openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Rotation /CircleRationalGraphTransfer.lean
| import OAI.Analysis.IrrationalRotation.Model.CircleRationalGraphs | |
| import OAI.Analysis.IrrationalRotation.Rotation.CircleRationalGraphCaps | |
| import OAI.Analysis.IrrationalRotation.Operators.OperatorCompactTransfer | |
| import OAI.Analysis.IrrationalRotation.Analysis.CountableGraphTransfer | |
| import OAI.Analysis.IrrationalRotation.Analysis.UniformAbsoluteContinuity | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped Topology | |
| namespace Invariant233.CircleParameterLaws | |
| variable {θ : ℝ} {p : CircleParameterData} (hp : CircleParameterLaws θ p) | |
| include hp | |
| theorem rational_graph_measure_ratio_cap (A : ℕ) (hA : 1 ≤ A) (r k : ℤ) : | |
| ∀ᵐ z ∂(circleTranslationGraph (p.rationalTranslation A r) : Measure (Circle × Circle)), | |
| p.floorCocycle θ k z.2 / p.floorCocycle θ k z.1 ≤ p.rationalRatioBound A := by | |
| change ∀ᵐ z ∂circleMeasure.map (fun x => (x, x + p.rationalTranslation A r)), _ | |
| apply (ae_map_iff (measurable_id.prodMk | |
| (measurable_id.add measurable_const)).aemeasurable ?_).mpr | |
| · filter_upwards [hp.ae_rational_graph_ratio_cap] with x hx | |
| exact hx A hA r k | |
| · exact measurableSet_le | |
| ((hp.measurable_parameter_cocycle k).comp measurable_snd |>.div | |
| ((hp.measurable_parameter_cocycle k).comp measurable_fst)) measurable_const | |
| theorem rational_graph_identity_of_approximants | |
| (P : Circle → SequenceHilbert ℤ →L[ℂ] SequenceHilbert ℤ) | |
| (hP : ∀ x, IsStarProjection (P x)) | |
| (hPm : ∀ k : ℤ, Measurable (fun x => P x (lp.single 2 k 1))) | |
| (A : ℕ) (hA : 1 ≤ A) (r : ℤ) | |
| (ν : ℕ → FiniteMeasure (Circle × Circle)) | |
| (hν : Tendsto ν atTop (𝓝 (circleTranslationGraph (p.rationalTranslation A r)))) | |
| (hfirst : UniformlyAbsolutelyContinuous circleMeasure | |
| (fun j => (ν j : Measure (Circle × Circle)).map Prod.fst)) | |
| (hsecond : UniformlyAbsolutelyContinuous circleMeasure | |
| (fun j => (ν j : Measure (Circle × Circle)).map Prod.snd)) | |
| (hcap : ∀ j k, ∀ᵐ z ∂(ν j : Measure (Circle × Circle)), | |
| p.floorCocycle θ k z.2 / p.floorCocycle θ k z.1 ≤ p.rationalRatioBound A) | |
| (hzero : ∀ j, ∀ᵐ z ∂(ν j : Measure (Circle × Circle)), | |
| (1 - P z.2).comp ((ratioDiagonal (p.floorCocycle θ) z).comp (P z.1)) = 0) : | |
| ∀ᵐ x ∂circleMeasure, | |
| (1 - P (x + p.rationalTranslation A r)) * | |
| ratioDiagonal (p.floorCocycle θ) (x, x + p.rationalTranslation A r) * P x = 0 := by | |
| let : IsFiniteMeasure circleMeasure := by | |
| change IsFiniteMeasure (volume : Measure Circle) | |
| infer_instance | |
| have hlim := CompactTransfer.compact_operator_transfer circleMeasure P hP | |
| (p.floorCocycle θ) (fun k x => p.floorCocycle_pos θ k x) | |
| hp.measurable_parameter_cocycle hPm hν hfirst hsecond | |
| (p.rationalRatioBound A) (Real.exp_pos _) | |
| hcap (fun k => hp.rational_graph_measure_ratio_cap A hA r k) hzero | |
| have h := ae_of_ae_map | |
| (measurable_id.prodMk (measurable_id.add measurable_const)).aemeasurable hlim | |
| filter_upwards [h] with x hx | |
| exact hx | |
| theorem rational_graph_identity_of_orbit_approximants | |
| (P : Circle → SequenceHilbert ℤ →L[ℂ] SequenceHilbert ℤ) | |
| (hP : ∀ x, IsStarProjection (P x)) | |
| (hPm : ∀ k : ℤ, Measurable (fun x => P x (lp.single 2 k 1))) | |
| (horbit : ∀ n : ℕ, 0 < n → ∀ᵐ x ∂circleMeasure, | |
| (1 - P (CircleWeights.rotate θ (n : ℤ) x)).comp | |
| ((ratioDiagonal (p.floorCocycle θ) | |
| (x, CircleWeights.rotate θ (n : ℤ) x)).comp (P x)) = 0) | |
| (A : ℕ) (hA : 1 ≤ A) (r : ℤ) | |
| (ν : ℕ → FiniteMeasure (Circle × Circle)) | |
| (hν : Tendsto ν atTop (𝓝 (circleTranslationGraph (p.rationalTranslation A r)))) | |
| (hfirst : UniformlyAbsolutelyContinuous circleMeasure | |
| (fun j => (ν j : Measure (Circle × Circle)).map Prod.fst)) | |
| (hsecond : UniformlyAbsolutelyContinuous circleMeasure | |
| (fun j => (ν j : Measure (Circle × Circle)).map Prod.snd)) | |
| (hcap : ∀ j k, ∀ᵐ z ∂(ν j : Measure (Circle × Circle)), | |
| p.floorCocycle θ k z.2 / p.floorCocycle θ k z.1 ≤ p.rationalRatioBound A) | |
| (hsupport : ∀ j, ∀ᵐ z ∂(ν j : Measure (Circle × Circle)), | |
| ∃ n : ℕ, 0 < n ∧ z.2 = CircleWeights.rotate θ (n : ℤ) z.1) : | |
| ∀ᵐ x ∂circleMeasure, | |
| (1 - P (x + p.rationalTranslation A r)) * | |
| ratioDiagonal (p.floorCocycle θ) (x, x + p.rationalTranslation A r) * P x = 0 := by | |
| apply hp.rational_graph_identity_of_approximants P hP hPm A hA r ν hν hfirst hsecond hcap | |
| intro j | |
| apply Fields.ae_pair_relation_of_countable_graphs | |
| (fun x y => (1 - P y).comp ((ratioDiagonal (p.floorCocycle θ) (x, y)).comp (P x)) = 0) | |
| (fun n : ℕ+ => CircleWeights.rotate θ (n : ℤ)) | |
| (hfirst.absolutelyContinuous j) | |
| · intro n | |
| exact horbit n n.property | |
| · filter_upwards [hsupport j] with z hz | |
| obtain ⟨n, hn, he⟩ := hz | |
| exact ⟨⟨n, hn⟩, he⟩ | |
| omit hp in | |
| theorem rationalGraphIdentities_of_forall (_hp : CircleParameterLaws θ p) | |
| (P : Circle → SequenceHilbert ℤ →L[ℂ] SequenceHilbert ℤ) | |
| (h : ∀ A : ℕ, 1 ≤ A → ∀ r : ℤ, ∀ᵐ x ∂circleMeasure, | |
| (1 - P (x + p.rationalTranslation A r)) * | |
| ratioDiagonal (p.floorCocycle θ) (x, x + p.rationalTranslation A r) * P x = 0) : | |
| p.RationalGraphIdentities θ P := by | |
| apply ae_all_iff.mpr | |
| intro A | |
| by_cases hA : 1 ≤ A | |
| · filter_upwards [ae_all_iff.mpr (h A hA)] with x hx | |
| exact fun _ => hx | |
| · exact Filter.Eventually.of_forall fun _ hAx => (hA hAx).elim | |
| end Invariant233.CircleParameterLaws | |
| end | |
| end OAI | |