| import OAI.Analysis.IrrationalRotation.Model.CircleTruncationEvent |
| import OAI.Analysis.IrrationalRotation.Model.ConditionalConvolution |
| import OAI.Analysis.IrrationalRotation.Model.Rotation |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory |
| open scoped BigOperators ComplexConjugate |
|
|
| namespace Invariant233.CircleCoupling |
|
|
| def projectionDiagonal (F : Circle → Rotation.Fiber →L[ℂ] Rotation.Fiber) |
| (x : Circle) : ℝ := ((F x (lp.single 2 0 1)) 0).re |
|
|
| def projectionCoefficientA (θ : ℝ) (p : CircleParameterData) |
| (F : Circle → Rotation.Fiber →L[ℂ] Rotation.Fiber) (x : Circle) (v : ℤ) : ℂ := |
| (F x (lp.single 2 0 1)) v / (p.floorCocycle θ v x : ℂ) |
|
|
| def projectionCoefficientB (θ : ℝ) (p : CircleParameterData) |
| (F : Circle → Rotation.Fiber →L[ℂ] Rotation.Fiber) (y : Circle) (v : ℤ) : ℂ := |
| (p.floorCocycle θ (-v) y : ℂ) * conj (((1 - F y) (lp.single 2 0 1)) (-v)) |
|
|
| def truncationSum (θ : ℝ) (p : CircleParameterData) (j : ℕ) |
| (hr : ∀ i, 0 < MixedRadix.radices p.q i) |
| (F : Circle → Rotation.Fiber →L[ℂ] Rotation.Fiber) (ω : Sample) : ℝ := |
| ∑ n ∈ Finset.range (2 * p.K j + 1), |
| ‖InvariantProjection.ConditionalConvolution.truncated (p.K j) n |
| (projectionCoefficientA θ p F (point p hr false ω)) |
| (projectionCoefficientB θ p F (point p hr true ω))‖ |
|
|
| def TruncationBounds (θ : ℝ) (p : CircleParameterData) |
| (hr : ∀ i, 0 < MixedRadix.radices p.q i) |
| (F : Circle → Rotation.Fiber →L[ℂ] Rotation.Fiber) : Prop := |
| ∃ J : ℕ, 1 ≤ J ∧ ∀ j : ℕ, J ≤ j → |
| Real.exp (-90) / 2 ≤ (law p j).real (truncationEvent p j hr (projectionDiagonal F)) ∧ |
| (Real.exp (-90) / 2) * p.ε j ≤ |
| ∫ ω in truncationEvent p j hr (projectionDiagonal F), |
| truncationSum θ p j hr F ω ∂law p j |
|
|
| end Invariant233.CircleCoupling |
|
|
| end |
|
|
| end OAI |
|
|
