| import OAI.Analysis.FiniteFactor.OrbitSampling.SelfMoment |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory Filter |
| open scoped ENNReal |
| noncomputable section |
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| / |
| 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 _ _ |
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| / |
| 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) |
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| end |
| end FiniteFactor |
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| end OAI |
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