| import OAI.MeasureTheory.Falconer.Shells.HighShellParentEnergy |
| import OAI.MeasureTheory.Falconer.Shells.HighShellProfileMass |
| import OAI.MeasureTheory.Falconer.Shells.HighShellMeasurable |
| import OAI.MeasureTheory.Falconer.Shells.HighShellErrorPower |
| import OAI.MeasureTheory.Falconer.Shells.ShellApproximationError |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set MeasureTheory Filter |
| namespace PositiveDistanceData |
|
|
| theorem uniform_high_shell_parent_bound {sig lam eps T : ℝ} |
| (hsig : 0 < sig) (heps : 0 < eps) : |
| ∃ n N₀ : ℕ, ∀ d : PositiveDistanceData, ∀ (B : ℕ) (δ : ℝ), |
| RegularizedProfileData d.Q d.ρ d.N B δ d.lam d.mass → |
| N₀ ≤ d.N → d.sig = sig → d.lam = lam → |
| 0 < d.Q.side → d.StronglySeparated → 0 < d.q → |
| d.sig*d.N ≤ (d.q:ℝ) → d.a+100*d.q < d.t → d.t ≤ d.N → |
| d.X.length = d.a → d.Y.length = d.a → |
| ∀ v : ℝ, (d.t:ℝ)-d.sig*d.N ≤ v → v ≤ d.t → |
| ∀ Ψ : ℝ → ℝ, Measurable Ψ → (∀ x, 0 ≤ Ψ x) → |
| (∀ x, Ψ x ≠ 0 → x ∈ Ioo (1/4:ℝ) 4) → |
| ∀ W : ℝ, 0 ≤ W → (∀ x, Ψ x ≤ W) → |
| (∀ x ∈ Icc (1/2:ℝ) 2, 1 ≤ Ψ x) → |
| ∀ H : ℝ, 0 ≤ H → |
| (∀ j ∈ Finset.range (n+1), |
| ∀ b ∈ inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList), ∀ ξ, |
| (d.inversePhasedHF v Ψ b.2 ξ).toReal ≤ H) → |
| (2:ℝ)^(-(d.a:ℝ)) * |
| (∫ r in Icc ((2:ℝ)^(v-1)) ((2:ℝ)^(v+1)), ‖charFun d.weightedDistance r‖^2) ≤ |
| ((2:ℝ)^d.N)^eps*H+((2:ℝ)^d.N)^(-T) := by |
| obtain ⟨n,N₁,h₁⟩ := uniform_high_shell_profile_error (lam := lam) (T := (T+3)/2) hsig |
| obtain ⟨N₂,h₂⟩ := uniform_spatialLeibniz_sum_subpower 96 2004 (by norm_num) (by norm_num) |
| (show 0 < eps/4 by linarith) n |
| obtain ⟨N₃,h₃⟩ := fixed_constant_absorbed (half_pos heps) 4 |
| refine ⟨n,max 1 (max N₁ (max N₂ N₃)),?_⟩ |
| intro d B δ hd hN hsig' hlam hQ hsep hq hqs hw ht hX hY v hv hvt Ψ hΨ hpos hs W hW hΨW hone H hH0 hH |
| let := hd.probability |
| let := d.weightedDistance_isFinite hQ hsep |
| have hN1 : N₁ ≤ d.N := (le_max_left _ _).trans ((le_max_right _ _).trans hN) |
| have hN2 : N₂ ≤ d.N := (le_max_left _ _).trans ((le_max_right _ _).trans ((le_max_right _ _).trans hN)) |
| have hN3 : N₃ ≤ d.N := (le_max_right _ _).trans ((le_max_right _ _).trans ((le_max_right _ _).trans hN)) |
| let R : ℝ := (2:ℝ)^d.N |
| let μ := volume.restrict (Icc ((2:ℝ)^(v-1)) ((2:ℝ)^(v+1))) |
| let e := R^(-((T+3)/2)) |
| have herr : ∀ᵐ r ∂μ, ‖charFun d.weightedDistance r-d.normalizedHighShellF n v r‖ ≤ e := by |
| filter_upwards [ae_restrict_mem measurableSet_Icc] with r hr |
| exact h₁ d B δ hd hN1 hsig' hlam hQ hsep hq hqs hw ht hX hY v r hv hr.1 |
| have hi : Integrable (fun r => ‖charFun d.weightedDistance r‖^2) μ := |
| integrable_norm_square_bounded μ _ measurable_charFun measureReal_nonneg |
| (Eventually.of_forall (fun r => norm_charFun_le r)) |
| have hsq := (shell_square_approximation μ _ _ (d.normalizedHighShellF_measurable hQ hsep n v) |
| hi (show 0 ≤ e by dsimp [e]; positivity) herr).2 |
| have hmain := d.normalizedHighShellF_parent_energy hQ hsep (by rw [hsig']; exact hsig.le) |
| hq hqs hw hv n Ψ hΨ hpos hs hW hΨW hone hH0 hH |
| let S := ∑ j ∈ Finset.range (n+1), spatialLeibnizCost 96 2004 (by norm_num) (by norm_num) j |
| (fun k => d.mass k-(k:ℝ)/(d.N:ℝ)) d.N d.q d.a d.t |
| have hS : S ≤ R^(eps/4) := h₂ d.N hN2 _ d.q d.a d.t hq |
| have hS0 : 0 ≤ S := by |
| apply Finset.sum_nonneg |
| intro j hj |
| apply mul_nonneg (spatialFourierCost_nonneg ..) |
| apply mul_nonneg _ (by positivity) |
| apply List.sum_nonneg |
| intro x hx |
| obtain ⟨b,hb,rfl⟩ := List.mem_map.mp hx |
| positivity |
| have hcost : 4*S^2 ≤ R^eps := by |
| calc |
| _ ≤ R^(eps/2)*(R^(eps/4))^2 := |
| mul_le_mul (h₃ d.N hN3) (pow_le_pow_left₀ hS0 hS 2) (sq_nonneg _) (by positivity) |
| _ = _ := by |
| rw [← Real.rpow_mul_natCast (show 0 ≤ R by dsimp [R]; positivity), |
| ← Real.rpow_add (show 0 < R by dsimp [R]; positivity)] |
| congr 1 |
| ring |
| have herror := high_shell_error_power d.N d.a ((le_max_left _ _).trans hN) |
| (hvt.trans (Nat.cast_le.mpr ht)) T |
| calc |
| _ ≤ (2:ℝ)^(-(d.a:ℝ)) * (2*(∫ r, ‖d.normalizedHighShellF n v r‖^2 ∂μ)+2*e^2*μ.real univ) := |
| mul_le_mul_of_nonneg_left hsq (by positivity) |
| _ = 2*((2:ℝ)^(-(d.a:ℝ))*(∫ r, ‖d.normalizedHighShellF n v r‖^2 ∂μ))+ |
| (2:ℝ)^(-(d.a:ℝ))*(2*e^2*μ.real univ) := by ring |
| _ ≤ 2*(2*S^2*H)+R^(-T) := |
| add_le_add (mul_le_mul_of_nonneg_left hmain (by norm_num)) herror |
| _ ≤ R^eps*H+R^(-T) := by nlinarith [mul_le_mul_of_nonneg_right hcost hH0] |
|
|
| end PositiveDistanceData |
|
|
| end PlanarFalconer |
|
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| end OAI |
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