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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 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 | 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 constructed power is long enough for density at most two on
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
/-- Rectangle discrepancy for the actual pair law of the deterministic
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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