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67aee1d | 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 | import OAI.Analysis.FiniteFactor.OrbitSampling.SelfMoment
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter
open scoped ENNReal
noncomputable section
/-- Marginal phase domination controls the joint self-pair integral.
This is the source's one-phase step before taking coordinate products. -/
theorem sampling_phase_square_bound {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → ℝ) (hX : Measurable X)
(hdom : μ.map X ≤ (2 : ℝ≥0∞) • unitIntervalMeasure)
(u : ℝ → ℝ → ℝ) (C : ℝ) (hu : Measurable (fun p : ℝ × ℝ => u p.1 p.2))
(hb : ∀ s t, 0 ≤ u s t ∧ u s t ≤ C) :
(∫ s, ∫ ω, u s (X ω)^2 ∂μ ∂unitIntervalMeasure) ≤
2*(∫ s, ∫ t, u s t^2 ∂unitIntervalMeasure ∂unitIntervalMeasure) := by
have hΩ : Measurable (fun p : ℝ × Ω => u p.1 (X p.2)) :=
hu.comp (measurable_fst.prodMk (hX.comp measurable_snd))
have hiΩ := sampling_square_integrable (unitIntervalMeasure.prod μ) _ C hΩ
(fun p => hb p.1 (X p.2))
have hiU := sampling_square_integrable (unitIntervalMeasure.prod unitIntervalMeasure) _ C hu
(fun p => hb p.1 p.2)
have hpoint (s : ℝ) : (∫ ω, u s (X ω)^2 ∂μ) ≤ 2*(∫ t, u s t^2 ∂unitIntervalMeasure) := by
have hm : Measurable (fun t => u s t) := hu.comp (measurable_const.prodMk measurable_id)
have hi := sampling_square_integrable unitIntervalMeasure _ C hm (hb s)
have hbig := hi.smul_measure (c := (2 : ℝ≥0∞)) (by norm_num)
have hle := integral_mono_measure hdom (Eventually.of_forall (fun t => sq_nonneg (u s t))) hbig
rw [integral_map hX.aemeasurable (hm.pow_const 2).aestronglyMeasurable,
integral_smul_measure] at hle
simpa using hle
calc
_ ≤ ∫ s, 2*(∫ t, u s t^2 ∂unitIntervalMeasure) ∂unitIntervalMeasure :=
integral_mono_ae hiΩ.integral_prod_left (hiU.integral_prod_left.const_mul 2)
(Eventually.of_forall hpoint)
_ = _ := integral_const_mul _ _
/-- Actual angle expectation of the tested self-pair, at every admissible
selected power, for all source orientations and coordinate types. -/
theorem sourceSampling_phase_self_moment (stage : ℕ) (hs : 200000 ≤ sourceBlockSize stage)
(i : SourceCoordinate) (hi : i.1 ≤ stage) (n : ℤ) (hn : n ∈ sourceSamplingPowers stage)
(r b : Bool) (E G : Set ℝ) (hE : MeasurableSet E) (hG : MeasurableSet G) :
(∫ s, ∫ θ, samplingOrientedWeight r b (sourceSamplingWidth stage) (sourceSamplingCutoff stage)
E G s (Int.fract ((n : ℝ)*θ i)) ^ 2 ∂sourceAngleLaw ∂unitIntervalMeasure) ≤
2+8*(sourceBlockSize stage : ℝ)^(3/5 : ℝ)*Real.log (sourceBlockSize stage : ℝ) := by
have hp := sourceSampling_parameters stage
have hbound := sampling_phase_square_bound sourceAngleLaw (fun θ => Int.fract ((n : ℝ)*θ i))
(sourceFract_measurable.comp (measurable_const.mul (measurable_pi_apply i)))
(sourceSampling_phase_domination stage hs i hi n hn)
(samplingOrientedWeight r b (sourceSamplingWidth stage) (sourceSamplingCutoff stage) E G)
(max 1 (1/(sourceSamplingWidth stage*sourceSamplingCutoff stage)))
(samplingOrientedWeight_measurable r b _ _ E G hE hG)
(samplingOrientedWeight_bounds r b _ _ E G hp.1 hp.2.2)
have hself := sourceSampling_oriented_self_moment stage r b E G hE hG
exact hbound.trans (by nlinarith)
end
end FiniteFactor
end OAI
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