| import OAI.Analysis.FiniteFactor.OrbitSampling.UniformEvent |
| import OAI.Analysis.FiniteFactor.OrbitSampling.RotationTransport |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter Finset |
| open scoped Classical ENNReal |
| noncomputable section |
|
|
| /-- The actual second marginal of one orbit summand, obtained by weighted |
| Haar transport and not supplied as an assumption. -/ |
| theorem sourceSamplingSummand_second_density (I : Finset SourceCoordinate) (stage : ℕ) |
| (θ : SourceCoordinate → ℝ) (n : ℤ) : |
| (sourceSamplingSummand I stage θ n).map Prod.snd = |
| sourceSamplingHaar.withDensity (fun y => ENNReal.ofReal |
| (sourceSamplingWeight I stage θ n (coordinateRotate θ (-n) y))) := by |
| unfold sourceSamplingSummand |
| have hp : Measurable (fun x : SourceCoordinate → ℝ => (x,coordinateRotate θ n x)) := |
| measurable_id.prodMk (coordinateRotate_preserving θ n).measurable |
| rw [Measure.map_map measurable_snd hp] |
| exact sourceSamplingHaar_rotate_withDensity θ n _ |
| (sourceSamplingWeight_measurable I stage θ n).ennreal_ofReal |
|
|
| /-- The actual second marginal is the second-oriented empirical density. -/ |
| theorem sourceSamplingUntrimmed_second_density (I : Finset SourceCoordinate) (stage : ℕ) |
| (θ : SourceCoordinate → ℝ) : |
| (sourceSamplingUntrimmed I stage θ).map Prod.snd = |
| sourceSamplingHaar.withDensity (fun x => ENNReal.ofReal |
| (sourceSamplingEmpiricalDensity stage true (fun i => decide (i ∈ I)) |
| (fun _ => Set.Icc 0 1) (fun _ => Set.Icc 0 1) θ x)) := by |
| let S := sourceSamplingPowers stage |
| let c : ℝ≥0∞ := ENNReal.ofReal (1/((sourceSamplingPowers stage).card : ℝ)) |
| let w (n : ℤ) := fun x => sourceSamplingWeight I stage θ n (coordinateRotate θ (-n) x) |
| have hrep : (fun x => ENNReal.ofReal (sourceSamplingEmpiricalDensity stage true (fun i => decide (i ∈ I)) |
| (fun _ => Set.Icc 0 1) (fun _ => Set.Icc 0 1) θ x)) =ᵐ[sourceSamplingHaar] |
| (fun x => c*(∑ n ∈ S, ENNReal.ofReal (w n x))) := by |
| filter_upwards [sourceSamplingHaar_cube] with x hx |
| have hw (n : ℤ) : sourceSamplingProduct stage true (fun i => decide (i ∈ I)) |
| (fun _ => Set.Icc 0 1) (fun _ => Set.Icc 0 1) n θ x = w n x := by |
| apply Finset.prod_congr rfl |
| intro i _ |
| have hx' : x i ∈ Set.Icc (0 : ℝ) 1 := ⟨(hx i).1,(hx i).2.le⟩ |
| have hz : Int.fract (x i-Int.fract ((n : ℝ)*θ i)) ∈ Set.Icc (0 : ℝ) 1 := |
| ⟨Int.fract_nonneg _,(Int.fract_lt_one _).le⟩ |
| simp only [samplingOrientedWeight,ite_true,hz,hx',and_self] |
| rw [coordinateRotate_neg_phase] |
| simp only [sourceSamplingEmpiricalDensity,hw] |
| rw [div_eq_mul_inv,mul_comm,← one_div, |
| ENNReal.ofReal_mul (by positivity),ENNReal.ofReal_sum_of_nonneg (fun n _ => (sourceSamplingWeight_bounds I stage θ n (coordinateRotate θ (-n) x)).1)] |
| rw [withDensity_congr_ae hrep] |
| ext A hA |
| rw [Measure.map_apply measurable_snd hA] |
| unfold sourceSamplingUntrimmed |
| rw [Measure.smul_apply,Measure.finsetSum_apply,withDensity_apply _ hA] |
| have hterm (n : ℤ) : sourceSamplingSummand I stage θ n (Prod.snd ⁻¹' A) = |
| ∫⁻ x in A, ENNReal.ofReal (w n x) ∂sourceSamplingHaar := by |
| rw [← Measure.map_apply measurable_snd hA,sourceSamplingSummand_second_density] |
| exact withDensity_apply _ hA |
| simp only [hterm,smul_eq_mul] |
| rw [lintegral_const_mul' c _ ENNReal.ofReal_ne_top] |
| congr 1 |
| exact (lintegral_finsetSum S (fun n _ => |
| ((sourceSamplingWeight_measurable I stage θ n).comp (coordinateRotate_preserving θ (-n)).measurable).ennreal_ofReal)).symm |
|
|
| /-- Uniform L2 densities for the actual untrimmed second marginals. -/ |
| theorem sourceSamplingUntrimmed_second_L2 (I : Finset SourceCoordinate) : |
| ∀ᵐ θ ∂sourceAngleLaw, ∃ C : ℝ, ∀ stage, |
| ∃ f : (SourceCoordinate → ℝ) → ℝ, |
| (sourceSamplingUntrimmed I stage θ).map Prod.snd = sourceSamplingHaar.withDensity (fun x => ENNReal.ofReal (f x)) ∧ |
| MemLp f 2 sourceSamplingHaar ∧ (∫ x, f x^2 ∂sourceSamplingHaar) ≤ C := by |
| filter_upwards [sourceSamplingEmpiricalDensity_uniform true (fun i => decide (i ∈ I)) |
| (fun _ => 0) (fun _ => 1) (fun _ => 0) (fun _ => 1)] with θ hθ |
| obtain ⟨C,hC⟩ := hθ |
| refine ⟨C,fun stage => ⟨_,sourceSamplingUntrimmed_second_density I stage θ,(hC stage).1,(hC stage).2⟩⟩ |
|
|
| end |
| end FiniteFactor |
|
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| end OAI |
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