| import OAI.Geometry.Kahler.BaseLaterMoments |
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| namespace OAI |
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| open Complex |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise ComplexOrder |
| open scoped ContDiff ComplexOrder |
| open Set Filter Topology |
| open scoped ContDiff ENNReal Pointwise |
| open scoped ContDiff ENNReal |
| open Set Filter Topology MeasureTheory |
| open scoped ContDiff |
| noncomputable section |
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| open Set Filter Topology MeasureTheory |
| open scoped ContDiff |
| namespace PinchedHartogs.BaseConstruction |
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| lemma regularizedLog_scale (a ε : ℝ) (v : ℂ) : regularizedLog a ε v=regularizedLog 1 ε ((a:ℂ)*v) := by |
| simp only [regularizedLog,Complex.ofReal_one,one_mul] |
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| lemma density_log_comparison {a : ℝ} (pr : RadialProfiles a) {Q M i N : ℕ} |
| (hQ : 2 ≤ Q) (hM : M+2 < Q) (hiN : i ≤ N) (P : ℕ → Finset Sphere) |
| {ε θ : ℝ} (he : ε ≠ 0) (hθ : 0 ≤ θ) (hθ1 : θ < 1) (c : ℂ) |
| (hratio : ∀ ξ : Sphere, ‖logNormalized ε (c*peakPolynomial (P (Q^i)) (Q^i) ξ)‖ ≤ θ) |
| (hposi : ∀ ξ : Sphere, 0 ≤ density Q pr.R pr.f pr.b P i ξ) |
| (hposN : ∀ ξ : Sphere, 0 ≤ density Q pr.R pr.f pr.b P N ξ) |
| [IsProbabilityMeasure (densityMeasure (density Q pr.R pr.f pr.b P i))] |
| [IsProbabilityMeasure (densityMeasure (density Q pr.R pr.f pr.b P N))] : |
| |(∫ ξ : Sphere, regularizedLog 1 ε (c*peakPolynomial (P (Q^i)) (Q^i) ξ) ∂densityMeasure (density Q pr.R pr.f pr.b P N))- |
| (∫ ξ : Sphere, regularizedLog 1 ε (c*peakPolynomial (P (Q^i)) (Q^i) ξ) ∂densityMeasure (density Q pr.R pr.f pr.b P i))| ≤ |
| 4*θ^(M+1)/(1-θ) := by |
| let V : Sphere → ℂ := fun ξ => c*peakPolynomial (P (Q^i)) (Q^i) ξ |
| have hV : Continuous V := continuous_const.mul ((peakPolynomial_analytic _ _).continuous.comp continuous_subtype_val) |
| let f : Sphere → ℝ := fun ξ => regularizedLog 1 ε (V ξ) |
| let g : Sphere → ℝ := fun ξ => logTruncation ε M (V ξ) |
| have hf : Continuous f := (regularizedLog_contDiff 1 he).continuous.comp hV |
| have hg : Continuous g := (logTruncation_continuous ε M).comp hV |
| have hb : ∀ ξ, |f ξ-g ξ| ≤ 2*θ^(M+1)/(1-θ) := fun ξ => logTruncation_error he hθ hθ1 (hratio ξ) M |
| have h1 := probability_integral_error (μ := densityMeasure (density Q pr.R pr.f pr.b P N)) |
| (compact_continuous_integrable hf) (compact_continuous_integrable hg) hb |
| have h2 := probability_integral_error (μ := densityMeasure (density Q pr.R pr.f pr.b P i)) |
| (compact_continuous_integrable hf) (compact_continuous_integrable hg) hb |
| have hW (j : ℕ) : Continuous (density Q pr.R pr.f pr.b P j) := |
| (density_smooth (by omega : 0 < Q) pr.cutoff_lt pr.smooth_f pr.smooth_b pr.tail P j).continuous |
| have hpersist : (∫ ξ, g ξ ∂densityMeasure (density Q pr.R pr.f pr.b P N))= |
| ∫ ξ, g ξ ∂densityMeasure (density Q pr.R pr.f pr.b P i) := by |
| rw [densityMeasure_integral (hW N) hposN,densityMeasure_integral (hW i) hposi] |
| simp only [smul_eq_mul] |
| simpa only [mul_comm] using logTruncation_density_persist pr hQ hM hiN P ε c |
| rw [hpersist] at h1 |
| have hh := abs_sub_le (∫ ξ, f ξ ∂densityMeasure (density Q pr.R pr.f pr.b P N)) |
| (∫ ξ, g ξ ∂densityMeasure (density Q pr.R pr.f pr.b P i)) |
| (∫ ξ, f ξ ∂densityMeasure (density Q pr.R pr.f pr.b P i)) |
| rw [abs_sub_comm (∫ ξ, g ξ ∂densityMeasure (density Q pr.R pr.f pr.b P i))] at hh |
| exact hh.trans (by convert add_le_add h1 h2 using 1; ring) |
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| lemma logarithmicTest_later_threshold {a : ℝ} (ha : 0 ≤ a) {ε : ℝ} (he : ε ≠ 0) {d : ℝ} (hd : 0 < d) : |
| ∃ M : ℕ, ∃ θ : ℝ, 0 < θ ∧ θ < 1 ∧ 4*θ^(M+1)/(1-θ) ≤ d ∧ |
| ∀ q ∈ Icc 0 (a*gaussianConstant), q/logFactor q ε ≤ θ := by |
| obtain ⟨θ,hθ,hθ1,hb⟩ := logFactor_ratio_bound he (mul_nonneg ha gaussianConstant_pos.le) |
| have ht : Tendsto (fun M : ℕ => 4*θ^(M+1)/(1-θ)) atTop (𝓝 0) := by |
| have hh := (tendsto_pow_atTop_nhds_zero_of_lt_one hθ.le hθ1).mul_const θ |
| simpa only [← pow_succ,mul_zero,zero_mul,zero_div] using (hh.const_mul 4).div_const (1-θ) |
| obtain ⟨M,hM⟩ := eventually_atTop.mp (ht.eventually (gt_mem_nhds hd)) |
| exact ⟨M,θ,hθ,hθ1,(hM M le_rfl).le,hb⟩ |
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| end PinchedHartogs.BaseConstruction |
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| end |
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| end OAI |
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