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f1f7953 | 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 | 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
end
end OAI
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