| 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 |
|
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| end OAI |
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