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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 | import OAI.MeasureTheory.Falconer.Estimates.DescriptorInverseRegime
import OAI.MeasureTheory.Falconer.Radial.RadialShellMeasure
namespace OAI
/-! # The actual source shell satisfies the descriptor stationary regime -/
namespace Falconer
open Set MeasureTheory
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
theorem eventual_descriptor_source_shell_regime {σ : ℝ} (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
theorem eventually_separated_inverse_frequency {σ δ : ℝ} (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⟩
theorem eventual_descriptor_pair_shell_regime {d : ℕ} {σ δ : ℝ}
(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
end Falconer
end OAI
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