openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Product /ProductConvolutionIdentity.lean
| import OAI.Analysis.IrrationalRotation.Couplings.ProductPairIdentity | |
| import OAI.Analysis.IrrationalRotation.Rotation.CocycleWindowBounds | |
| import OAI.Analysis.IrrationalRotation.Analysis.ConvolutionRatios | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped ENNReal BigOperators ComplexConjugate | |
| namespace Invariant233.ProductModel | |
| theorem ae_antitone_cocycleVersion (θ c : ℕ → ℝ) | |
| (hθ0 : ∀ i, 0 ≤ θ i) (hc : ∀ i, 0 ≤ c i) (hθ : Summable θ) : | |
| ∀ᵐ x ∂baseMeasure, Antitone (fun k : ℤ => ProductCocycle.cocycleVersion θ c k x) := by | |
| obtain ⟨s, _hsm, hfull, _hinv, _hs, hlaw⟩ := | |
| ProductCocycle.exists_invariant_cocycle_core θ hθ0 hθ | |
| filter_upwards [hfull] with x hx | |
| intro a b hab | |
| have heq := hlaw x hx c a (b - a) | |
| rw [add_sub_cancel] at heq | |
| calc | |
| ProductCocycle.cocycleVersion θ c b x = | |
| ProductCocycle.cocycleVersion θ c (b - a) (rotationIterate θ a x) * | |
| ProductCocycle.cocycleVersion θ c a x := heq | |
| _ ≤ 1 * ProductCocycle.cocycleVersion θ c a x := | |
| mul_le_mul_of_nonneg_right | |
| (ProductCocycle.cocycleVersion_le_one_of_nonneg_time θ c hθ0 hc (b - a) | |
| (sub_nonneg.mpr hab) _) (ProductCocycle.cocycleVersion_pos θ c a x).le | |
| _ = _ := one_mul _ | |
| theorem ae_convolution_two_ratio_bounds (θ c : ℕ → ℝ) | |
| (hθ0 : ∀ i, 0 ≤ θ i) (hc : ∀ i, 0 ≤ c i) (hθ : Summable θ) | |
| (F : Base → Fiber →L[ℂ] Fiber) (ν : Measure EndpointPair) | |
| (hfst : ν.map Prod.fst ≪ baseMeasure) (hsnd : ν.map Prod.snd ≪ baseMeasure) | |
| (hpair : ∀ᵐ xy ∂ν, PairIdentity θ c F xy) : | |
| ∀ᵐ xy ∂ν, ∀ (n : ℕ) (v : ℤ), | |
| 0 < ProductCocycle.cocycleVersion θ c ((n : ℤ) + v) xy.2 / | |
| ProductCocycle.cocycleVersion θ c v xy.1 ∧ | |
| ProductCocycle.cocycleVersion θ c ((n : ℤ) + v) xy.2 / | |
| ProductCocycle.cocycleVersion θ c v xy.1 ≤ | |
| min (cocycleRatio θ c xy.1 xy.2 v) | |
| (cocycleRatio θ c xy.1 xy.2 ((n : ℤ) + v)) ∧ | |
| min (cocycleRatio θ c xy.1 xy.2 v) | |
| (cocycleRatio θ c xy.1 xy.2 ((n : ℤ) + v)) ≤ 1 := by | |
| have hmono := ae_antitone_cocycleVersion θ c hθ0 hc hθ | |
| have hx := ae_of_ae_map measurable_fst.aemeasurable (hfst.ae_le hmono) | |
| have hy := ae_of_ae_map measurable_snd.aemeasurable (hsnd.ae_le hmono) | |
| filter_upwards [hx, hy, hpair] with xy hx hy hp n v | |
| exact InvariantProjection.cocycle_two_ratio_bounds | |
| (fun k => ProductCocycle.cocycleVersion θ c k xy.1) | |
| (fun k => ProductCocycle.cocycleVersion θ c k xy.2) | |
| (fun k => ProductCocycle.cocycleVersion_pos θ c k xy.1) | |
| (fun k => ProductCocycle.cocycleVersion_pos θ c k xy.2) | |
| hx hy (fun k => (hp.1 k).2) n v | |
| theorem coefficients_absolute_and_zero_of_pair (θ c : ℕ → ℝ) | |
| (F : Base → Fiber →L[ℂ] Fiber) (x y : Base) | |
| (hproj : IsStarProjection (F y)) | |
| (hinv : (1 - F y) * WeightedShift.operator (actualWeightField θ c y) * F y = 0) | |
| (hpair : PairIdentity θ c F (x, y)) (n : ℕ) | |
| (hprod : ∀ k : ℤ, | |
| (∏ j ∈ Finset.range n, weight θ c (rotationIterate θ (k + j) y)) = | |
| ProductCocycle.cocycleVersion θ c n (rotationIterate θ k y)) | |
| (hlaw : ∀ k : ℤ, | |
| ProductCocycle.cocycleVersion θ c (k + n) y = | |
| ProductCocycle.cocycleVersion θ c n (rotationIterate θ k y) * | |
| ProductCocycle.cocycleVersion θ c k y) | |
| (ξ η : Fiber) : | |
| let A : ℤ → ℂ := fun v => F x ξ v / (ProductCocycle.cocycleVersion θ c v x : ℂ) | |
| let B : ℤ → ℂ := fun v => (ProductCocycle.cocycleVersion θ c (-v) y : ℂ) * | |
| star (((1 - F y) η) (-v)) | |
| (Summable (fun v : ℤ => ‖A v * B (-(n : ℤ) - v)‖)) ∧ | |
| (∑' v : ℤ, A v * B (-(n : ℤ) - v)) = 0 := by | |
| let r : WeightedShift.Weight := ⟨fun k => (cocycleRatio θ c x y k : ℂ), | |
| memℓp_infty ⟨1, by | |
| rintro z ⟨k, rfl⟩ | |
| simpa only [Complex.norm_real, Real.norm_eq_abs, abs_of_pos (hpair.1 k).1] | |
| using (hpair.1 k).2⟩⟩ | |
| have hr : WeightedShift.diagonal r = ratioDiagonal θ c x y := | |
| (boundedDiagonal_weight r).symm | |
| have hp : (1 - F y) * WeightedShift.diagonal r * F x = 0 := by | |
| rw [hr] | |
| exact hpair.2 | |
| apply OrbitInvariance.cocycle_coefficients_absolute_and_zero | |
| (actualWeightField θ c y) r | |
| (fun k => ProductCocycle.cocycleVersion θ c k x) | |
| (fun k => ProductCocycle.cocycleVersion θ c k y) | |
| hproj hinv hp n ξ η | |
| intro v | |
| change (∏ j ∈ Finset.range n, (weight θ c (rotationIterate θ (v + j) y) : ℂ)) * | |
| ((ProductCocycle.cocycleVersion θ c v y / | |
| ProductCocycle.cocycleVersion θ c v x : ℝ) : ℂ) = _ | |
| rw [← Complex.ofReal_prod, hprod v, Complex.ofReal_div, ← mul_div_assoc, | |
| ← Complex.ofReal_mul, ← hlaw v] | |
| theorem ae_coefficients_absolute_and_zero (θ c : ℕ → ℝ) | |
| (hθ0 : ∀ i, 0 ≤ θ i) (hc : ∀ i, 0 ≤ c i) (hθ : Summable θ) | |
| (p : Hilbert →L[ℂ] Hilbert) (F : Base → Fiber →L[ℂ] Fiber) | |
| (hproj : ∀ x, IsStarProjection (F x)) | |
| (hrep : ∀ f : Hilbert, p f =ᵐ[baseMeasure] fun x => F x (f x)) | |
| (hinv : (1 - p) * weightedOperator θ c * p = 0) | |
| (ν : Measure EndpointPair) (hsnd : ν.map Prod.snd ≪ baseMeasure) | |
| (hpair : ∀ᵐ xy ∂ν, PairIdentity θ c F xy) (ξ η : Fiber) : | |
| ∀ᵐ xy ∂ν, ∀ n : ℕ, | |
| let A : ℤ → ℂ := fun v => F xy.1 ξ v / | |
| (ProductCocycle.cocycleVersion θ c v xy.1 : ℂ) | |
| let B : ℤ → ℂ := fun v => (ProductCocycle.cocycleVersion θ c (-v) xy.2 : ℂ) * | |
| star (((1 - F xy.2) η) (-v)) | |
| (Summable (fun v : ℤ => ‖A v * B (-(n : ℤ) - v)‖)) ∧ | |
| (∑' v : ℤ, A v * B (-(n : ℤ) - v)) = 0 := by | |
| have hInv := ae_of_ae_map measurable_snd.aemeasurable | |
| (hsnd.ae_le (ae_actualWeightField_invariant θ c p F hrep hinv)) | |
| obtain ⟨s, _hsm, hfull, _hinv, _hs, hlaw⟩ := | |
| ProductCocycle.exists_invariant_cocycle_core θ hθ0 hθ | |
| have hcore := ae_of_ae_map measurable_snd.aemeasurable (hsnd.ae_le hfull) | |
| apply ae_all_iff.mpr | |
| intro n | |
| have hProd := ae_of_ae_map measurable_snd.aemeasurable | |
| (hsnd.ae_le (ae_weight_orbit_product_eq_cocycleVersion θ c hθ0 hc hθ n)) | |
| filter_upwards [hInv, hcore, hProd, hpair] with xy hi hs hp hxy | |
| exact coefficients_absolute_and_zero_of_pair θ c F xy.1 xy.2 (hproj xy.2) | |
| hi hxy n hp (fun k => hlaw xy.2 hs c k n) ξ η | |
| theorem ae_coefficients_absolute_sum_le (θ c : ℕ → ℝ) | |
| (hθ0 : ∀ i, 0 ≤ θ i) (hc : ∀ i, 0 ≤ c i) (hθ : Summable θ) | |
| (F : Base → Fiber →L[ℂ] Fiber) (hproj : ∀ x, IsStarProjection (F x)) | |
| (ν : Measure EndpointPair) (hfst : ν.map Prod.fst ≪ baseMeasure) | |
| (hsnd : ν.map Prod.snd ≪ baseMeasure) | |
| (hpair : ∀ᵐ xy ∂ν, PairIdentity θ c F xy) (ξ η : Fiber) : | |
| ∀ᵐ xy ∂ν, ∀ n : ℕ, | |
| let A : ℤ → ℂ := fun v => F xy.1 ξ v / | |
| (ProductCocycle.cocycleVersion θ c v xy.1 : ℂ) | |
| let B : ℤ → ℂ := fun v => (ProductCocycle.cocycleVersion θ c (-v) xy.2 : ℂ) * | |
| star (((1 - F xy.2) η) (-v)) | |
| (∑' v : ℤ, ‖A v * B (-(n : ℤ) - v)‖) ≤ ‖ξ‖ * ‖η‖ := by | |
| filter_upwards [ae_convolution_two_ratio_bounds θ c hθ0 hc hθ F ν hfst hsnd hpair] | |
| with xy hratio n | |
| dsimp only | |
| have hbound := InvariantProjection.l2_reindexed_multiplier_bound | |
| (F xy.1 ξ) ((1 - F xy.2) η) (Equiv.addLeft (n : ℤ)) | |
| (fun v => ProductCocycle.cocycleVersion θ c ((n : ℤ) + v) xy.2 / | |
| ProductCocycle.cocycleVersion θ c v xy.1) | |
| (fun v => ⟨(hratio n v).1.le, (hratio n v).2.1.trans (hratio n v).2.2⟩) | |
| have hξ : ‖F xy.1 ξ‖ ≤ ‖ξ‖ := by | |
| simpa only [one_mul] using (F xy.1).le_of_opNorm_le | |
| (IsStarProjection.norm_le _ (hproj xy.1)) ξ | |
| have hη : ‖(1 - F xy.2) η‖ ≤ ‖η‖ := by | |
| simpa only [one_mul] using (1 - F xy.2).le_of_opNorm_le | |
| (IsStarProjection.norm_le _ (hproj xy.2).one_sub) η | |
| have hsum := hbound.2.trans (mul_le_mul hξ hη (norm_nonneg _) (norm_nonneg _)) | |
| convert hsum using 1 | |
| try rfl | |
| apply tsum_congr | |
| intro v | |
| congr 1 | |
| rw [show -(-(n : ℤ) - v) = (n : ℤ) + v by omega] | |
| exact InvariantProjection.convolution_coefficient_factorization | |
| (fun k => ProductCocycle.cocycleVersion θ c k xy.1) | |
| (fun k => ProductCocycle.cocycleVersion θ c k xy.2) | |
| (F xy.1 ξ) ((1 - F xy.2) η) n v | |
| end Invariant233.ProductModel | |
| end | |
| end OAI | |