| 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 |
|
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| end OAI |
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|
