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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 | import OAI.Analysis.FiniteFactor.Sampling.Fourier
import OAI.Analysis.FiniteFactor.OrbitSampling.Powers
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter Finset
noncomputable section
theorem sourceBlockSize_monotone : Monotone sourceBlockSize := by
apply monotone_nat_of_le_succ
intro s
have hp := (sourceStage (s+1)).previous_size
have hd : (sourceBlockSize s : ℝ) ≤ (sourceStageState (s+1)).d := by
change (sourceBlockSize s : ℝ) ≤ ((sourceStageState s).d+(sourceStage s).m : ℕ)
change ((sourceStage s).m : ℝ) ≤ ((sourceStageState s).d+(sourceStage s).m : ℕ)
push_cast
exact le_add_of_nonneg_left (Nat.cast_nonneg _)
have hm : (sourceBlockSize s : ℝ) ≤ sourceBlockSize (s+1) := by
change ((sourceStageState (s+1)).d : ℝ) ≤ (sourceBlockSize (s+1) : ℝ)/100 at hp
have := Nat.cast_nonneg (α := ℝ) (sourceBlockSize (s+1))
linarith
exact_mod_cast hm
theorem sourceFrequencyScale_monotone :
Monotone (fun s => sourceBlockSize s * (sourceStageState s).K) := by
have hK : Monotone (fun s => (sourceStageState s).K) := by
apply monotone_nat_of_le_succ
intro s
have := sourceStage_cutoff_growth s
omega
intro s t hst
exact Nat.mul_le_mul (sourceBlockSize_monotone hst) (hK hst)
/-- The full infinite-product angle expectation has the same coordinate
Fourier estimate; no independence between two phases is asserted. -/
theorem sourceAngleLaw_pair_fourier_le (i : SourceCoordinate) (v v' : ℤ) (n n' : ℝ)
(hw : (v : ℝ)*n+(v' : ℝ)*n' ≠ 0) :
‖∫ θ, Complex.exp (2 * Real.pi * Complex.I *
((v : ℂ) * (Int.fract (n*θ i) : ℝ) + (v' : ℂ) * (Int.fract (n'*θ i) : ℝ)))
∂sourceAngleLaw‖ ≤
1/(Real.pi*sourceAngleUnit i*|(v : ℝ)*n+(v' : ℝ)*n'|) := by
have hh := sourceAngle_pair_fourier_le i v v' n n' hw
simp_rw [pair_fourier_fract] at hh ⊢
rw [← sourceAngleLaw_coordinate i,
integral_map (measurable_pi_apply i).aemeasurable (by fun_prop)] at hh
exact hh
theorem sourceSampling_pair_fourier_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')
(v v' : ℤ) (hv : |v| ≤ (samplingH (sourceBlockSize s) : ℤ))
(hv' : |v'| ≤ (samplingH (sourceBlockSize s) : ℤ)) (hvv' : v ≠ 0 ∨ v' ≠ 0) :
‖∫ θ, Complex.exp (2 * Real.pi * Complex.I *
((v : ℂ) * (Int.fract ((n : ℝ)*θ i) : ℝ) +
(v' : ℂ) * (Int.fract ((n' : ℝ)*θ i) : ℝ))) ∂sourceAngleLaw‖ ≤
Real.pi⁻¹*(sourceBlockSize s : ℝ)^(-1000 : ℝ) := by
have hsep := (sourceSamplingPowers_spec s hs).2.2 hn hn' hne
have hfreq : (samplingR (sourceBlockSize s) (sourceStageState s).K : ℤ) < |v*n+v'*n'| := by
by_contra hh
exact hsep ⟨v,mem_Icc.mpr (abs_le.mp hv),v',mem_Icc.mpr (abs_le.mp hv'),hvv',le_of_not_gt hh⟩
let w : ℝ := (v : ℝ)*(n : ℝ)+(v' : ℝ)*(n' : ℝ)
have hwR : (samplingR (sourceBlockSize s) (sourceStageState s).K : ℝ) < |w| := by
dsimp [w]
exact_mod_cast hfreq
have hwpos : 0 < |w| := lt_of_le_of_lt (Nat.cast_nonneg _) hwR
have hw : w ≠ 0 := abs_pos.mp hwpos
apply (sourceAngleLaw_pair_fourier_le i v v' (n : ℝ) (n' : ℝ) hw).trans
have hN : (sourceBlockSize i.1 : ℝ)*(sourceStageState i.1).K ≤
(sourceBlockSize s : ℝ)*(sourceStageState s).K := by
exact_mod_cast sourceFrequencyScale_monotone hi
have hmpos := sourceBlockSize_pos s
have hNpos := mul_pos (sourceBlockSize_pos i.1) (sourceCutoff_pos i.1)
have hscale : (sourceBlockSize s : ℝ)^1000 ≤ sourceAngleUnit i * |w| := by
unfold samplingR at hwR
push_cast at hwR
unfold sourceAngleUnit
rw [one_div, inv_mul_eq_div, le_div_iff₀ hNpos]
have hh := mul_le_mul_of_nonneg_right hN (pow_nonneg hmpos.le 1000)
nlinarith
change 1/(Real.pi*sourceAngleUnit i*|w|) ≤ _
calc
_ = Real.pi⁻¹*(1/(sourceAngleUnit i*|w|)) := by ring
_ ≤ Real.pi⁻¹*(1/(sourceBlockSize s : ℝ)^1000) := by
exact mul_le_mul_of_nonneg_left
(one_div_le_one_div_of_le (pow_pos hmpos 1000) hscale) (inv_nonneg.mpr Real.pi_pos.le)
_ = _ := by rw [Real.rpow_neg, Real.rpow_ofNat, one_div]; positivity
end
end FiniteFactor
end OAI
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