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9eeb9c7 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 | import OAI.Analysis.IrrationalRotation.Model.ConditionalConvolution
import OAI.Analysis.IrrationalRotation.Couplings.ConditionalL2Tail
import OAI.Analysis.IrrationalRotation.Analysis.OmittedEvents
import OAI.Analysis.IrrationalRotation.Analysis.FiniteWindowIntegralLower
import OAI.Analysis.IrrationalRotation.Couplings.ConditionalConvolutionPointwise
namespace OAI
open MeasureTheory
open scoped ENNReal lp ComplexConjugate
noncomputable section
namespace InvariantProjection.ConditionalConvolution
variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω}
variable {K : ℕ} {D a c η δ : ℝ}
def exceptionalTail (H : ℤ → Set Ω) (K n : ℕ)
(ξ γ : Ω → ℓ²(ℤ, ℂ)) (ω : Ω) : ℝ :=
∑' v : ℤ, (omittedEvent H K n v).indicator
(fun ω => ‖ξ ω v‖ * ‖γ ω ((n : ℤ) + v)‖) ω
omit mΩ in
theorem exceptionalTail_nonneg (H : ℤ → Set Ω) (K n : ℕ)
(ξ γ : Ω → ℓ²(ℤ, ℂ)) (ω : Ω) :
0 ≤ exceptionalTail H K n ξ γ ω := by
apply tsum_nonneg
intro v
exact Set.indicator_nonneg (fun _ _ => mul_nonneg (norm_nonneg _) (norm_nonneg _)) ω
omit mΩ in
theorem exceptionalTail_eq_productMask (H : ℤ → Set Ω) (K n : ℕ)
(ξ γ : Ω → ℓ²(ℤ, ℂ)) :
exceptionalTail H K n ξ γ = fun ω => ∑' v : ℤ,
({z : Ω × ℤ | z.1 ∈ omittedEvent H K n z.2}).indicator
(fun z => ‖ξ z.1 z.2‖ * ‖γ z.1 ((n : ℤ) + z.2)‖) (ω, v) := by
funext ω
unfold exceptionalTail
apply tsum_congr
intro v
by_cases hv : ω ∈ omittedEvent H K n v <;> simp [hv]
omit mΩ in
theorem exceptionalTail_eq_indicatorOne (H : ℤ → Set Ω) (K n : ℕ)
(ξ γ : Ω → ℓ²(ℤ, ℂ)) (ω : Ω) :
exceptionalTail H K n ξ γ ω = ∑' v : ℤ,
(omittedEvent H K n v).indicator (fun _ => (1 : ℝ)) ω *
(‖ξ ω v‖ * ‖γ ω ((n : ℤ) + v)‖) := by
unfold exceptionalTail
apply tsum_congr
intro v
by_cases hv : ω ∈ omittedEvent H K n v <;> simp [hv]
namespace Data
variable (d : Data Ω μ K D a c η δ)
theorem coeffA_measurable (v : ℤ) : Measurable (fun ω => coeffA d.ξ d.Fx ω v) := by
have hx : StronglyMeasurable (fun ω => d.ξ ω v) :=
(lp.evalCLM ℂ (fun _ : ℤ => ℂ) 2 v).continuous.comp_stronglyMeasurable
(d.ξ_measurable.mono d.mX_le)
exact hx.measurable.div (Complex.continuous_ofReal.measurable.comp (d.Fx_measurable v))
theorem coeffB_measurable (v : ℤ) : Measurable (fun ω => coeffB d.γ d.Fy ω v) := by
have hy : StronglyMeasurable (fun ω => d.γ ω (-v)) :=
(lp.evalCLM ℂ (fun _ : ℤ => ℂ) 2 (-v)).continuous.comp_stronglyMeasurable
(d.γ_measurable.mono d.mY_le)
exact (Complex.continuous_ofReal.measurable.comp (d.Fy_measurable (-v))).mul
(Complex.continuous_conj.measurable.comp hy.measurable)
theorem integrableOn_norm_truncated [IsFiniteMeasure μ] (n : ℕ) :
IntegrableOn (fun ω => ‖truncated K n (coeffA d.ξ d.Fx ω) (coeffB d.γ d.Fy ω)‖) d.E μ := by
exact Invariant233.FiniteWindow.integrableOn_norm_convolution μ d.E
(measure_lt_top μ d.E) K n D (coeffA d.ξ d.Fx) (coeffB d.γ d.Fy)
d.coeffA_measurable d.coeffB_measurable
(d.coefficient_bounds.mono fun _ h => h.1)
(d.coefficient_bounds.mono fun _ h => h.2.1)
theorem first_tail_bound [IsProbabilityMeasure μ] (n : ℕ) (hδ : 0 ≤ δ) :
Integrable (exceptionalTail d.H₁ K n d.ξ d.γ) μ ∧
(∫ ω, exceptionalTail d.H₁ K n d.ξ d.γ ω ∂μ) ≤ Real.sqrt δ := by
have h := integral_l2_masked_tail_le_of_condExp (mΩ := mΩ)
d.ξ d.γ (d.γ_measurable.mono d.mY_le) d.ξ_unit d.γ_unit
(Equiv.addLeft (n : ℤ))
(measurableSet_omittedMask d.H₁ K n (fun k _ => d.H₁_measurable k))
d.mX_le d.ξ_measurable hδ
(condExp_omittedMask_section_le μ d.mX d.H₁ K n hδ d.conditional₁)
rw [exceptionalTail_eq_productMask]
exact h.2
theorem second_tail_bound [IsProbabilityMeasure μ] (n : ℕ) (hδ : 0 ≤ δ) :
Integrable (exceptionalTail d.H₂ K n d.ξ d.γ) μ ∧
(∫ ω, exceptionalTail d.H₂ K n d.ξ d.γ ω ∂μ) ≤ Real.sqrt δ := by
have h := integral_l2_masked_tail_le_of_condExp_right (mΩ := mΩ)
d.ξ d.γ (d.ξ_measurable.mono d.mX_le) d.ξ_unit d.γ_unit
(Equiv.addLeft (n : ℤ))
(measurableSet_omittedMask d.H₂ K n (fun k _ => d.H₂_measurable k))
d.mY_le d.γ_measurable hδ
(condExp_omittedMask_section_le μ d.mY d.H₂ K n hδ d.conditional₂)
rw [exceptionalTail_eq_productMask]
exact h
theorem integral_truncated_le_of_pointwise [IsProbabilityMeasure μ]
(n : ℕ) (hη : 0 ≤ η) (hδ : 0 ≤ δ)
(hpoint : ∀ᵐ ω ∂μ.restrict d.E,
‖truncated K n (coeffA d.ξ d.Fx ω) (coeffB d.γ d.Fy ω)‖ ≤
η + exceptionalTail d.H₁ K n d.ξ d.γ ω + exceptionalTail d.H₂ K n d.ξ d.γ ω) :
(∫ ω in d.E, ‖truncated K n (coeffA d.ξ d.Fx ω) (coeffB d.γ d.Fy ω)‖ ∂μ) ≤
η + 2 * Real.sqrt δ := by
obtain ⟨ht₁, hb₁⟩ := d.first_tail_bound n hδ
obtain ⟨ht₂, hb₂⟩ := d.second_tail_bound n hδ
have hi : Integrable (d.E.indicator (fun ω =>
‖truncated K n (coeffA d.ξ d.Fx ω) (coeffB d.γ d.Fy ω)‖)) μ :=
(integrable_indicator_iff d.E_measurable).mpr (d.integrableOn_norm_truncated n)
have hfull : ∀ᵐ ω ∂μ,
d.E.indicator (fun ω => ‖truncated K n (coeffA d.ξ d.Fx ω) (coeffB d.γ d.Fy ω)‖) ω ≤
η + exceptionalTail d.H₁ K n d.ξ d.γ ω + exceptionalTail d.H₂ K n d.ξ d.γ ω := by
filter_upwards [(ae_restrict_iff' d.E_measurable).mp hpoint] with ω hω
by_cases hE : ω ∈ d.E
· simpa only [Set.indicator_of_mem hE] using hω hE
· rw [Set.indicator_of_notMem hE]
exact add_nonneg (add_nonneg hη (exceptionalTail_nonneg _ _ _ _ _ _))
(exceptionalTail_nonneg _ _ _ _ _ _)
have hiR := ((integrable_const η).add ht₁).add ht₂
have hbound := integral_mono_ae hi hiR hfull
rw [integral_indicator d.E_measurable] at hbound
have heval : (∫ ω, ((fun _ : Ω => η) + exceptionalTail d.H₁ K n d.ξ d.γ +
exceptionalTail d.H₂ K n d.ξ d.γ) ω ∂μ) =
η + (∫ ω, exceptionalTail d.H₁ K n d.ξ d.γ ω ∂μ) +
(∫ ω, exceptionalTail d.H₂ K n d.ξ d.γ ω ∂μ) := by
calc
_ = (∫ ω, ((fun _ : Ω => η) + exceptionalTail d.H₁ K n d.ξ d.γ) ω ∂μ) +
∫ ω, exceptionalTail d.H₂ K n d.ξ d.γ ω ∂μ :=
integral_add ((integrable_const η).add ht₁) ht₂
_ = ((∫ _ : Ω, η ∂μ) + ∫ ω, exceptionalTail d.H₁ K n d.ξ d.γ ω ∂μ) +
∫ ω, exceptionalTail d.H₂ K n d.ξ d.γ ω ∂μ := by
rw [show (∫ ω, ((fun _ : Ω => η) + exceptionalTail d.H₁ K n d.ξ d.γ) ω ∂μ) =
(∫ _ : Ω, η ∂μ) + ∫ ω, exceptionalTail d.H₁ K n d.ξ d.γ ω ∂μ from
integral_add (integrable_const η) ht₁]
_ = _ := by simp
rw [heval] at hbound
linarith
theorem pointwise_truncated_bound (n : ℕ) (hn : n ≤ 2 * K) (hη : 0 ≤ η) :
∀ᵐ ω ∂μ.restrict d.E,
‖truncated K n (coeffA d.ξ d.Fx ω) (coeffB d.γ d.Fy ω)‖ ≤
η + exceptionalTail d.H₁ K n d.ξ d.γ ω + exceptionalTail d.H₂ K n d.ξ d.γ ω := by
filter_upwards [ae_restrict_of_ae d.ξ_unit, ae_restrict_of_ae d.γ_unit,
ae_restrict_of_ae (d.ratio_bounds n hn), ae_restrict_of_ae (d.zero_convolution n hn),
d.attenuation] with ω hξ hγ hr hz ht
have hz' : (∑' v : ℤ, (d.ξ ω v / (d.Fx ω v : ℂ)) *
((d.Fy ω ((n : ℤ) + v) : ℂ) * conj (d.γ ω ((n : ℤ) + v)))) = 0 := by
simpa only [coeffA, coeffB, neg_sub, neg_neg, sub_neg_eq_add, add_comm] using hz
have h := pointwise_bound K n (d.Fx ω) (d.Fy ω) (d.ξ ω) (d.γ ω)
hξ hγ η hη d.H₁ d.H₂ ω hr hz' ht
rw [← exceptionalTail_eq_indicatorOne, ← exceptionalTail_eq_indicatorOne] at h
exact h
theorem obstruction_of_pointwise [IsProbabilityMeasure μ]
{ε : ℝ} (hε : 0 ≤ ε)
(hcoeff : ∀ A B : ℤ → ℂ,
(∀ v : ℤ, |v| ≤ (K : ℤ) → ‖A v‖ ≤ D) →
(∀ v : ℤ, |v| ≤ (K : ℤ) → ‖B v‖ ≤ D) →
a ≤ ‖A 0‖ → a ≤ ‖B 0‖ →
ε ≤ ∑ n ∈ Finset.range (2 * K + 1), ‖truncated K n A B‖)
(hη : 0 ≤ η) (hδ : 0 ≤ δ)
(hpoint : ∀ n : ℕ, n ≤ 2 * K → ∀ᵐ ω ∂μ.restrict d.E,
‖truncated K n (coeffA d.ξ d.Fx ω) (coeffB d.γ d.Fy ω)‖ ≤
η + exceptionalTail d.H₁ K n d.ξ d.γ ω + exceptionalTail d.H₂ K n d.ξ d.γ ω) :
c * ε ≤ ((2 * K + 1 : ℕ) : ℝ) * (η + 2 * Real.sqrt δ) := by
apply Invariant233.FiniteWindow.finite_sum_obstruction μ d.E (2 * K + 1)
(fun n ω => ‖truncated K n (coeffA d.ξ d.Fx ω) (coeffB d.γ d.Fy ω)‖)
c ε (η + 2 * Real.sqrt δ) hε d.E_mass
· intro n _
exact d.integrableOn_norm_truncated n
· exact Invariant233.FiniteWindow.integral_lower_of_sum_bound μ K D a ε hcoeff
d.E (coeffA d.ξ d.Fx) (coeffB d.γ d.Fy)
d.coeffA_measurable d.coeffB_measurable d.coefficient_bounds
· intro n hn
exact d.integral_truncated_le_of_pointwise n hη hδ
(hpoint n (Nat.lt_succ_iff.mp (Finset.mem_range.mp hn)))
end Data
theorem conditional_convolution_obstruction
(K : ℕ) (D a : ℝ) (ha : 0 < a) :
∃ ε > 0, ∀ {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
[IsProbabilityMeasure μ] (c η δ : ℝ),
0 ≤ η → 0 ≤ δ → ∀ _ : Data Ω μ K D a c η δ,
c * ε ≤ ((2 * K + 1 : ℕ) : ℝ) * (η + 2 * Real.sqrt δ) := by
obtain ⟨ε, hε, hcoeff⟩ := Invariant233.FiniteWindow.uniform_convolution_sum_lower_bound K D a ha
refine ⟨ε, hε, ?_⟩
intro Ω mΩ μ hμ c η δ hη hδ d
exact d.obstruction_of_pointwise hε.le hcoeff hη hδ
(fun n hn => d.pointwise_truncated_bound n hn hη)
end InvariantProjection.ConditionalConvolution
end
end OAI
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