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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 | import OAI.Analysis.IrrationalRotation.Analysis.BlockedSourceEvent
import OAI.Analysis.IrrationalRotation.Product.ProductScaleCocycleWindow
import OAI.Analysis.IrrationalRotation.Rotation.ProjectionCocycleWindow
import OAI.Analysis.IrrationalRotation.Analysis.FiniteWindowIntegralLower
namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped Topology BigOperators ComplexConjugate lp
namespace Invariant233.BlockedProductCoupling
theorem measurable_sourceEvent (p : ProductScaleData)
(e : ℕ ≃ Σ s, Fin (p.m s)) {r : ProductModel.Base → ℝ}
(hr : Measurable r) (j : ℕ) : MeasurableSet (sourceEvent p e r j) := by
have hx : Measurable (circleLeftEndpoint e) := by
unfold circleLeftEndpoint toCircle leftEndpoint flatten OrderedProductCoupling.leftEndpoint
fun_prop
have hy : Measurable (circleRightEndpoint e) := by
unfold circleRightEndpoint toCircle rightEndpoint flatten OrderedProductCoupling.rightEndpoint
fun_prop
have hc : MeasurableSet (currentEvent p.m j) := by
change MeasurableSet ({z : Sample p.m | ∃ i, (z j i).1 = some (4 : Fin 5)} ∩
{z | ∀ i, (z j i).2.1 < 1 - 2 / (p.m j : ℝ) ∧
2 / (p.m j : ℝ) < (z j i).2.2})
apply MeasurableSet.inter
· simp only [ofPred_exists]
exact MeasurableSet.iUnion fun i => (by fun_prop :
Measurable (fun z : Sample p.m => (z j i).1)) (measurableSet_singleton _)
· simp only [ofPred_forall]
apply MeasurableSet.iInter
intro i
have hxi : Measurable (fun z : Sample p.m => (z j i).2.1) := by fun_prop
have hyi : Measurable (fun z : Sample p.m => (z j i).2.2) := by fun_prop
exact (measurableSet_lt hxi measurable_const).inter
(measurableSet_lt measurable_const hyi)
have hf : MeasurableSet (futureBad p.m (fun s => (p.K s : ℝ)) j) := by
apply MeasurableSet.iUnion
intro n
apply MeasurableSet.iUnion
intro i
have hxi : Measurable (fun z : Sample p.m => (z (j + 1 + n) i).2.1) := by fun_prop
have hyi : Measurable (fun z : Sample p.m => (z (j + 1 + n) i).2.2) := by fun_prop
exact (measurableSet_le measurable_const hxi).union
(measurableSet_le hyi measurable_const)
exact hc.diff (hf.union
((measurableSet_lt (hr.comp hx) measurable_const).union
(measurableSet_lt (measurable_const.sub (hr.comp hy)) measurable_const)))
theorem ae_sourceEvent_coefficient_properties {p : ProductScaleData}
(hp : ProductScaleLaws p) (e : ℕ ≃ Σ s, Fin (p.m s)) (θ : ℕ → ℝ)
(hphase : ∀ i, θ i ∈ Icc (1 / p.N (e i).1) (2 / p.N (e i).1))
(P : ProjectionField) (hP : ∀ᵐ x ∂ProductModel.baseMeasure, IsStarProjection (P x)) (j : ℕ) :
∀ᵐ z ∂sourceLaw p j,
z ∈ sourceEvent p e (projectionDiagonal P) j →
(∀ v : ℤ, |v| ≤ (p.K j : ℤ) →
‖sourceCoefficientA p e θ P z v‖ ≤ p.coefficientBound j) ∧
(∀ v : ℤ, |v| ≤ (p.K j : ℤ) →
‖sourceCoefficientB p e θ P z v‖ ≤ p.coefficientBound j) ∧
1 / ((j + 1 : ℕ) : ℝ) ≤ ‖sourceCoefficientA p e θ P z 0‖ ∧
1 / ((j + 1 : ℕ) : ℝ) ≤ ‖sourceCoefficientB p e θ P z 0‖ := by
have hrep := ae_circle_representatives hp.m_ge_100 source_kappa_nonneg source_kappa_le
(fun s _ => (hp.a_bounds s).1) (fun s _ => (hp.a_bounds s).2)
(fun s _ => (hp.b_bounds s).1) (fun s _ => (hp.b_bounds s).2) e j
filter_upwards [hrep, (sourceLaw_quasiMeasurePreserving_left hp e j).ae hP,
(sourceLaw_quasiMeasurePreserving_right hp e j).ae hP] with z hz hPx hPy
intro he
obtain ⟨_, hcurrent, hfuture, hdiagx, hdiagy⟩ := mem_sourceRetainedEvent_iff.mp he
have hcur : ∀ i, (e i).1 = j →
ProductCocycle.representative (circleLeftEndpoint e z i) < 1 - 2 / (p.m j : ℝ) ∧
2 / (p.m j : ℝ) < ProductCocycle.representative (circleRightEndpoint e z i) := by
intro i hi
change circleRepresentative _ < _ ∧ _ < circleRepresentative _
rw [(hz i).1, (hz i).2]
change (z (e i).1 (e i).2).2.1 < _ ∧ _ < (z (e i).1 (e i).2).2.2
subst j
exact hcurrent (e i).2
have hfut : ∀ i, j < (e i).1 →
ProductCocycle.representative (circleLeftEndpoint e z i) <
1 - 2 * (p.K j : ℝ) / p.N (e i).1 ∧
2 * (p.K j : ℝ) / p.N (e i).1 <
ProductCocycle.representative (circleRightEndpoint e z i) := by
intro i hi
change circleRepresentative _ < _ ∧ _ < circleRepresentative _
rw [(hz i).1, (hz i).2]
exact hfuture (e i).1 hi (e i).2
obtain ⟨hA, hB⟩ := hp.product_finite_window_coefficients e θ hphase j
(circleLeftEndpoint e z) (circleRightEndpoint e z) hcur hfut
(P (circleLeftEndpoint e z) sourceBasisZero)
((1 - P (circleRightEndpoint e z)) sourceBasisZero)
(ProjectionCocycleWindow.projection_vector_norm_le _ hPx)
(ProjectionCocycleWindow.complement_vector_norm_le _ hPy)
obtain ⟨hA0, hB0⟩ := ProjectionCocycleWindow.central_lower_bounds
(P (circleLeftEndpoint e z)) (P (circleRightEndpoint e z))
(1 / ((j + 1 : ℕ) : ℝ)) hdiagx (by
change 1 / ((j + 1 : ℕ) : ℝ) ≤
1 - ((P (circleRightEndpoint e z) ProjectionCocycleWindow.basisZero) 0).re at hdiagy
linarith)
exact ⟨hA, hB,
by simpa [sourceCoefficientA, sourceBasisZero, ProjectionCocycleWindow.basisZero] using hA0,
by simpa [sourceCoefficientB, sourceBasisZero, ProjectionCocycleWindow.basisZero] using hB0⟩
theorem measurable_sourceCoefficientA (p : ProductScaleData)
(e : ℕ ≃ Σ s, Fin (p.m s)) (θ : ℕ → ℝ) (P : ProjectionField)
(hPm : ∀ v, Measurable (fun x => (P x sourceBasisZero) v)) (v : ℤ) :
Measurable (fun z => sourceCoefficientA p e θ P z v) := by
have hx : Measurable (circleLeftEndpoint e) := by
unfold circleLeftEndpoint toCircle leftEndpoint flatten OrderedProductCoupling.leftEndpoint
fun_prop
exact ((hPm v).comp hx).div (Complex.measurable_ofReal.comp
((ProductCocycle.measurable_cocycleVersion θ (fun i => p.C (e i).1) v).comp hx))
theorem measurable_sourceCoefficientB (p : ProductScaleData)
(e : ℕ ≃ Σ s, Fin (p.m s)) (θ : ℕ → ℝ) (P : ProjectionField)
(hPm : ∀ v, Measurable (fun x => (P x sourceBasisZero) v)) (v : ℤ) :
Measurable (fun z => sourceCoefficientB p e θ P z v) := by
have hy : Measurable (circleRightEndpoint e) := by
unfold circleRightEndpoint toCircle rightEndpoint flatten OrderedProductCoupling.rightEndpoint
fun_prop
have hvec : Measurable (fun z : Sample p.m =>
sourceBasisZero (-v) - (P (circleRightEndpoint e z) sourceBasisZero) (-v)) :=
measurable_const.sub ((hPm (-v)).comp hy)
have hvec' : Measurable (fun z : Sample p.m =>
((1 - P (circleRightEndpoint e z)) sourceBasisZero) (-v)) := by
simpa only [sub_apply, one_apply_eq_self, lp.coeFn_sub, Pi.sub_apply] using hvec
exact (Complex.measurable_ofReal.comp
((ProductCocycle.measurable_cocycleVersion θ (fun i => p.C (e i).1) (-v)).comp hy)).mul
(Complex.continuous_conj.measurable.comp hvec')
theorem source_polynomial_sum_lower {p : ProductScaleData} (hp : ProductScaleLaws p)
(j : ℕ) (A B : ℤ → ℂ)
(hA : ∀ v, |v| ≤ (p.K j : ℤ) → ‖A v‖ ≤ p.coefficientBound j)
(hB : ∀ v, |v| ≤ (p.K j : ℤ) → ‖B v‖ ≤ p.coefficientBound j)
(hA0 : 1 / ((j + 1 : ℕ) : ℝ) ≤ ‖A 0‖)
(hB0 : 1 / ((j + 1 : ℕ) : ℝ) ≤ ‖B 0‖) :
p.ε j ≤ ∑ n ∈ Finset.range (2 * p.K j + 1),
‖FiniteWindow.convolution (p.K j) A B n‖ := by
obtain ⟨n, hn, hε⟩ := hp.polynomial_lower j A B hA hB hA0 hB0
exact hε.trans (Finset.single_le_sum (fun k _ => norm_nonneg _)
(Finset.mem_range.mpr (by omega)))
theorem sourceEvent_integral_lower {p : ProductScaleData}
(hp : ProductScaleLaws p) (e : ℕ ≃ Σ s, Fin (p.m s)) (θ : ℕ → ℝ)
(hphase : ∀ i, θ i ∈ Icc (1 / p.N (e i).1) (2 / p.N (e i).1))
(P : ProjectionField) (hP : ∀ᵐ x ∂ProductModel.baseMeasure, IsStarProjection (P x))
(hPm : ∀ v, Measurable (fun x => (P x sourceBasisZero) v)) (j : ℕ) :
p.ε j * (sourceLaw p j).real (sourceEvent p e (projectionDiagonal P) j) ≤
∫ z in sourceEvent p e (projectionDiagonal P) j,
∑ n ∈ Finset.range (2 * p.K j + 1),
‖FiniteWindow.convolution (p.K j)
(sourceCoefficientA p e θ P z) (sourceCoefficientB p e θ P z) n‖
∂sourceLaw p j := by
have := sourceLaw_isProbabilityMeasure hp j
have hr : Measurable (projectionDiagonal P) := Complex.measurable_re.comp (hPm 0)
apply FiniteWindow.integral_lower_of_sum_bound (sourceLaw p j)
(p.K j) (p.coefficientBound j) (1 / ((j + 1 : ℕ) : ℝ)) (p.ε j)
(source_polynomial_sum_lower hp j)
(sourceEvent p e (projectionDiagonal P) j)
(sourceCoefficientA p e θ P) (sourceCoefficientB p e θ P)
(measurable_sourceCoefficientA p e θ P hPm)
(measurable_sourceCoefficientB p e θ P hPm)
filter_upwards [ae_restrict_of_ae (ae_sourceEvent_coefficient_properties hp e θ hphase P hP j),
ae_restrict_mem (measurable_sourceEvent p e hr j)] with z hz he
exact hz he
theorem eventually_sourceEvent_and_integral_lower {p : ProductScaleData}
(hp : ProductScaleLaws p) (e : ℕ ≃ Σ s, Fin (p.m s)) (θ : ℕ → ℝ)
(hphase : ∀ i, θ i ∈ Icc (1 / p.N (e i).1) (2 / p.N (e i).1))
(P : ProjectionField) (hP : ∀ᵐ x ∂ProductModel.baseMeasure, IsStarProjection (P x))
(hPm : ∀ v, Measurable (fun x => (P x sourceBasisZero) v))
(hi : ∀ᵐ x ∂ProductModel.baseMeasure,
0 < projectionDiagonal P x ∧ projectionDiagonal P x < 1) :
∀ᶠ j in atTop,
Real.exp (-82) / 2 ≤ (sourceLaw p j).real (sourceEvent p e (projectionDiagonal P) j) ∧
(Real.exp (-82) / 2) * p.ε j ≤
∫ z in sourceEvent p e (projectionDiagonal P) j,
∑ n ∈ Finset.range (2 * p.K j + 1),
‖FiniteWindow.convolution (p.K j)
(sourceCoefficientA p e θ P z) (sourceCoefficientB p e θ P z) n‖
∂sourceLaw p j := by
have hr : Measurable (projectionDiagonal P) := Complex.measurable_re.comp (hPm 0)
filter_upwards [eventually_sourceEvent_mass hp e hr hi] with j hj
refine ⟨hj, ?_⟩
calc
(Real.exp (-82) / 2) * p.ε j ≤
p.ε j * (sourceLaw p j).real (sourceEvent p e (projectionDiagonal P) j) := by
simpa only [mul_comm] using mul_le_mul_of_nonneg_right hj (hp.epsilon_pos j).le
_ ≤ _ := sourceEvent_integral_lower hp e θ hphase P hP hPm j
end Invariant233.BlockedProductCoupling
end
end OAI
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