| import OAI.Analysis.FiniteFactor.Sampling.TestedSecondMoment |
| import OAI.Analysis.FiniteFactor.OrbitSampling.Weights |
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|
| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory Filter |
| noncomputable section |
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|
| theorem sourceSamplingWidth_le_half (s : ℕ) : sourceSamplingWidth s ≤ 1/2 := by |
| have hm : (100 : ℝ) ≤ sourceBlockSize s := by exact_mod_cast (sourceStage s).m_ge_100 |
| have hm0 : 0 ≤ (sourceBlockSize s : ℝ) := by positivity |
| have he : sourceSamplingWidth s ≤ (sourceBlockSize s : ℝ)^(-1/2 : ℝ) := |
| Real.rpow_le_rpow_of_exponent_le (by linarith) (by norm_num) |
| have hs : 2 ≤ Real.sqrt (sourceBlockSize s : ℝ) := by |
| nlinarith [Real.sq_sqrt hm0,Real.sqrt_nonneg (sourceBlockSize s : ℝ)] |
| apply he.trans |
| rw [show (-1/2 : ℝ) = -(1/2) by ring,Real.rpow_neg hm0,← Real.sqrt_eq_rpow] |
| simpa only [one_div] using (one_div_le_one_div_of_le (by norm_num : (0 : ℝ) < 2) hs) |
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|
| / |
| source's tested second-diagonal one-coordinate moment bound. -/ |
| theorem sourceSampling_tested_second_moment (s : ℕ) (E G : Set ℝ) |
| (hE : MeasurableSet E) (hG : MeasurableSet G) : |
| (∀ᵐ x ∂unitIntervalMeasure, |
| 0 ≤ samplingTestedSecondDensity (sourceSamplingWidth s) (sourceSamplingCutoff s) E G x ∧ |
| samplingTestedSecondDensity (sourceSamplingWidth s) (sourceSamplingCutoff s) E G x ≤ 2) ∧ |
| (∫ x, samplingTestedSecondDensity (sourceSamplingWidth s) (sourceSamplingCutoff s) E G x^2 |
| ∂unitIntervalMeasure) ≤ 1+4*(sourceBlockSize s : ℝ)^(-6/5 : ℝ) := by |
| have hp := sourceSampling_parameters s |
| have hhalf := sourceSamplingWidth_le_half s |
| have hden : 0 < 1-sourceSamplingWidth s := by linarith [hp.2.1] |
| have hb := samplingTestedSecondDensity_square (sourceSamplingWidth s) (sourceSamplingCutoff s) |
| E G hp.1 hp.2.1 hp.2.2 hE hG |
| have hi : 1/(1-sourceSamplingWidth s) ≤ 2 := (div_le_iff₀ hden).mpr (by linarith) |
| have herror : (sourceSamplingWidth s)^2/(1-sourceSamplingWidth s)^2 ≤ 4*(sourceSamplingWidth s)^2 := by |
| apply (div_le_iff₀ (sq_pos_of_pos hden)).mpr |
| have hsq : (1/4 : ℝ) ≤ (1-sourceSamplingWidth s)^2 := by nlinarith |
| nlinarith [sq_nonneg (sourceSamplingWidth s),mul_nonneg (sq_nonneg (sourceSamplingWidth s)) |
| (show 0 ≤ (1-sourceSamplingWidth s)^2-1/4 by linarith)] |
| have he : (sourceSamplingWidth s)^2 = (sourceBlockSize s : ℝ)^(-6/5 : ℝ) := by |
| unfold sourceSamplingWidth |
| rw [← Real.rpow_natCast,← Real.rpow_mul (sourceBlockSize_pos s).le] |
| norm_num |
| exact ⟨hb.1.mono (fun x hx => ⟨hx.1,hx.2.trans hi⟩),hb.2.trans (by rw [← he]; linarith)⟩ |
|
|
| end |
| end FiniteFactor |
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| end OAI |
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