| import OAI.Analysis.IrrationalRotation.Rotation.CircleQuantizedMarginals |
| import OAI.Analysis.IrrationalRotation.Couplings.ProductDensityLimit |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal BigOperators |
| namespace Invariant233.CircleCoupling |
| open CircleParameterData |
|
|
| theorem hybrid_density_of_coordinate_budget {θ : ℝ} {p : CircleParameterData} |
| (hp : CircleParameterLaws θ p) (j : ℕ) |
| (hr : ∀ i, 0 < MixedRadix.radices p.q i) (ε : ℕ → Bool) {C : ℝ} |
| (hsum : Summable (fun i : ℕ => if i = 0 then 0 else 50 * |
| (kappa (p.blockIndex i) ^ 2 / (p.m (p.blockIndex i) : ℝ) + |
| tag j (p.blockIndex i) / (p.m (p.blockIndex i) : ℝ)))) |
| (hC : (∑' i : ℕ, if i = 0 then 0 else 50 * |
| (kappa (p.blockIndex i) ^ 2 / (p.m (p.blockIndex i) : ℝ) + |
| tag j (p.blockIndex i) / (p.m (p.blockIndex i) : ℝ))) ≤ C) : |
| ∃ H : Circle → ℝ, MemLp H 2 volume ∧ (0 ≤ᵐ[volume] H) ∧ |
| (∫ x, H x) = 1 ∧ eLpNorm H 2 volume ≤ ENNReal.ofReal (Real.exp (C / 2)) ∧ |
| (law p j).map (hybridPoint p hr ε) = |
| volume.withDensity (fun x => ENNReal.ofReal (H x)) := by |
| let : ∀ i, IsProbabilityMeasure (digitReference p hr i) := |
| digitReference_isProbabilityMeasure p hr |
| have he (i : ℕ) := (digitDensity_laws hp j hr (ε i) i).2.2 |
| have hsE : Summable (fun i => |
| ∫ d, (digitDensity p hr j (ε i) i d - 1) ^ 2 ∂digitReference p hr i) := |
| Summable.of_nonneg_of_le (fun i => integral_nonneg (fun d => sq_nonneg _)) he hsum |
| obtain ⟨H, hH, hnonneg, hmean, hnorm, hmeasure⟩ := |
| ProductDensity.exists_product_density |
| (μ := fun i => digitReference p hr i) |
| (h := fun i => digitDensity p hr j (ε i) i) (C := C) |
| (fun i => digitDensity_memLp hp j hr (ε i) i) |
| (fun i d => ((digitDensity_laws hp j hr (ε i) i).1 d).1) |
| (fun i => (digitDensity_laws hp j hr (ε i) i).2.1) |
| hsE ((hsE.tsum_le_tsum he hsum).trans hC) |
| have htwo := MixedRadix.radices_ge_two hp.q_pos hp.frequency_increasing hp.frequency_divisible |
| change MemLp H 2 (MixedRadix.digitLaw (MixedRadix.radices p.q) hr) at hH |
| change (0 ≤ᵐ[MixedRadix.digitLaw (MixedRadix.radices p.q) hr] H) at hnonneg |
| change (∫ d, H d ∂MixedRadix.digitLaw (MixedRadix.radices p.q) hr) = 1 at hmean |
| change eLpNorm H 2 (MixedRadix.digitLaw (MixedRadix.radices p.q) hr) ≤ _ at hnorm |
| change (MixedRadix.digitLaw (MixedRadix.radices p.q) hr).withDensity |
| (fun d => ENNReal.ofReal (H d)) = _ at hmeasure |
| let G : Circle → ℝ := H ∘ MixedRadix.circleCanonical (MixedRadix.radices p.q) hr |
| have hcan := MixedRadix.circleCanonical_measurePreserving htwo hr |
| have htransport := MixedRadix.digitLaw_circle_density_transport_ae htwo hr |
| hH.aestronglyMeasurable.aemeasurable |
| refine ⟨G, ?_, ?_, ?_, ?_, ?_⟩ |
| · exact hH.comp_measurePreserving hcan |
| · change ∀ᵐ x ∂(volume : Measure Circle), |
| 0 ≤ H (MixedRadix.circleCanonical (MixedRadix.radices p.q) hr x) |
| exact hcan.quasiMeasurePreserving.ae (p := fun d => 0 ≤ H d) hnonneg |
| · have hHm : AEStronglyMeasurable H |
| ((volume : Measure Circle).map (MixedRadix.circleCanonical (MixedRadix.radices p.q) hr)) := by |
| rw [hcan.map_eq] |
| exact hH.aestronglyMeasurable |
| have hi := integral_map hcan.measurable.aemeasurable hHm |
| rw [hcan.map_eq] at hi |
| exact hi.symm.trans hmean |
| · exact htransport.2.le.trans hnorm |
| · change (law p j).map |
| (MixedRadix.circleValue ∘ hybridDigits p hr ε) = _ |
| rw [← Measure.map_map (MixedRadix.measurable_circleValue htwo) |
| (measurable_hybridDigits hr ε), map_hybridDigits hp, ← hmeasure] |
| exact htransport.1 |
|
|
| end Invariant233.CircleCoupling |
|
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| end |
|
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| end OAI |
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