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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 | import OAI.Analysis.IrrationalRotation.Rotation.CircleTruncationIntegral
import OAI.Analysis.IrrationalRotation.Rotation.CircleTailExceptionalFamilies
import OAI.Analysis.IrrationalRotation.Rotation.CircleOrderedTransfer
import OAI.Analysis.IrrationalRotation.Analysis.ParameterConvolutionIdentity
import OAI.Analysis.IrrationalRotation.Rotation.CircleObstruction
namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped lp ComplexConjugate
namespace Invariant233.CircleCoupling
open InvariantProjection.ConditionalConvolution
variable {θ : ℝ} {p : CircleParameterData} (hp : CircleParameterLaws θ p)
include hp
theorem conditionalData_of_truncation_mass
(j : ℕ) (hj : 1 ≤ j) (hr : ∀ i, 0 < MixedRadix.radices p.q i)
{P : Rotation.Hilbert →L[ℂ] Rotation.Hilbert}
(hmem : P ∈ Rotation.algebra θ) (hP : IsStarProjection P)
(hT : (1 - P) * p.measurableOperator θ hp * P = 0)
(F : Circle → Rotation.Fiber →L[ℂ] Rotation.Fiber)
(hF : ∀ z, IsStarProjection (F z))
(hFm : ∀ v : Rotation.Fiber, StronglyMeasurable (fun z => F z v))
(hrep : ∀ ξ : Rotation.Hilbert, P ξ =ᵐ[circleMeasure] fun z => F z (ξ z))
(c : ℝ) (hmass : c ≤ (law p j).real (truncationEvent p j hr
(fun z => ((F z ProjectionCocycleWindow.basisZero) 0).re))) :
Nonempty (Data Sample (law p j) (p.K j) (p.coefficientBound j)
(1 / (j : ℝ)) c (Real.exp (-p.L j)) (Real.exp (-p.L j / 400))) := by
classical
let := law_isProbabilityMeasure hp j
let r : Circle → ℝ := fun z => ((F z ProjectionCocycleWindow.basisZero) 0).re
let E := truncationEvent p j hr r
let ξ : Sample → Rotation.Fiber := fun ω =>
F (point p hr false ω) ProjectionCocycleWindow.basisZero
let γ : Sample → Rotation.Fiber := fun ω =>
(1 - F (point p hr true ω)) ProjectionCocycleWindow.basisZero
let Fx : Sample → ℤ → ℝ := fun ω k => p.floorCocycle θ k (point p hr false ω)
let Fy : Sample → ℤ → ℝ := fun ω k => p.floorCocycle θ k (point p hr true ω)
have hrm : Measurable r := Complex.continuous_re.measurable.comp
(((lp.evalCLM ℂ (fun _ : ℤ => ℂ) 2 0).continuous.comp_stronglyMeasurable
(hFm ProjectionCocycleWindow.basisZero)).measurable)
have hEm : MeasurableSet E := measurableSet_truncationEvent hp j hr hrm
obtain ⟨H₁, H₂, hm₁, hm₂, hc₁, hc₂, hatt⟩ :=
tail_exceptional_families_of_subset hp j hj hr E hEm
(truncationEvent_subset_tagEvent p j hr r)
have hordered := orderedCouplingIdentities_of_rational hp hr F hF
(fun k => (hFm (lp.single 2 k 1)).measurable)
(hp.rational_graphs_of_representation hmem hP hT F hrep) j
have hbound := hordered.mono (fun _ h k => (h.2 k).2)
have hzero := hp.ae_parameter_convolution_absolute_and_zero (law p j)
(point p hr false) (point p hr true) (measurable_point hp hr true).aemeasurable
(endpoint_absolutelyContinuous hp j hr true) F hF hrep hT
(hordered.mono (fun _ h => h.1)) hbound
have hratio := hp.ae_parameter_convolution_ratio_bounds (law p j)
(point p hr false) (point p hr true) (measurable_point hp hr false).aemeasurable
(measurable_point hp hr true).aemeasurable
(endpoint_absolutelyContinuous hp j hr false)
(endpoint_absolutelyContinuous hp j hr true) hbound
have hγm : StronglyMeasurable (fun z =>
(1 - F z) ProjectionCocycleWindow.basisZero) := by
have h : StronglyMeasurable
((fun _ : Circle => ProjectionCocycleWindow.basisZero) -
fun z => F z ProjectionCocycleWindow.basisZero) :=
stronglyMeasurable_const.sub (hFm ProjectionCocycleWindow.basisZero)
convert h using 1
funext z
simp only [Pi.sub_apply, sub_apply, one_apply_eq_self]
refine ⟨{
mX := endpointSigma p hr false
mY := endpointSigma p hr true
mX_le := (measurable_point hp hr false).comap_le
mY_le := (measurable_point hp hr true).comap_le
ξ := ξ
γ := γ
ξ_measurable := (hFm ProjectionCocycleWindow.basisZero).comp_measurable
(measurable_iff_comap_le.mpr le_rfl)
γ_measurable := hγm.comp_measurable (measurable_iff_comap_le.mpr le_rfl)
ξ_unit := Eventually.of_forall (fun ω =>
ProjectionCocycleWindow.projection_vector_norm_le _ (hF _))
γ_unit := Eventually.of_forall (fun ω =>
ProjectionCocycleWindow.complement_vector_norm_le _ (hF _))
Fx := Fx
Fy := Fy
Fx_measurable := fun k => (hp.measurable_parameter_cocycle k).comp
(measurable_point hp hr false)
Fy_measurable := fun k => (hp.measurable_parameter_cocycle k).comp
(measurable_point hp hr true)
Fx_pos := Eventually.of_forall (fun ω k => p.floorCocycle_pos θ k _)
Fy_pos := Eventually.of_forall (fun ω k => p.floorCocycle_pos θ k _)
Fx_zero := Eventually.of_forall (fun ω => p.floorCocycle_zero θ _)
Fy_zero := Eventually.of_forall (fun ω => p.floorCocycle_zero θ _)
zero_convolution := ?_
ratio_bounds := fun n _ => hratio.mono (fun _ h => h n)
E := E
E_measurable := hEm
E_mass := hmass
coefficient_bounds := ?_
H₁ := H₁
H₂ := H₂
H₁_measurable := hm₁
H₂_measurable := hm₂
attenuation := hatt
conditional₁ := hc₁
conditional₂ := hc₂ }⟩
· intro n _
filter_upwards [hzero] with ω hω
exact (hω n ProjectionCocycleWindow.basisZero ProjectionCocycleWindow.basisZero).2
· filter_upwards [ae_restrict_mem hEm,
ae_restrict_of_ae (ae_truncationEvent_exclusions hp j hj hr r)] with ω hω he
have he' := he hω
have hdiag := hω.2.2.2.2
have hright : ((F (point p hr true ω) ProjectionCocycleWindow.basisZero) 0).re ≤
1 - 1 / (j : ℝ) := by
change 1 / (j : ℝ) ≤ 1 - ((F (point p hr true ω)
ProjectionCocycleWindow.basisZero) 0).re at hdiag
linarith
exact hp.canonical_projection_coefficient_properties j hj _ _
(fun s hs i hi => (he' s hs i hi).1) (fun s hs i hi => (he' s hs i hi).2)
_ _ (hF _) (hF _) (1 / (j : ℝ)) hω.2.2.2.1 hright
end Invariant233.CircleCoupling
end
end OAI
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