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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 | import OAI.Analysis.IrrationalRotation.Model.CharacterSampling
import OAI.Analysis.IrrationalRotation.Sampling.CharacterSamplingLimit
import OAI.Analysis.IrrationalRotation.Sampling.ScalarSamplingContinuity
import OAI.Analysis.IrrationalRotation.Rotation.CircleRepresentative
namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped BigOperators Topology
namespace Invariant233.CharacterSampling
lemma fract_sub_fract (z v : ℝ) : Int.fract (z - Int.fract v) = Int.fract (z - v) := by
have heq : z - Int.fract v = z - v + (⌊v⌋ : ℤ) := by
rw [Int.fract]
ring
rw [heq, Int.fract_add_intCast]
theorem kernel_fract_right (endpoint : Bool) (m : ℕ) (z v : ℝ) :
kernel endpoint m z (Int.fract v) = kernel endpoint m z v := by
cases endpoint <;>
simp only [kernel, Bool.false_eq_true, ↓reduceIte, ScalarSampling.manuscriptEndpointKernel,
ScalarSampling.manuscriptKernel, ScalarSampling.circleKernel, Int.fract_fract,
fract_sub_fract]
local instance : IsProbabilityMeasure (volume : Measure UnitAddCircle) :=
⟨by simp [AddCircle.measure_univ]⟩
lemma measurable_representative : Measurable representative :=
measurable_subtype_coe.comp (AddCircle.measurableEquivIco (1 : ℝ) 0).measurable
lemma continuousAt_representative {x : UnitAddCircle} (hx : x ≠ 0) :
ContinuousAt representative x :=
continuousAt_subtype_val.comp (AddCircle.continuousAt_equivIco (1 : ℝ) 0 hx)
lemma measurable_kernel (endpoint : Bool) (m : ℕ) :
Measurable (fun p : ℝ × ℝ => kernel endpoint m p.1 p.2) := by
cases endpoint
· exact ScalarSampling.measurable_circleKernel _ _
· exact (ScalarSampling.measurable_circleKernel _ _).comp
((measurable_fst.sub measurable_snd).prodMk measurable_snd)
lemma kernel_nonneg (endpoint : Bool) {m : ℕ} (hm : 200 ≤ m) (z v : ℝ) :
0 ≤ kernel endpoint m z v := by
have hh := (ScalarSampling.manuscript_parameters hm).1.le
cases endpoint <;> exact ScalarSampling.rawKernel_nonneg hh
⟨Int.fract_nonneg _, (Int.fract_lt_one _).le⟩
lemma kernel_le_height (endpoint : Bool) {m : ℕ} (hm : 200 ≤ m) (z v : ℝ) :
kernel endpoint m z v ≤ (m : ℝ) ^ (13 / 5 : ℝ) := by
cases endpoint <;> exact ScalarSampling.manuscriptKernel_le_height hm _ _
lemma kernel_abs_le_height (endpoint : Bool) {m : ℕ} (hm : 200 ≤ m) (z v : ℝ) :
|kernel endpoint m z v| ≤ (m : ℝ) ^ (13 / 5 : ℝ) := by
rw [abs_of_nonneg (kernel_nonneg endpoint hm z v)]
exact kernel_le_height endpoint hm z v
theorem ae_joint_continuousAt_kernel {m : ℕ} (hm : 200 ≤ m)
(endpoint : Bool) (n τ : ℝ) (hphase : Int.fract (n * τ) ≠ 0) :
∀ᵐ x : UnitAddCircle,
ContinuousAt (fun p : ℝ × UnitAddCircle =>
kernel endpoint m (representative p.2) (n * p.1)) (τ, x) := by
have hreal : ∀ᵐ z : ℝ ∂volume,
ContinuousAt (fun p : ℝ × ℝ => kernel endpoint m p.1 (n * p.2)) (z, τ) := by
cases endpoint
· exact ScalarSampling.ae_joint_continuousAt_manuscriptPower hm n τ hphase
· exact ScalarSampling.ae_joint_continuousAt_manuscriptEndpointPower hm n τ hphase
have hpull := CocycleSupport.measurePreserving_circleRepresentative.quasiMeasurePreserving.ae
(ae_restrict_of_ae (s := Ico (0 : ℝ) 1) hreal)
have hnotzero := CocycleSupport.measurePreserving_circleRepresentative.quasiMeasurePreserving.ae
(ae_restrict_of_ae (s := Ico (0 : ℝ) 1) ((volume : Measure ℝ).ae_ne 0))
filter_upwards [hpull, hnotzero] with x hx hrepzero
have hzero : x ≠ 0 := by
intro heq
subst x
apply hrepzero
exact AddCircle.equivIco_coe_of_mem (p := (1 : ℝ)) (a := 0) (by norm_num)
have hc : ContinuousAt
(fun p : ℝ × UnitAddCircle => (representative p.2, p.1)) (τ, x) :=
((continuousAt_representative hzero).comp continuous_snd.continuousAt).prodMk
continuous_fst.continuousAt
exact hx.comp_of_eq hc rfl
lemma measurable_squaredError (m : ℕ) (powers : Finset ℕ) (endpoint : Bool) :
Measurable (squaredError m powers endpoint) := by
apply Measurable.pow_const
apply Measurable.sub _ measurable_const
apply measurable_const.mul
apply Finset.measurable_sum
intro n hn
apply Finset.measurable_prod
intro i hi
exact (measurable_kernel endpoint m).comp
((measurable_representative.comp ((measurable_pi_apply i).comp measurable_snd)).prodMk
(measurable_const.mul ((measurable_pi_apply i).comp measurable_fst)))
theorem ae_continuousAt_coordinateFactor {m : ℕ} (hm : 200 ≤ m)
(endpoint : Bool) (n : ℕ) (phase : Fin m → ℝ) (i : Fin m)
(hphase : Int.fract ((n : ℝ) * phase i) ≠ 0) :
∀ᵐ y : UnitAddTorus (Fin m), ContinuousAt
(fun p : (Fin m → ℝ) × UnitAddTorus (Fin m) =>
kernel endpoint m (representative (p.2 i)) ((n : ℝ) * p.1 i))
(phase, y) := by
have heval : ∀ᵐ y : UnitAddTorus (Fin m), ContinuousAt
(fun p : ℝ × UnitAddCircle => kernel endpoint m (representative p.2) ((n : ℝ) * p.1))
(phase i, y i) :=
(measurePreserving_eval (fun _ : Fin m => (volume : Measure UnitAddCircle)) i).quasiMeasurePreserving.ae
(ae_joint_continuousAt_kernel hm endpoint (n : ℝ) (phase i) hphase)
filter_upwards [heval] with y hy
have hc : ContinuousAt
(fun p : (Fin m → ℝ) × UnitAddTorus (Fin m) => (p.1 i, p.2 i)) (phase,y) :=
((continuous_apply i).continuousAt.comp continuous_fst.continuousAt).prodMk
((continuous_apply i).continuousAt.comp continuous_snd.continuousAt)
exact ContinuousAt.comp (x := (phase,y)) hy hc
theorem ae_continuousAt_squaredError {m : ℕ} (hm : 200 ≤ m)
(powers : Finset ℕ) (endpoint : Bool) (phase : Fin m → ℝ)
(hphase : ∀ n ∈ powers, ∀ i, Int.fract ((n : ℝ) * phase i) ≠ 0) :
∀ᵐ y : UnitAddTorus (Fin m),
ContinuousAt (squaredError m powers endpoint) (phase, y) := by
have hall : ∀ᵐ y : UnitAddTorus (Fin m), ∀ n ∈ powers, ∀ i : Fin m,
ContinuousAt
(fun p : (Fin m → ℝ) × UnitAddTorus (Fin m) =>
kernel endpoint m (representative (p.2 i)) ((n : ℝ) * p.1 i))
(phase, y) := by
rw [Finset.eventually_all]
intro n hn
exact Filter.eventually_all.mpr (fun i => ae_continuousAt_coordinateFactor hm endpoint n phase i (hphase n hn i))
filter_upwards [hall] with y hy
apply ContinuousAt.pow
apply ContinuousAt.sub _ continuousAt_const
apply ContinuousAt.const_mul
exact tendsto_finsetSum powers (fun n hn =>
tendsto_finsetProd Finset.univ (fun i _ => hy n hn i))
lemma average_nonneg {m : ℕ} (hm : 200 ≤ m) (powers : Finset ℕ)
(endpoint : Bool) (phase : Fin m → ℝ) (spatial : UnitAddTorus (Fin m)) :
0 ≤ average m powers endpoint phase spatial := by
apply mul_nonneg (inv_nonneg.mpr (Nat.cast_nonneg _))
apply Finset.sum_nonneg
intro n hn
exact Finset.prod_nonneg (fun i _ => kernel_nonneg endpoint hm _ _)
lemma average_le_height {m : ℕ} (hm : 200 ≤ m) (powers : Finset ℕ)
(endpoint : Bool) (phase : Fin m → ℝ) (spatial : UnitAddTorus (Fin m)) :
average m powers endpoint phase spatial ≤ ((m : ℝ) ^ (13 / 5 : ℝ)) ^ m := by
by_cases hp : powers.Nonempty
· have hcard : (0 : ℝ) < powers.card := by exact_mod_cast hp.card_pos
have hs : (∑ n ∈ powers, ∏ i : Fin m,
kernel endpoint m (representative (spatial i)) ((n : ℝ) * phase i)) ≤
(powers.card : ℝ) * ((m : ℝ) ^ (13 / 5 : ℝ)) ^ m := by
calc
_ ≤ ∑ _n ∈ powers, ((m : ℝ) ^ (13 / 5 : ℝ)) ^ m := by
apply Finset.sum_le_sum
intro n hn
calc
_ ≤ ∏ _i : Fin m, (m : ℝ) ^ (13 / 5 : ℝ) :=
Finset.prod_le_prod₀ (fun i _ => kernel_nonneg endpoint hm _ _)
(fun i _ => kernel_le_height endpoint hm _ _)
_ = _ := by simp
_ = _ := by simp
calc
average m powers endpoint phase spatial ≤
(powers.card : ℝ)⁻¹ * ((powers.card : ℝ) * ((m : ℝ) ^ (13 / 5 : ℝ)) ^ m) :=
mul_le_mul_of_nonneg_left hs (inv_nonneg.mpr hcard.le)
_ = _ := by rw [← mul_assoc, inv_mul_cancel₀ hcard.ne', one_mul]
· have hempty : powers = ∅ := Finset.not_nonempty_iff_eq_empty.mp hp
simp only [hempty, average, Finset.card_empty, Nat.cast_zero, inv_zero, Finset.sum_empty, mul_zero]
positivity
theorem squaredError_abs_le {m : ℕ} (hm : 200 ≤ m) (powers : Finset ℕ)
(endpoint : Bool) (p : (Fin m → ℝ) × UnitAddTorus (Fin m)) :
|squaredError m powers endpoint p| ≤ (((m : ℝ) ^ (13 / 5 : ℝ)) ^ m + 1) ^ 2 := by
rw [squaredError, abs_pow]
apply pow_le_pow_left₀ (abs_nonneg _)
calc
|average m powers endpoint p.1 p.2 - 1| ≤
|average m powers endpoint p.1 p.2| + |(1 : ℝ)| := by simpa using (abs_sub_le (average m powers endpoint p.1 p.2) 0 1)
_ = average m powers endpoint p.1 p.2 + 1 := by
rw [abs_of_nonneg (average_nonneg hm powers endpoint p.1 p.2), abs_one]
_ ≤ _ := by linarith [average_le_height hm powers endpoint p.1 p.2]
theorem tendsto_integral_squaredError {m : ℕ} (hm : 200 ≤ m)
(powers : Finset ℕ) (endpoint : Bool)
(q : ℕ → ℕ → ℕ) (hq : ∀ j i, 0 < q j i) (hmono : ∀ j, Monotone (q j))
(hgap : ∀ i, i + 1 < m → Tendsto
(fun j => (q j (i + 1) : ℝ) / (q j i : ℝ)) atTop atTop)
{phase : ℕ → Fin m → ℝ} {target : Fin m → ℝ}
(hphase : Tendsto phase atTop (𝓝 target))
(hcut : ∀ n ∈ powers, ∀ i, Int.fract ((n : ℝ) * target i) ≠ 0) :
Tendsto (fun j => ∫ x : UnitAddCircle,
squaredError m powers endpoint
(phase j, fun i : Fin m => (q j i.val : ℤ) • x)) atTop
(𝓝 (∫ y : UnitAddTorus (Fin m), squaredError m powers endpoint (target, y))) := by
exact CharacterLaws.tendsto_integral_character_phase_of_ae_continuous q m hq hmono hgap
hphase (measurable_squaredError m powers endpoint)
(ae_continuousAt_squaredError hm powers endpoint target hcut)
(sq_nonneg _) (squaredError_abs_le hm powers endpoint)
theorem integrable_squaredError_comp {X : Type*} [MeasurableSpace X]
{μ : Measure X} [IsFiniteMeasure μ] {m : ℕ} (hm : 200 ≤ m)
(powers : Finset ℕ) (endpoint : Bool)
{F : X → (Fin m → ℝ) × UnitAddTorus (Fin m)} (hF : Measurable F) :
Integrable (fun x => squaredError m powers endpoint (F x)) μ := by
apply Integrable.of_bound ((measurable_squaredError m powers endpoint).comp hF).aestronglyMeasurable
((((m : ℝ) ^ (13 / 5 : ℝ)) ^ m + 1) ^ 2)
exact Eventually.of_forall (fun x => by
simpa only [Real.norm_eq_abs, Function.comp_apply] using squaredError_abs_le hm powers endpoint (F x))
theorem lintegral_squaredError_eq {X : Type*} [MeasurableSpace X]
{μ : Measure X} [IsFiniteMeasure μ] {m : ℕ} (hm : 200 ≤ m)
(powers : Finset ℕ) (endpoint : Bool)
{F : X → (Fin m → ℝ) × UnitAddTorus (Fin m)} (hF : Measurable F) :
(∫⁻ x, ENNReal.ofReal (squaredError m powers endpoint (F x)) ∂μ) =
ENNReal.ofReal (∫ x, squaredError m powers endpoint (F x) ∂μ) := by
apply (ofReal_integral_eq_lintegral_ofReal
(integrable_squaredError_comp hm powers endpoint hF) _).symm
exact Eventually.of_forall (fun x => sq_nonneg _)
end Invariant233.CharacterSampling
end
end OAI
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