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f1f7953 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 | import OAI.Analysis.IrrationalRotation.Sampling.CircleSamplingGraph
import OAI.Analysis.IrrationalRotation.Analysis.GraphMeasureConvergence
import OAI.Analysis.IrrationalRotation.Analysis.L2UniformAbsoluteContinuity
import OAI.Analysis.IrrationalRotation.Sampling.CirclePhaseTails
namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped Topology ENNReal
namespace Invariant233.CircleParameterData
def circleFiniteBase : FiniteMeasure Circle :=
⟨circleMeasure, by unfold circleMeasure; infer_instance⟩
instance circleFiniteBase_probability : IsProbabilityMeasure (circleFiniteBase : Measure Circle) := by
constructor
change volume (Set.univ : Set Circle) = 1
exact UnitAddCircle.measure_univ
def samplingGraphFinite (p : CircleParameterData) (θ : ℝ) (j r : ℕ)
(hm : 0 < p.m j) : FiniteMeasure (Circle × Circle) :=
⟨p.samplingGraphAverage θ j r, p.samplingGraphAverage_finite θ j r hm⟩
end Invariant233.CircleParameterData
namespace Invariant233.CircleParameterLaws
open CircleParameterData
variable {θ : ℝ} {p : CircleParameterData} (hp : CircleParameterLaws θ p)
include hp
theorem samplingAverage_error_tendsto_zero
(j r : ℕ → ℕ) (hj : ∀ n, 1 ≤ j n)
(hr : ∀ n, r n < p.q (p.blockEnd (j n - 1)))
(hjt : Tendsto j atTop atTop) (endpoint : Bool) :
Tendsto (fun n => eLpNorm (fun x => p.samplingAverage θ (j n) (r n) endpoint x - 1)
2 circleMeasure) atTop (𝓝 0) := by
have hlim := ENNReal.continuous_ofReal.continuousAt.tendsto.comp
(tendsto_one_div_atTop_nhds_zero_nat.comp hjt)
apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds
(show Tendsto (fun n => ENNReal.ofReal (1 / (j n : ℝ))) atTop (𝓝 0) from by
simpa only [Function.comp_def, ENNReal.ofReal_zero] using hlim)
(fun _ => bot_le)
(fun n => (hp.sampling_L2 (j n) (hj n) (r n) (hr n) endpoint).le)
theorem tendsto_samplingGraph_first_marginal
(j r : ℕ → ℕ) (hj : ∀ n, 1 ≤ j n)
(hr : ∀ n, r n < p.q (p.blockEnd (j n - 1)))
(hjt : Tendsto j atTop atTop) :
Tendsto (fun n => (p.samplingGraphFinite θ (j n) (r n) (hp.m_pos _ (hj n))).map Prod.fst)
atTop (𝓝 circleFiniteBase) := by
apply GraphMeasureConvergence.tendsto_of_density_L2 circleFiniteBase _
(fun n => p.samplingAverage θ (j n) (r n) false)
· intro n
exact p.memLp_samplingAverage θ (j n) (r n) false (hp.m_pos _ (hj n)) 2
· intro n
exact Eventually.of_forall fun x =>
(p.samplingAverage_bounds θ (j n) (r n) false (hp.m_pos _ (hj n)) x).1
· intro n
exact p.samplingGraphAverage_first_marginal θ (j n) (r n) (hp.m_pos _ (hj n))
· exact hp.samplingAverage_error_tendsto_zero j r hj hr hjt false
theorem tendsto_samplingGraph_second_marginal
(j r : ℕ → ℕ) (hj : ∀ n, 1 ≤ j n)
(hr : ∀ n, r n < p.q (p.blockEnd (j n - 1)))
(hjt : Tendsto j atTop atTop) :
Tendsto (fun n => (p.samplingGraphFinite θ (j n) (r n) (hp.m_pos _ (hj n))).map Prod.snd)
atTop (𝓝 circleFiniteBase) := by
apply GraphMeasureConvergence.tendsto_of_density_L2 circleFiniteBase _
(fun n => p.samplingAverage θ (j n) (r n) true)
· intro n
exact p.memLp_samplingAverage θ (j n) (r n) true (hp.m_pos _ (hj n)) 2
· intro n
exact Eventually.of_forall fun x =>
(p.samplingAverage_bounds θ (j n) (r n) true (hp.m_pos _ (hj n)) x).1
· intro n
exact p.samplingGraphAverage_second_marginal θ (j n) (r n) (hp.m_pos _ (hj n))
· exact hp.samplingAverage_error_tendsto_zero j r hj hr hjt true
theorem samplingGraph_marginals_uniformlyAbsolutelyContinuous
(j r : ℕ → ℕ) (hj : ∀ n, 1 ≤ j n)
(hr : ∀ n, r n < p.q (p.blockEnd (j n - 1))) :
UniformlyAbsolutelyContinuous circleMeasure
(fun n => (p.samplingGraphAverage θ (j n) (r n)).map Prod.fst) ∧
UniformlyAbsolutelyContinuous circleMeasure
(fun n => (p.samplingGraphAverage θ (j n) (r n)).map Prod.snd) := by
let : IsFiniteMeasure circleMeasure := by unfold circleMeasure; infer_instance
have hden (endpoint : Bool) : UniformlyAbsolutelyContinuous circleMeasure
(fun n => circleMeasure.withDensity
(fun x => ENNReal.ofReal (p.samplingAverage θ (j n) (r n) endpoint x))) := by
apply EndpointLoss.uniformlyAbsolutelyContinuous_withDensity circleMeasure
(fun n => p.samplingAverage θ (j n) (r n) endpoint)
(fun n => p.memLp_samplingAverage θ (j n) (r n) endpoint (hp.m_pos _ (hj n)) 2)
(fun n => Eventually.of_forall fun x =>
(p.samplingAverage_bounds θ (j n) (r n) endpoint (hp.m_pos _ (hj n)) x).1) 2
intro n
rw [EndpointLoss.sqrt_integral_sq_eq_toReal_eLpNorm
(p.memLp_samplingAverage θ (j n) (r n) endpoint (hp.m_pos _ (hj n)) 2)]
have h := ENNReal.toReal_mono (by norm_num : (2 : ENNReal) ≠ ∞)
(hp.samplingAverage_eLpNorm_le_two (j n) (hj n) (r n) (hr n) endpoint)
simpa only [ENNReal.toReal_ofNat] using h
constructor
· convert hden false using 1
funext n
exact p.samplingGraphAverage_first_marginal θ (j n) (r n) (hp.m_pos _ (hj n))
· convert hden true using 1
funext n
exact p.samplingGraphAverage_second_marginal θ (j n) (r n) (hp.m_pos _ (hj n))
theorem ell_nonneg_and_le (j : ℕ) : 0 ≤ p.ell j ∧
p.ell j ≤ 1 / ((j + 1 : ℕ) : ℝ)^2 := by
have hq : (1 : ℝ) ≤ p.q (p.blockEnd (j - 1)) := by
exact_mod_cast hp.q_pos (p.blockEnd (j - 1))
have ha : 0 ≤ (p.blockEnd (j - 1) : ℝ) := Nat.cast_nonneg _
have hj : (0 : ℝ) < ((j + 1 : ℕ) : ℝ)^2 := by positivity
have hprod : 1 ≤ (1 + (p.blockEnd (j - 1) : ℝ)) *
(p.q (p.blockEnd (j - 1)) : ℝ) :=
one_le_mul_of_one_le_of_one_le (by linarith) hq
unfold ell
constructor
· positivity
· apply one_div_le_one_div_of_le hj
nlinarith
theorem tendsto_ell_zero : Tendsto p.ell atTop (𝓝 0) := by
have h : Tendsto (fun n : ℕ => 1 / ((n + 1 : ℕ) : ℝ)) atTop (𝓝 (0 : ℝ)) :=
(tendsto_add_atTop_iff_nat 1).mpr tendsto_one_div_atTop_nhds_zero_nat
have hsq : Tendsto (fun n : ℕ => 1 / ((n + 1 : ℕ) : ℝ)^2) atTop (𝓝 (0 : ℝ)) := by
simpa only [one_div_pow, zero_pow (by decide : 2 ≠ 0)] using h.pow 2
exact squeeze_zero (fun n => (hp.ell_nonneg_and_le n).1)
(fun n => (hp.ell_nonneg_and_le n).2) hsq
theorem samplingGraphAverage_ae_dist_lt
(j : ℕ) (hj : 1 ≤ j) (r : ℕ)
(hr : r < p.q (p.blockEnd (j - 1))) (t : Circle)
(hshift : (p.shift j r : Circle) = t) :
∀ᵐ z ∂p.samplingGraphAverage θ j r, dist z.2 (z.1 + t) < p.ell j := by
filter_upwards [p.samplingGraphAverage_ae_graph θ j r] with z hz
obtain ⟨k, hk, heq⟩ := hz
have hphase := (hp.samples_phase j hj r hr k hk).2
rw [heq]
change dist (z.1 + (((k : ℝ) * θ : ℝ) : Circle)) (z.1 + t) < p.ell j
rw [dist_add_left, ← hshift, dist_eq_norm, ← AddCircle.coe_sub]
calc
‖(((k : ℝ) * θ - p.shift j r : ℝ) : Circle)‖ =
‖((Int.fract ((k : ℝ) * θ - p.shift j r) : ℝ) : Circle)‖ := by
rw [AddCircle.coe_fract]
_ ≤ ‖Int.fract ((k : ℝ) * θ - p.shift j r)‖ := QuotientAddGroup.norm_mk_le_norm
_ = Int.fract ((k : ℝ) * θ - p.shift j r) :=
Real.norm_of_nonneg (Int.fract_nonneg _)
_ < p.ell j := hphase
theorem tendsto_samplingGraph_translation
(j r : ℕ → ℕ) (hj : ∀ n, 1 ≤ j n)
(hr : ∀ n, r n < p.q (p.blockEnd (j n - 1)))
(hjt : Tendsto j atTop atTop) (t : Circle)
(hshift : ∀ n, (p.shift (j n) (r n) : Circle) = t) :
Tendsto (fun n => p.samplingGraphFinite θ (j n) (r n) (hp.m_pos _ (hj n)))
atTop (𝓝 (circleFiniteBase.map (fun x => (x, x + t)))) := by
apply GraphMeasureConvergence.tendsto_graph_of_first_marginal _ circleFiniteBase
(fun x => x + t) (continuous_id.add continuous_const)
(hp.tendsto_samplingGraph_first_marginal j r hj hr hjt)
intro δ hδ
filter_upwards [(hp.tendsto_ell_zero.comp hjt).eventually_lt_const hδ] with n hn
exact (hp.samplingGraphAverage_ae_dist_lt (j n) (hj n) (r n) (hr n) t (hshift n)).mono
(fun _ hz => hz.trans hn)
theorem tendsto_samplingGraph_translation_trimmed
(j r : ℕ → ℕ) (hj : ∀ n, 1 ≤ j n)
(hr : ∀ n, r n < p.q (p.blockEnd (j n - 1)))
(hjt : Tendsto j atTop atTop) (t : Circle)
(hshift : ∀ n, (p.shift (j n) (r n) : Circle) = t)
(s : ℕ → Set Circle) (hs : ∀ n, MeasurableSet (s n))
(hbase : Tendsto (fun n => circleMeasure.real (s n)ᶜ) atTop (𝓝 0)) :
Tendsto (fun n => (p.samplingGraphFinite θ (j n) (r n) (hp.m_pos _ (hj n))).restrict
(Prod.fst ⁻¹' s n)) atTop (𝓝 (circleFiniteBase.map (fun x => (x, x + t)))) := by
apply GraphMeasureConvergence.tendsto_trimmed_graph_of_uniform_first_marginal
_ circleFiniteBase (fun x => x + t) (continuous_id.add continuous_const)
(hp.tendsto_samplingGraph_first_marginal j r hj hr hjt) _
(hp.samplingGraph_marginals_uniformlyAbsolutelyContinuous j r hj hr).1 s hs hbase
intro δ hδ
filter_upwards [(hp.tendsto_ell_zero.comp hjt).eventually_lt_const hδ] with n hn
exact (hp.samplingGraphAverage_ae_dist_lt (j n) (hj n) (r n) (hr n) t (hshift n)).mono
(fun _ hz => hz.trans hn)
end Invariant233.CircleParameterLaws
end
end OAI
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