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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 | import OAI.Analysis.IrrationalRotation.Sampling.CharacterSamplingTests
import OAI.Analysis.IrrationalRotation.Analysis.SquaredIntegralLp
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped ENNReal Topology BigOperators
namespace Invariant233.CharacterSampling
local instance characterSamplingLpCircleProbability :
IsProbabilityMeasure (volume : Measure UnitAddCircle) :=
⟨by simp [AddCircle.measure_univ]⟩
lemma measurable_average (m : ℕ) (powers : Finset ℕ) (endpoint : Bool) :
Measurable (fun p : (Fin m → ℝ) × UnitAddTorus (Fin m) =>
average m powers endpoint p.1 p.2) := by
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 memLp_samplingError_comp {X : Type*} [MeasurableSpace X]
{μ : Measure X} [IsFiniteMeasure μ] {m : ℕ} (hm : 200 ≤ m)
(powers : Finset ℕ) (endpoint : Bool) (p : ℝ≥0∞)
{F : X → (Fin m → ℝ) × UnitAddTorus (Fin m)} (hF : Measurable F) :
MemLp (fun x => average m powers endpoint (F x).1 (F x).2 - 1) p μ := by
apply MemLp.of_bound
(((measurable_average m powers endpoint).comp hF).sub_const 1).aestronglyMeasurable
(((m : ℝ) ^ (13 / 5 : ℝ)) ^ m + 1)
apply Eventually.of_forall
intro x
rw [Real.norm_eq_abs]
calc
_ ≤ |average m powers endpoint (F x).1 (F x).2| + |(1 : ℝ)| := abs_sub _ _
_ = average m powers endpoint (F x).1 (F x).2 + 1 := by
rw [abs_of_nonneg (average_nonneg hm powers endpoint _ _), abs_one]
_ ≤ _ := add_le_add (average_le_height hm powers endpoint _ _) le_rfl
theorem tendsto_eLpNorm_samplingError {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 => eLpNorm (fun x : UnitAddCircle =>
average m powers endpoint (phase j)
(fun i : Fin m => (q j i.val : ℤ) • x) - 1) 2 volume) atTop
(𝓝 (eLpNorm (fun y : UnitAddTorus (Fin m) =>
average m powers endpoint target y - 1) 2 volume)) := by
apply SquaredIntegralLp.tendsto_eLpNorm_two_of_integral_sq
· intro index
have continuousPair : Continuous (fun x : UnitAddCircle =>
(phase index, fun i : Fin m => (q index i.val : ℤ) • x)) :=
continuous_const.prodMk (continuous_pi fun i => continuous_const_smul _)
exact (((measurable_average m powers endpoint).comp
continuousPair.measurable).sub_const 1).aestronglyMeasurable
· exact (((measurable_average m powers endpoint).comp
(measurable_const.prodMk measurable_id)).sub_const 1).aestronglyMeasurable
· intro j
have hc : Continuous (fun x : UnitAddCircle =>
(phase j, fun i : Fin m => (q j i.val : ℤ) • x)) :=
continuous_const.prodMk (continuous_pi fun i => continuous_const_smul _)
exact integrable_squaredError_comp (μ := volume) hm powers endpoint
(F := fun x : UnitAddCircle =>
(phase j, fun i : Fin m => (q j i.val : ℤ) • x)) hc.measurable
· exact integrable_squaredError_comp (μ := volume) hm powers endpoint
(F := fun y : UnitAddTorus (Fin m) => (target, y))
(measurable_const.prodMk measurable_id)
· exact tendsto_integral_squaredError hm powers endpoint q hq hmono hgap hphase hcut
theorem eventually_eLpNorm_samplingError_lt {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)
{δ : ℝ} (hδ : eLpNorm (fun y : UnitAddTorus (Fin m) =>
average m powers endpoint target y - 1) 2 volume < ENNReal.ofReal δ) :
∀ᶠ j in atTop, eLpNorm (fun x : UnitAddCircle =>
average m powers endpoint (phase j)
(fun i : Fin m => (q j i.val : ℤ) • x) - 1) 2 volume < ENNReal.ofReal δ :=
(tendsto_eLpNorm_samplingError hm powers endpoint q hq hmono hgap hphase hcut).eventually_lt_const hδ
end Invariant233.CharacterSampling
end
end OAI
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