| import OAI.MeasureTheory.Falconer.Estimates.DescriptorInverseRegime |
| import OAI.MeasureTheory.Falconer.Radial.RadialShellMeasure |
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| namespace OAI |
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| /-! # The actual source shell satisfies the descriptor stationary regime -/ |
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| namespace Falconer |
| open Set MeasureTheory |
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| theorem source_shell_frequency_bounds (v : ℝ) (Ψ : ℝ → ℝ) (hΨ : Measurable Ψ) |
| (hsupp : ∀ r, Ψ r ≠ 0 → r ∈ Ioo (1/4:ℝ) 4) : |
| ∀ᵐ r ∂PlanarFalconer.sourceAngularShellMeasure v Ψ, |
| (2:ℝ)^v/4 < r ∧ r < 4*(2:ℝ)^v := by |
| have hn : ∀ᵐ r ∂PlanarFalconer.sourceAngularShellMeasure v Ψ, |
| Ψ (r/(2:ℝ)^v) ≠ 0 := by |
| unfold PlanarFalconer.sourceAngularShellMeasure |
| apply (ae_withDensity_iff (by fun_prop)).mpr |
| filter_upwards with r hr |
| intro hz |
| apply hr |
| simp only [hz,zero_div,ENNReal.ofReal_zero] |
| filter_upwards [hn] with r hr |
| have h := hsupp _ hr |
| constructor |
| · have hl := (lt_div_iff₀ (show 0 < (2:ℝ)^v by positivity)).mp h.1 |
| linarith |
| · exact (div_lt_iff₀ (show 0 < (2:ℝ)^v by positivity)).mp h.2 |
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| theorem eventual_descriptor_source_shell_regime (hσ : 0 < σ) (hσ1 : σ ≤ 1) |
| (Ψ : ℝ → ℝ) (hΨ : Measurable Ψ) |
| (hsupp : ∀ r, Ψ r ≠ 0 → r ∈ Ioo (1/4:ℝ) 4) : |
| ∃ N₀ : ℕ, ∀ N ≥ N₀, ∀ (L : ℕ) (a v : ℝ), L ≤ N → |
| 20*σ*(N:ℝ) ≤ v-a → (L:ℝ) ≤ (v-a)/2+σ*(N:ℝ) → |
| ∀ᵐ r ∂PlanarFalconer.sourceAngularShellMeasure v Ψ, |
| 1 ≤ descriptorDerivativeScale N σ L ∧ |
| descriptorDerivativeScale N σ L ≤ ((2:ℝ)^N)^2 ∧ |
| 1 ≤ r*(2:ℝ)^(-a) ∧ |
| (descriptorDerivativeScale N σ L)^2 ≤ r*(2:ℝ)^(-a) ∧ |
| r*(2:ℝ)^(-a) ≤ 4*(2:ℝ)^(v-a) ∧ |
| (descriptorDerivativeScale N σ L)^2/(r*(2:ℝ)^(-a)) ≤ ((2:ℝ)^N)^(-17*σ) := by |
| obtain ⟨N₀,hN₀⟩ := eventual_descriptor_inverse_frequency hσ hσ1 |
| refine ⟨N₀,?_⟩ |
| intro N hN L a v hLN hq hL |
| filter_upwards [source_shell_frequency_bounds v Ψ hΨ hsupp] with r hr |
| obtain ⟨hD,hDu,hΛ,hR⟩ := hN₀ N hN L (v-a) (r*(2:ℝ)^(-a)) hLN hq hL |
| (dyadic_shell_frequency_lower hr.1.le) |
| have hΛpos : 0 < r*(2:ℝ)^(-a) := zero_lt_one.trans_le hΛ |
| have hsmall : ((2:ℝ)^N)^(-17*σ) ≤ 1 := |
| Real.rpow_le_one_of_one_le_of_nonpos (one_le_pow₀ (by norm_num)) (by linarith) |
| refine ⟨hD,hDu,hΛ,(div_le_one hΛpos).mp (hR.trans hsmall),?_,hR⟩ |
| calc |
| _ ≤ (4*(2:ℝ)^v)*(2:ℝ)^(-a) := |
| mul_le_mul_of_nonneg_right hr.2.le (by positivity) |
| _ = 4*(2:ℝ)^(v-a) := by |
| rw [mul_assoc,← Real.rpow_add (by norm_num : (0:ℝ) < 2)] |
| congr 2 |
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| theorem eventually_separated_inverse_frequency (hσ : 0 < σ) (hδ : 0 < δ) : |
| ∃ N₀ : ℕ, ∀ N ≥ N₀, ∀ D r s t : ℝ, 1 ≤ D → 0 < s → δ ≤ t/s → |
| 0 < r*s → D^2/(r*s) ≤ ((2:ℝ)^N)^(-17*σ) → |
| 1 ≤ r*t ∧ D^2/(r*t) ≤ ((2:ℝ)^N)^(-16*σ) := by |
| obtain ⟨N₀,hN₀⟩ := Regularization.fixed_constant_absorbed hσ δ⁻¹ |
| refine ⟨N₀,?_⟩ |
| intro N hN D r s t hD hs ht hΛ hratio |
| have hr : 0 < r := (mul_pos_iff_of_pos_right hs).mp hΛ |
| have hst : δ*s ≤ t := (le_div_iff₀ hs).mp ht |
| have hδΛ : 0 < (r*s)*δ := mul_pos hΛ hδ |
| have hlower : (r*s)*δ ≤ r*t := by nlinarith [mul_le_mul_of_nonneg_left hst hr.le] |
| have hrt : 0 < r*t := hδΛ.trans_le hlower |
| have hbound : D^2/(r*t) ≤ ((2:ℝ)^N)^(-16*σ) := by |
| calc |
| _ ≤ D^2/((r*s)*δ) := div_le_div_of_nonneg_left (sq_nonneg _) hδΛ hlower |
| _ = δ⁻¹*(D^2/(r*s)) := by ring |
| _ ≤ δ⁻¹*((2:ℝ)^N)^(-17*σ) := |
| mul_le_mul_of_nonneg_left hratio (inv_nonneg.mpr hδ.le) |
| _ ≤ ((2:ℝ)^N)^σ*((2:ℝ)^N)^(-17*σ) := |
| mul_le_mul_of_nonneg_right (hN₀ N hN) (by positivity) |
| _ = ((2:ℝ)^N)^(-16*σ) := by |
| rw [← Real.rpow_add (by positivity)] |
| congr 1 |
| ring |
| have hsmall : ((2:ℝ)^N)^(-16*σ) ≤ 1 := |
| Real.rpow_le_one_of_one_le_of_nonpos (one_le_pow₀ (by norm_num)) (by linarith) |
| exact ⟨(one_le_pow₀ hD).trans ((div_le_one hrt).mp (hbound.trans hsmall)),hbound⟩ |
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| theorem eventual_descriptor_pair_shell_regime |
| (hσ : 0 < σ) (hσ1 : σ ≤ 1) (hδ : 0 < δ) |
| (Ψ : ℝ → ℝ) (hΨ : Measurable Ψ) |
| (hsupp : ∀ r, Ψ r ≠ 0 → r ∈ Ioo (1/4:ℝ) 4) : |
| ∃ N₀ : ℕ, ∀ N ≥ N₀, ∀ (L : ℕ) (a v : ℝ), L ≤ N → |
| 20*σ*(N:ℝ) ≤ v-a → (L:ℝ) ≤ (v-a)/2+σ*(N:ℝ) → |
| ∀ (μ ν : Measure (Space d)), |
| (∀ᵐ z ∂μ.prod ν, δ ≤ dist z.1 z.2/(2:ℝ)^(-a)) → |
| ∀ᵐ r ∂PlanarFalconer.sourceAngularShellMeasure v Ψ, |
| 1 ≤ descriptorDerivativeScale N σ L ∧ |
| descriptorDerivativeScale N σ L ≤ ((2:ℝ)^N)^2 ∧ |
| (0 < r*(2:ℝ)^(-a) ∧ |
| (descriptorDerivativeScale N σ L)^2 ≤ r*(2:ℝ)^(-a) ∧ |
| r*(2:ℝ)^(-a) ≤ 4*(2:ℝ)^(v-a)) ∧ |
| ∀ᵐ z ∂μ.prod ν, 1 ≤ r*dist z.1 z.2 ∧ |
| (descriptorDerivativeScale N σ L)^2/(r*dist z.1 z.2) ≤ ((2:ℝ)^N)^(-16*σ) := by |
| obtain ⟨N₁,hN₁⟩ := eventual_descriptor_source_shell_regime hσ hσ1 Ψ hΨ hsupp |
| obtain ⟨N₂,hN₂⟩ := eventually_separated_inverse_frequency hσ hδ |
| refine ⟨max N₁ N₂,?_⟩ |
| intro N hN L a v hLN hq hL μ ν hsep |
| filter_upwards [hN₁ N ((le_max_left _ _).trans hN) L a v hLN hq hL] with r hr |
| obtain ⟨hD,hDu,hΛ,hDΛ,hΛu,hratio⟩ := hr |
| have hΛpos : 0 < r*(2:ℝ)^(-a) := zero_lt_one.trans_le hΛ |
| refine ⟨hD,hDu,⟨hΛpos,hDΛ,hΛu⟩,?_⟩ |
| filter_upwards [hsep] with z hz |
| exact hN₂ N ((le_max_right _ _).trans hN) _ r ((2:ℝ)^(-a)) (dist z.1 z.2) |
| hD (by positivity) hz hΛpos hratio |
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| end Falconer |
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| end OAI |
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