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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.OrbitSampling.DiagonalIntegral
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter Finset
open scoped Classical ENNReal Topology
noncomputable section
/-- Exact whole-circle first density, including the retained cutoff. -/
theorem samplingTestedFirstDensity_whole (b : Bool) (h α s : ℝ)
(hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) (hs : s ∈ Set.Ico (0 : ℝ) 1) :
samplingTestedFirstDensity b h α (Set.Icc 0 1) (Set.Icc 0 1) s =
if b then 1-s else if α ≤ samplingDistance s then 1 else 0 := by
have he : samplingTestedFirstDensity b h α (Set.Icc 0 1) (Set.Icc 0 1) s =
∫ t, samplingWeight b h α s t ∂unitIntervalMeasure := by
apply integral_congr_ae
filter_upwards [] with t
have hs' : s ∈ Set.Icc (0 : ℝ) 1 := ⟨hs.1,hs.2.le⟩
have ht' : Int.fract (s+t) ∈ Set.Icc (0 : ℝ) 1 := ⟨Int.fract_nonneg _,(Int.fract_lt_one _).le⟩
simp only [samplingTestedFirstWeight,hs',ht',and_self,ite_true]
exact he.trans (samplingWeight_first_integral b h α s hh hh1 hα hs)
/-- The exact lost mass of one untested diagonal coordinate is 2α. -/
theorem sampling_diagonal_whole_mass (h α : ℝ)
(hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) (hα1 : α ≤ 1/2) :
(∫ s, samplingTestedFirstDensity false h α (Set.Icc 0 1) (Set.Icc 0 1) s ∂unitIntervalMeasure) = 1-2*α := by
have he : samplingTestedFirstDensity false h α (Set.Icc 0 1) (Set.Icc 0 1) =ᵐ[unitIntervalMeasure]
(Set.Icc α (1-α)).indicator (fun _ => (1 : ℝ)) := by
filter_upwards [ae_restrict_mem measurableSet_Ico] with s hs
rw [samplingTestedFirstDensity_whole false h α s hh hh1 hα hs]
have hc : α ≤ samplingDistance s ↔ s ∈ Set.Icc α (1-α) := by
simp only [samplingDistance,le_min_iff,Set.mem_Icc]
constructor <;> intro h <;> constructor <;> linarith [h.1,h.2]
simp only [Bool.false_eq_true,ite_false,hc,Set.indicator_apply]
rw [integral_congr_ae he,integral_indicator measurableSet_Icc,integral_const]
have hsub : Set.Icc α (1-α) ⊆ Set.Ico (0 : ℝ) 1 := by
intro s hs
exact ⟨hα.le.trans hs.1,hs.2.trans_lt (by linarith)⟩
simp only [measureReal_def,Measure.restrict_apply MeasurableSet.univ,Set.univ_inter,
smul_eq_mul,mul_one]
unfold unitIntervalMeasure
rw [Measure.restrict_apply measurableSet_Icc,Set.inter_eq_left.mpr hsub,Real.volume_Icc,
ENNReal.toReal_ofReal (by linarith)]
ring
/-- The total cutoff budget tends to zero despite the growing dimension. -/
theorem sourceSampling_cutoff_budget_tendsto :
Tendsto (fun stage => 2*((sourceSamplingPrefix stage).card : ℝ)*sourceSamplingCutoff stage)
atTop (𝓝 0) := by
have ht : Tendsto (fun stage => (101/50 : ℝ)*(sourceBlockSize stage : ℝ)⁻¹) atTop (𝓝 0) := by
simpa using (tendsto_inv_atTop_zero.comp
(tendsto_natCast_atTop_atTop.comp sourceBlockSize_tendsto)).const_mul (101/50 : ℝ)
apply squeeze_zero (fun stage => by have := (sourceSampling_parameters stage).2.2; positivity) _ ht
intro stage
have hm := sourceBlockSize_pos stage
calc
_ ≤ 2*((101/100 : ℝ)*sourceBlockSize stage)*sourceSamplingCutoff stage := by
exact mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_left (sourceSamplingPrefix_card_bound stage) (by norm_num))
(sourceSampling_parameters stage).2.2.le
_ = (101/50 : ℝ)*(sourceBlockSize stage : ℝ)⁻¹ := by
simp only [sourceSamplingCutoff,Real.rpow_neg hm.le,Real.rpow_two]
field_simp
ring
/-- Any subset of untested coordinates retains asymptotically all its mass.
The estimate uses its actual varying cardinality, bounded by the prefix. -/
theorem sourceSampling_untested_mass_tendsto (k : ℕ → ℕ)
(hk : ∀ stage, k stage ≤ (sourceSamplingPrefix stage).card) :
Tendsto (fun stage => (1-2*sourceSamplingCutoff stage)^(k stage)) atTop (𝓝 1) := by
have he : ∀ stage, |(1-2*sourceSamplingCutoff stage)^(k stage)-1| ≤
2*((sourceSamplingPrefix stage).card : ℝ)*sourceSamplingCutoff stage := by
intro stage
have hα := (sourceSampling_parameters stage).2.2
have hα1 := sourceSamplingCutoff_le_half stage
have hlo : 0 ≤ 1-2*sourceSamplingCutoff stage := by linarith
have hup : (1-2*sourceSamplingCutoff stage)^(k stage) ≤ 1 := pow_le_one₀ hlo (by linarith)
have hb := one_add_mul_le_pow (show (-2 : ℝ) ≤ -(2*sourceSamplingCutoff stage) by linarith) (k stage)
have hk' : (k stage : ℝ) ≤ (sourceSamplingPrefix stage).card := by exact_mod_cast hk stage
rw [abs_of_nonpos (by linarith)]
have hm := mul_le_mul_of_nonneg_right hk' (show 0 ≤ 2*sourceSamplingCutoff stage by positivity)
simp only [mul_neg,← sub_eq_add_neg] at hb
nlinarith
apply tendsto_iff_norm_sub_tendsto_zero.mpr
simp only [Real.norm_eq_abs]
exact squeeze_zero (fun stage => abs_nonneg _) he sourceSampling_cutoff_budget_tendsto
end
end FiniteFactor
end OAI
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