| import OAI.Geometry.Kahler.BaseDensityThreshold |
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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 scoped ContDiff ENNReal |
| open Set Filter Topology MeasureTheory |
| open scoped ContDiff ENNReal Pointwise |
| open scoped ContDiff |
| open Set Filter Topology |
| noncomputable section |
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| open Set Filter Topology MeasureTheory |
| open scoped ContDiff |
| namespace PinchedHartogs.BaseConstruction |
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|
| theorem uniform_representing_densities {a : ℝ} (p : RadialProfiles a) : |
| ∃ c K : ℝ, 0 < c ∧ 0 < K ∧ ∀ D : ℝ, 2*Real.sqrt (2*p.R) < D → |
| ∀ kmin : ℕ, ∃ Q₀ : ℕ, ∀ Q : ℕ, Q₀ ≤ Q → |
| 2 ≤ Q ∧ kmin ≤ Q ∧ ∀ P : ℕ → Finset Sphere, |
| (∀ j, ProjectivelySeparated (D/Real.sqrt (Q^(j+1):ℕ)) (P (Q^(j+1)))) → |
| ∀ j, ContDiff ℝ ∞ (density Q p.R p.f p.b P j) ∧ |
| DensityStage (density Q p.R p.f p.b P) Q c K j ∧ |
| PhaseBandwidth (density Q p.R p.f p.b P j) (densityBandwidth Q j) ∧ |
| OrbitMeanOne (density Q p.R p.f p.b P j) ∧ |
| IsProbabilityMeasure (densityMeasure (density Q p.R p.f p.b P j)) ∧ |
| Represents (densityMeasure (density Q p.R p.f p.b P j)) := by |
| obtain ⟨B,F₁,B₁,hB,hF₁,hB₁,hbv,hfd,hbd⟩ := p.bounds |
| obtain ⟨c,K,N,hc,hc1,hK,hN,hgap,hder⟩ := exists_density_constants p hB hF₁ hB₁ |
| refine ⟨c,K,hc,hK,?_⟩ |
| intro D hD kmin |
| have hev := density_conditions_eventually (C₁ := 2*(Real.exp K*(2/c))*4^N*Real.exp 2) |
| (R := p.R) (B := B) (D := 1+2*Real.sqrt (2*p.R)) hgap hder |
| have hnat := tendsto_natCast_atTop_atTop.eventually hev |
| obtain ⟨Q₁,hQ₁⟩ := eventually_atTop.mp hnat |
| obtain ⟨Q₂,hQ₂⟩ := exists_nat_ge (2*p.R) |
| refine ⟨max (max Q₁ Q₂) (max 2 kmin),?_⟩ |
| intro Q hQ |
| have hQ1 : Q₁ ≤ Q := (le_max_left _ _).trans ((le_max_left _ _).trans hQ) |
| have hQ2 : Q₂ ≤ Q := (le_max_right _ _).trans ((le_max_left _ _).trans hQ) |
| have hQn : 2 ≤ Q := (le_max_left _ _).trans ((le_max_right _ _).trans hQ) |
| have hQk : kmin ≤ Q := (le_max_right _ _).trans ((le_max_right _ _).trans hQ) |
| have hQR : 2*p.R ≤ Q := hQ₂.trans (by exact_mod_cast hQ2) |
| obtain ⟨hH,hsmall,hphase,hsecond,hderH,hderT⟩ := hQ₁ Q hQ1 |
| refine ⟨hQn,hQk,?_⟩ |
| intro P hP j |
| have hq : 0 < Q := by omega |
| have hdis : ∀ i, (P (Q^(i+1)):Set Sphere).PairwiseDisjoint (peakPatch (Q^(i+1)) p.R) := |
| fun i => peakPatch_disjoint (pow_pos hq _) p.R_pos hD (hP i) |
| have hstage := density_induction hQn p.R_pos p.cutoff_lt hQR p.F_nonneg hB hF₁ hB₁ |
| (by nlinarith [Real.sqrt_nonneg (2*p.R)] : 1 ≤ 1+2*Real.sqrt (2*p.R)) (by linarith : 2*Real.sqrt (2*p.R) ≤ 1+2*Real.sqrt (2*p.R)) |
| hc hc1 hK.le hN p.smooth_f p.smooth_b p.tail p.small hbv hfd hbd P hdis |
| hH hsmall hphase hsecond hderH hderT j |
| have hsm := density_smooth hq p.cutoff_lt p.smooth_f p.smooth_b p.tail P j |
| refine ⟨hsm,hstage,density_bandwidth hq p.cutoff_lt p.smooth_f p.smooth_b p.tail P j, |
| density_orbit_mean hQn p.cutoff_lt p.smooth_f p.smooth_b p.tail P j,?_,?_⟩ |
| · exact densityMeasure_probability hsm.continuous (fun ξ => (hstage.positive ξ).le) |
| (density_mass_one hQn p.R_pos p.cutoff_lt p.smooth_f p.smooth_b p.ode p.b_zero p.tail P j) |
| · exact densityMeasure_represents hsm.continuous (fun ξ => (hstage.positive ξ).le) |
| (density_polynomial_moment hQn p.R_pos p.cutoff_lt p.smooth_f p.smooth_b p.ode p.b_zero p.tail P j) |
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| end PinchedHartogs.BaseConstruction |
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| end |
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| end OAI |
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