _ _ _ _ _ _ _ _ _) _ _).trans
exact descriptorGoodSet_exceptional Q ρ m N lam sig J ha hb
((mem_dyadicPaths _ P).mp (Finset.mem_filter.mp hP).1)
theorem listMask_failure_subset (Q : DyadicSquare) (ρ : Measure Plane)
(m : ℝ → ℝ) (N : ℕ) (lam sig : ℝ) (J i : ℕ) (S : Finset PolicyDescriptor) (x : Plane) :
{dir | descriptorListMask Q ρ m N lam sig J i S x dir ≠ 1} ⊆
⋃ b ∈ S, {dir | descriptorSpatialMask Q ρ m N lam sig J i b x dir ≠ 1} := by
intro dir hdir
by_contra hn
have hh : ∀ b ∈ S, descriptorSpatialMask Q ρ m N lam sig J i b x dir = 1 := by
intro b hb
by_contra h
exact hn (mem_iUnion.mpr ⟨b,mem_iUnion.mpr ⟨hb,h⟩⟩)
exact hdir (by simp only [descriptorListMask,Finset.prod_eq_one hh])
theorem ae_listMask_failure_bound (Q : DyadicSquare) (hQ : 0 < Q.side) (ρ : Measure Plane)
(hcarry : ρ Q.carrierᶜ = 0) (m : ℝ → ℝ) (N : ℕ) (lam sig : ℝ) (J : ℕ)
(ha : SourceAngularBounds Q ρ m N lam sig) (S : Finset PolicyDescriptor)
(hS : ∀ b ∈ S, ValidDescriptor N b) :
∀ᵐ x ∂ρ, sourceArcLength {dir | descriptorListMask Q ρ m N lam sig J 0 S x dir ≠ 1} ≤
(S.card:ENNReal)*ENNReal.ofReal (((2:ℝ)^N)^(-(lam+sig))) := by
have hx : ∀ᵐ x ∂ρ, ∀ n : ℕ, ∃ P ∈ occupiedAnchorPaths Q ρ n, x ∈ (Q.descendant P).carrier :=
ae_all_iff.mpr (fun n => ae_occupied_anchor Q ρ hcarry n)
filter_upwards [hx] with x hx
apply (measure_mono (listMask_failure_subset Q ρ m N lam sig J 0 S x)).trans
apply (measure_biUnion_finset_le _ _).trans
calc
_ ≤ ∑ _b ∈ S, ENNReal.ofReal (((2:ℝ)^N)^(-(lam+sig))) := by
apply Finset.sum_le_sum
intro b hb
exact descriptor_mask_failure_bound Q hQ ρ m N lam sig J ha (hS b hb) (hx _)
_ = _ := by simp
end PlanarFalconer
end OAI
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