| import OAI.Analysis.FiniteFactor.Sampling.Rectangle |
| import OAI.Analysis.FiniteFactor.OrbitSampling.Fourier |
| import OAI.Analysis.FiniteFactor.Sampling.Periodization |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Finset Filter |
| open scoped ENNReal Topology |
| noncomputable section |
|
|
| / |
| every coordinate in the sampling prefix. -/ |
| theorem sourceSampling_phase_length (s : ℕ) (hs : 200000 ≤ sourceBlockSize s) |
| (i : SourceCoordinate) (hi : i.1 ≤ s) (n : ℤ) (hn : n ∈ sourceSamplingPowers s) : |
| 1 ≤ (n : ℝ)*sourceAngleUnit i := by |
| have hm : (200000 : ℝ) ≤ sourceBlockSize s := by exact_mod_cast hs |
| have hm0 := sourceBlockSize_pos s |
| have hspec := (sourceSamplingPowers_spec s hs).1 n hn |
| have hQ : (sourceBlockSize s : ℝ)^3 ≤ samplingQ (sourceBlockSize s) := by |
| unfold samplingQ |
| rw [← Real.rpow_natCast] |
| exact Real.rpow_le_rpow_of_exponent_le (by linarith) (by norm_num; linarith) |
| have hN : (sourceBlockSize i.1 : ℝ)*(sourceStageState i.1).K ≤ (sourceBlockSize s : ℝ)^2 := by |
| have hh : (sourceBlockSize i.1 : ℝ)*(sourceStageState i.1).K ≤ |
| (sourceBlockSize s : ℝ)*(sourceStageState s).K := by exact_mod_cast sourceFrequencyScale_monotone hi |
| have hK : ((sourceStageState s).K : ℝ) ≤ sourceBlockSize s := by exact_mod_cast (sourceStage s).m_ge_K |
| nlinarith |
| have hnp : (sourceBlockSize i.1 : ℝ)*(sourceStageState i.1).K ≤ (n : ℝ) := by |
| nlinarith [hspec.1,sq_nonneg (sourceBlockSize s : ℝ)] |
| unfold sourceAngleUnit |
| rw [mul_one_div,le_div_iff₀ (mul_pos (sourceBlockSize_pos i.1) (sourceCutoff_pos i.1)),one_mul] |
| exact hnp |
|
|
| theorem sourceSampling_phase_domination (s : ℕ) (hs : 200000 ≤ sourceBlockSize s) |
| (i : SourceCoordinate) (hi : i.1 ≤ s) (n : ℤ) (hn : n ∈ sourceSamplingPowers s) : |
| sourceAngleLaw.map (fun θ => Int.fract ((n : ℝ)*θ i)) ≤ |
| (2 : ℝ≥0∞) • unitIntervalMeasure := by |
| have hp : Measurable (fun t : ℝ => Int.fract ((n : ℝ)*t)) := |
| sourceFract_measurable.comp (measurable_const.mul measurable_id) |
| have he : sourceAngleLaw.map (fun θ => Int.fract ((n : ℝ)*θ i)) = |
| (sourceAngleMarginal i).map (fun t => Int.fract ((n : ℝ)*t)) := by |
| rw [← sourceAngleLaw_coordinate i,Measure.map_map hp (measurable_pi_apply i)] |
| rfl |
| rw [he] |
| have hnp : (0 : ℝ) < n := by exact_mod_cast ((sourceSamplingPowers_spec s hs).1 n hn).2.2 |
| have hh := (sourceAngle_phase_sandwich i (n : ℝ) hnp).2 |
| apply hh.trans |
| have hlen := sourceSampling_phase_length s hs i hi n hn |
| have hiR : 1/(n*sourceAngleUnit i) ≤ (1 : ℝ) := by |
| simpa using one_div_le_one_div_of_le (by norm_num : (0 : ℝ) < 1) hlen |
| have heps : ENNReal.ofReal (1+1/(n*sourceAngleUnit i)) ≤ (2 : ℝ≥0∞) := by |
| exact_mod_cast ENNReal.ofReal_le_ofReal (show 1+1/(n*sourceAngleUnit i) ≤ (2 : ℝ) by linarith) |
| gcongr |
|
|
| theorem samplingCharacter_pair_expression (v w : ℤ) (a b : ℝ) : |
| samplingCharacter v a*samplingCharacter w b = |
| Complex.exp (2*Real.pi*Complex.I*((v : ℂ)*(a : ℂ)+(w : ℂ)*(b : ℂ))) := by |
| unfold samplingCharacter |
| rw [← Complex.exp_add] |
| congr 1 |
| ring |
|
|
| / |
| selected powers, with every probabilistic hypothesis discharged. -/ |
| theorem sourceSampling_rectangle_bound (s : ℕ) (hs : 200000 ≤ sourceBlockSize s) |
| (i : SourceCoordinate) (hi : i.1 ≤ s) (n n' : ℤ) |
| (hn : n ∈ sourceSamplingPowers s) (hn' : n' ∈ sourceSamplingPowers s) (hne : n ≠ n') |
| (a b a' b' : ℝ) : |
| |(sourceAngleLaw {θ | Int.fract ((n : ℝ)*θ i) ∈ Set.Icc a b ∧ |
| Int.fract ((n' : ℝ)*θ i) ∈ Set.Icc a' b'}).toReal - |
| (unitIntervalMeasure (Set.Icc a b)).toReal*(unitIntervalMeasure (Set.Icc a' b')).toReal| ≤ |
| 26*(sourceBlockSize s : ℝ)^(-80 : ℝ) := by |
| let m := sourceBlockSize s |
| have hm0 : (0 : ℝ) < m := sourceBlockSize_pos s |
| have hm1 : (1 : ℝ) ≤ m := by |
| have hm : (200000 : ℝ) ≤ m := by exact_mod_cast hs |
| linarith |
| let δ : ℝ := (m : ℝ)^(-80 : ℝ) |
| let ε : ℝ := Real.pi⁻¹*(m : ℝ)^(-1000 : ℝ) |
| have hH : 0 < samplingH m := pow_pos (by exact_mod_cast hm0) 300 |
| have hphase (n : ℤ) : Measurable (fun θ : SourceCoordinate → ℝ => Int.fract ((n : ℝ)*θ i)) := |
| sourceFract_measurable.comp (measurable_const.mul (measurable_pi_apply i)) |
| have hh := sampling_rectangle_discrepancy sourceAngleLaw _ _ (hphase n) (hphase n') |
| (sourceSampling_phase_domination s hs i hi n hn) (sourceSampling_phase_domination s hs i hi n' hn') |
| (samplingH m) hH a b a' b' δ ε (by dsimp [δ]; positivity) (by dsimp [ε]; positivity) |
| (fun v w hv hw hne' => by |
| simp_rw [samplingCharacter_pair_expression] |
| exact sourceSampling_pair_fourier_bound s hs i hi n n' hn hn' hne v w hv.le hw.le hne') |
| apply hh.trans |
| have htail : 1/(2*(samplingH m : ℝ)*δ^2) = (1/2 : ℝ)*(m : ℝ)^(-140 : ℝ) := by |
| dsimp [samplingH,δ] |
| norm_num [Real.rpow_neg,Real.rpow_ofNat,hm0.le] |
| field_simp |
| have hfour : (samplingH m : ℝ)^2*ε = Real.pi⁻¹*(m : ℝ)^(-400 : ℝ) := by |
| dsimp [samplingH,ε] |
| norm_num [Real.rpow_neg,Real.rpow_ofNat,hm0.le] |
| field_simp |
| rw [htail,hfour] |
| have h140 : (m : ℝ)^(-140 : ℝ) ≤ δ := Real.rpow_le_rpow_of_exponent_le hm1 (by norm_num) |
| have h400 : (m : ℝ)^(-400 : ℝ) ≤ δ := Real.rpow_le_rpow_of_exponent_le hm1 (by norm_num) |
| have hpi : Real.pi⁻¹ ≤ 1 := by simpa using one_div_le_one_div_of_le (by norm_num : (0 : ℝ) < 1) (show 1 ≤ Real.pi by linarith [Real.two_le_pi]) |
| have hδ : 0 ≤ δ := by dsimp [δ]; positivity |
| have hp0 : 0 ≤ (m : ℝ)^(-400 : ℝ) := by positivity |
| change 24*δ+1/2*(m : ℝ)^(-140 : ℝ)+Real.pi⁻¹*(m : ℝ)^(-400 : ℝ) ≤ 26*δ |
| nlinarith |
|
|
| end |
| end FiniteFactor |
|
|
| end OAI |
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