File size: 7,227 Bytes
7f37411 | 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 | import OAI.MeasureTheory.Falconer.Estimates.ComparableComplexRebin
import OAI.MeasureTheory.Falconer.Angular.ActualStationaryProductKernel
import OAI.MeasureTheory.Falconer.Angular.ActualFourStationarySupport
import OAI.MeasureTheory.Falconer.Angular.AngularComparableCarrier
namespace OAI
namespace PlanarFalconer
open Set MeasureTheory
open scoped ENNReal ContDiff
attribute [local instance] Classical.propDecidable
theorem uniform_actual_comparable_shell_rebin {Ψ : ℝ → ℝ}
(hΨ : ContDiff ℝ ∞ Ψ) (hsΨ : ∀ t, Ψ t ≠ 0 → t ∈ Ioo (1/4:ℝ) 4)
{sig : ℝ} (hsig : 0 < sig) (K o : ℕ) {J : ℕ} (hJ : 0 < J) :
∃ H : ℝ, 0 ≤ H ∧ ∃ N₀ : ℕ, ∀ N ≥ N₀,
∀ (Q : Fin 2 → DyadicSquare), (∀ l, (Q l).side = 1) →
∀ (ρ : Fin 2 → Measure Plane) [∀ l, IsFiniteMeasure (ρ l)],
(∀ l, ρ l (Q l).carrierᶜ = 0) →
∀ (m : Fin 2 → ℝ → ℝ) (q a₀ t₀ a d n v : ℕ),
0 < q → a₀ ≤ a+n → v ≤ t₀ → v ≤ N →
∀ (S : Fin 2 → Finset PolicyDescriptor) (k : Fin 2 → PolicyDescriptor → ℕ),
(∀ l, remainingPolicyDescriptors (fun t => m l t-(t:ℝ)/(N:ℝ)) N q a₀ t₀ a ⊆ S l) →
(∀ l, (S l).card ≤ (N+1)^3+1) → (∀ l, ∑ t ∈ S l, k l t ≤ K) →
∀ L : ℕ, (∀ l t, t ∈ S l → descriptorLength t ≤ L) →
∀ L₀ : ℝ, (v:ℝ)-(a+d:ℕ) = 2*L₀ → (L:ℝ) ≤ L₀ → 10*sig*(N:ℝ) ≤ L₀ →
∀ (lam : Fin 2 → ℝ) (i : Fin 2 → ℕ) (hsp : Fin 2 → Plane → ℂ),
(∀ l x, ‖hsp l x‖ ≤ 1) →
∀ A B : CellPath, A.length = a → B.length = a →
let β := fun l => inheritedCircleSymbol (S l) id (k l) (Q l) (ρ l) (m l)
N (lam l) sig J (i l) (hsp l)
let good := fun l => positiveDistanceGoodSet (Q l) (ρ l) (m l) N q (a+n) v (lam l) sig J (i l+1)
‖∑ row ∈ sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B d,
∑ col ∈ sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B d,
if ¬ angularNear (Q 0) (Q 1) (a+d) row col then
∫ x in angularLengthShell (a+n), angularStationarySpatialKernel (β 0) (β 1) o v Ψ x
∂angularFourCellMeasure (Q 0) (Q 1) (ρ 0) (ρ 1) row col
else 0‖ₑ ≤
(ENNReal.ofReal H * ENNReal.ofReal ((2:ℝ)^(-((v:ℝ)-(a+n:ℕ))))) *
∑ row ∈ sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B n,
∑ col ∈ sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B n,
if angularSeparated (Q 0) (Q 1) (a+n) row col then
∫⁻ x in angularLengthShell (a+n),
(fourPairGood (good 0) (good 1)).indicator (distanceDifferenceKernel v) x
∂angularFourCellMeasure (Q 0) (Q 1) (ρ 0) (ρ 1) row col
else 0 := by
obtain ⟨H,hH,N₁,h₁⟩ := uniform_actual_stationary_product_kernel hΨ hsΨ hsig K o
obtain ⟨N₂,h₂⟩ := uniform_actual_four_approximant_good hsig hJ
refine ⟨H,hH,max N₁ N₂,?_⟩
intro N hN Q hQ ρ inst hcarry m q a₀ t₀ a d n v hq ha₀ ht hv S k hS hcard hK L hL
L₀ hvh hLL hL₀ lam i hsp hspb A B hA hB
let β := fun l => inheritedCircleSymbol (S l) id (k l) (Q l) (ρ l) (m l)
N (lam l) sig J (i l) (hsp l)
let good := fun l => positiveDistanceGoodSet (Q l) (ρ l) (m l) N q (a+n) v (lam l) sig J (i l+1)
have hQp (l : Fin 2) : 0 < (Q l).side := by rw [hQ l]; norm_num
dsimp only
convert angular_complex_shell_rebin (Q 0) (Q 1) (hQ 0) (hQ 1) (ρ 0) (ρ 1) A B a d n hA hB
(fun row col => ¬ angularNear (Q 0) (Q 1) (a+d) row col)
(angularStationarySpatialKernel (β 0) (β 1) o v Ψ) (distanceDifferenceKernel v)
(fourPairGood (good 0) (good 1))
(measurable_fourPairGood (measurable_positiveDistanceGoodSet _ _ _ _ _ _ _ _ _ _ _)
(measurable_positiveDistanceGoodSet _ _ _ _ _ _ _ _ _ _ _))
(ENNReal.ofReal H * ENNReal.ofReal ((2:ℝ)^(-((v:ℝ)-(a+n:ℕ))))) (by finiteness) ?_ ?_ using 1
· congr 1
apply Finset.sum_congr rfl
intro row hr
apply Finset.sum_congr rfl
intro col hc
by_cases hg : ¬ angularNear (Q 0) (Q 1) (a+d) row col <;> simp only [hg,↓reduceIte,β]
· intro row hr col hc hfar
have hf := angular_comparable_far_ae (Q 0) (Q 1) (hQ 0) (hQ 1) (ρ 0) (ρ 1) (a+d) row col
(sourceProductVertices_subset _ _ _ _ _ _ a d hA hB hr)
(sourceProductVertices_subset _ _ _ _ _ _ a d hA hB hc) hfar
have hf' := ae_restrict_of_ae_restrict_of_subset (show angularLengthShell (a+n) ⊆
angularLengthComparable from inter_subset_left) hf
filter_upwards [hf',ae_restrict_mem (measurable_angularLengthShell (a+n))] with x hx hs
have hl := angularLengthShell_lengths (a+n) hs
have hd : ∀ l : Fin 2, (2:ℝ)^(-(a+n:ℕ):ℝ) ≤ ‖(![x 0,x 2] : Fin 2 → Plane) l-(![x 1,x 3] : Fin 2 → Plane) l‖ := by
intro l
have hp : 0 < (2:ℝ)^(-(a+n:ℕ):ℝ) := by positivity
fin_cases l
· change (2:ℝ)^(-((a+n:ℕ):ℝ)) ≤ ‖x 0-x 1‖
rw [← dist_eq_norm]
linarith [hl.1.1]
· change (2:ℝ)^(-((a+n:ℕ):ℝ)) ≤ ‖x 2-x 3‖
rw [← dist_eq_norm]
linarith [hl.2.1]
have hD : ∀ l : Fin 2, sourceSymbolScale N sig L^2 ≤
(2:ℝ)^(v:ℝ)*‖(![x 0,x 2] : Fin 2 → Plane) l-(![x 1,x 3] : Fin 2 → Plane) l‖ := by
intro l
apply sourceSymbolScale_far_frequency_le hsig.le N L hvh hLL hL₀
fin_cases l
· exact hx.1
· exact hx.2
have he := h₁ N ((le_max_left _ _).trans hN) S (fun _ => id) k hcard hK L hL
Q hQp ρ m lam (fun _ => J) i hsp ![x 0,x 2] ![x 1,x 3]
(fun l => hspb l _) (fun l => hspb l _) v (a+n:ℕ) hd hD
have hxvec : (![x 0,x 1,x 2,x 3] : Fin 4 → Plane) = x := by ext j; fin_cases j <;> rfl
simp only [Matrix.cons_val_zero,Matrix.cons_val_one,hxvec,← dist_eq_norm] at he
rw [← ofReal_norm]
have he' := ENNReal.ofReal_le_ofReal he
convert he' using 1
dsimp [distanceDifferenceKernel]
rw [← ENNReal.ofReal_mul hH,← ENNReal.ofReal_mul (mul_nonneg hH (by positivity))]
congr 1
ring
· intro row hr col hc hfar
let X : Fin 4 → Set Plane := ![((Q 0).descendant row.1).carrier,((Q 0).descendant col.1).carrier, ((Q 1).descendant row.2).carrier,((Q 1).descendant col.2).carrier]
have hs := h₂ N ((le_max_right _ _).trans hN) Q hQ ρ hcarry m q a₀ t₀ a (a+n) v
hq ha₀ ht (Nat.le_add_right _ _) hv S k hS lam i hsp X
have hs' : ∀ᵐ x ∂(angularFourCellMeasure (Q 0) (Q 1) (ρ 0) (ρ 1) row col).restrict
(angularLengthShell (a+n)), ∀ (o : ℕ) (w : ℝ) (Ψ : ℝ → ℝ),
(∀ t, Ψ t ≠ 0 → 0 < t) → x 0 ≠ x 1 → x 2 ≠ x 3 →
angularStationarySpatialKernel (β 0) (β 1) o w Ψ x ≠ 0 →
(x 0,x 1) ∈ good 0 ∧ (x 2,x 3) ∈ good 1 := ae_restrict_of_ae hs
filter_upwards [hs',ae_restrict_mem (measurable_angularLengthShell (a+n))] with x hx hshell
have hl := angularLengthShell_lengths (a+n) hshell
have hp : 0 < 256*(2:ℝ)^(-(a+n:ℕ):ℝ) := by positivity
exact hx o v Ψ (fun t ht => lt_trans (by norm_num) (hsΨ t ht).1)
(dist_pos.mp (hp.trans hl.1.1)) (dist_pos.mp (hp.trans hl.2.1))
end PlanarFalconer
end OAI
|