openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Sampling /CharacterSamplingLpRealization.lean
| import OAI.Analysis.IrrationalRotation.Sampling.CharacterSamplingLp | |
| import OAI.Analysis.IrrationalRotation.Sampling.CharacterSamplingRealization | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped ENNReal Topology | |
| namespace Invariant233.CharacterSampling | |
| local instance characterSamplingLpRealizationCircleProbability : | |
| IsProbabilityMeasure (volume : Measure UnitAddCircle) := | |
| ⟨by simp [AddCircle.measure_univ]⟩ | |
| theorem exists_realized_sampling_block_L2 {θ : ℝ} (hθ : Irrational θ) | |
| {m : ℕ} (hm : 200 ≤ m) (target : Fin m → ℝ) | |
| (htarget0 : ∀ i, 0 < target i) (htarget1 : ∀ i, target i < 1) | |
| {q₀ : ℕ} (hq₀ : 0 < q₀) (R : ℝ) | |
| (U : Set (Fin m → ℝ)) (hU : U ∈ 𝓝 target) | |
| (r : ℕ) (powers : Fin r → Finset ℕ) (δ : Fin r → Bool → ℝ) | |
| (hδ : ∀ k endpoint, 0 < δ k endpoint) | |
| (hcut : ∀ k, ∀ n ∈ powers k, ∀ i, Int.fract ((n : ℝ) * target i) ≠ 0) | |
| (hbound : ∀ k endpoint, | |
| eLpNorm (fun y : UnitAddTorus (Fin m) => | |
| average m (powers k) endpoint target y - 1) 2 volume < ENNReal.ofReal (δ k endpoint)) : | |
| ∃ q : ℕ → ℕ, | |
| q 0 = q₀ ∧ (∀ i, 0 < q i) ∧ StrictMono q ∧ | |
| (∀ i, q i ∣ q (i + 1)) ∧ | |
| (∀ i < m, R ≤ (q (i + 1) : ℝ) / (q i : ℝ)) ∧ | |
| (fun i : Fin m => Int.fract ((q (i.val + 1) : ℝ) * θ)) ∈ U ∧ | |
| ∀ k endpoint, | |
| eLpNorm (fun x : UnitAddCircle => average m (powers k) endpoint | |
| (fun i : Fin m => Int.fract ((q (i.val + 1) : ℝ) * θ)) | |
| (fun i : Fin m => (q (i.val + 1) : ℤ) • x) - 1) 2 volume < | |
| ENNReal.ofReal (δ k endpoint) := by | |
| have hsq : ∀ k endpoint, | |
| (∫ y : UnitAddTorus (Fin m), squaredError m (powers k) endpoint (target, y)) < | |
| (δ k endpoint) ^ 2 := by | |
| intro k endpoint | |
| have hint := integrable_squaredError_comp (μ := volume) hm (powers k) endpoint | |
| (F := fun y : UnitAddTorus (Fin m) => (target, y)) | |
| (measurable_const.prodMk measurable_id) | |
| exact (SquaredIntegralLp.eLpNorm_two_lt_ofReal_iff_integral_sq_lt volume | |
| (fun y : UnitAddTorus (Fin m) => average m (powers k) endpoint target y - 1) | |
| (((measurable_average m (powers k) endpoint).comp | |
| (measurable_const.prodMk measurable_id)).sub_const 1).aestronglyMeasurable | |
| hint (hδ k endpoint)).mp (hbound k endpoint) | |
| obtain ⟨q, hzero, hpos, hstrict, hdiv, hratio, hUq, hsqQ⟩ := | |
| exists_realized_sampling_block hθ hm target htarget0 htarget1 hq₀ R U hU | |
| r powers (fun k endpoint => (δ k endpoint) ^ 2) hcut hsq | |
| refine ⟨q, hzero, hpos, hstrict, hdiv, hratio, hUq, ?_⟩ | |
| intro k endpoint | |
| have hc : Continuous (fun x : UnitAddCircle => | |
| ((fun i : Fin m => Int.fract ((q (i.val + 1) : ℝ) * θ)), | |
| fun i : Fin m => (q (i.val + 1) : ℤ) • x)) := | |
| continuous_const.prodMk (continuous_pi fun i => continuous_zsmul _) | |
| have hint := integrable_squaredError_comp (μ := volume) hm (powers k) endpoint | |
| (F := fun x : UnitAddCircle => | |
| ((fun i : Fin m => Int.fract ((q (i.val + 1) : ℝ) * θ)), | |
| fun i : Fin m => (q (i.val + 1) : ℤ) • x)) hc.measurable | |
| exact (SquaredIntegralLp.eLpNorm_two_lt_ofReal_iff_integral_sq_lt volume | |
| (fun x : UnitAddCircle => average m (powers k) endpoint | |
| (fun i : Fin m => Int.fract ((q (i.val + 1) : ℝ) * θ)) | |
| (fun i : Fin m => (q (i.val + 1) : ℤ) • x) - 1) | |
| (((measurable_average m (powers k) endpoint).comp | |
| hc.measurable).sub_const 1).aestronglyMeasurable | |
| hint (hδ k endpoint)).mpr (hsqQ k endpoint) | |
| end Invariant233.CharacterSampling | |
| end | |
| end OAI | |