openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Shells /ApproximatingDistanceShellDecay.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.FalconerCompletion.Estimates.AssemblySmallParameters | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.AssemblyPairCoefficientSum | |
| import OAI.MeasureTheory.FalconerCompletion.Shells.TwoProfileComponentShell | |
| namespace OAI | |
| /-! Dyadic shells and prefix decompositions. -/ | |
| open _root_.OAI.PlanarFalconer | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport.CurrentFalconer.TreeND | |
| open Set MeasureTheory Filter PlanarFalconer | |
| open scoped ContDiff Classical Topology ENNReal NNReal | |
| variable {n : ℕ} | |
| local notation "V" => EuclideanSpace ℝ (Fin (n+1)) | |
| theorem actual_approximating_distance_shell_decay (hn : 1 ≤ n) (Ctest chi : ℝ) (hchi : 0 < chi) : | |
| ∃ beta : ℝ, 0 < beta ∧ beta < min (chi/10) (1/10) ∧ | |
| ∃ kap : ℝ, 0 < kap ∧ kap ≤ 1/1000 ∧ | |
| ∃ sig : ℝ, 0 < sig ∧ sig < kap/1000 ∧ beta+(n+4:ℝ)*sig < chi/4 ∧ | |
| ∃ omega gamma : ℝ, 0 < omega ∧ 4*(n+1:ℝ)*omega < beta/8 ∧ 0 < gamma ∧ | |
| ∀ (Ψ : ℝ → ℝ), ContDiff ℝ ∞ Ψ → tsupport Ψ ⊆ Ioo (1/4:ℝ) 4 → | |
| (∀ r, 0 ≤ Ψ r) → (∀ r ∈ Icc (1/2:ℝ) 2, 1 ≤ Ψ r) → | |
| ∀ (ψ : (EuclideanSpace ℝ (Fin (n+1))) → ℂ), ContDiff ℝ ∞ ψ → (∀ w, ‖w‖=1 → ‖ψ w‖ ≤ 1) → | |
| ∀ (δ K A : ℝ), 0 < δ → 0 < K → 0 < A → ∃ N₀ : ℕ, ∀ N ≥ N₀, | |
| ∀ {ι κ : Type} [Fintype ι] [Fintype κ] | |
| (μ ν : Measure (EuclideanSpace ℝ (Fin (n+1)))) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] | |
| (B : ℕ) (F : AssemblyClassFamily ι μ N B chi sig) | |
| (G : AssemblyClassFamily κ ν N B chi sig) | |
| (I : Finset ℕ) (X Y : ℕ → Set (EuclideanSpace ℝ (Fin (n+1)))), | |
| (∀ l ∈ I, MeasurableSet (X l) ∧ MeasurableSet (Y l)) → | |
| (∀ l ∈ I, X l ⊆ treeCell [] ∧ Y l ⊆ treeCell []) → | |
| (∀ l ∈ I, ∀ x ∈ X l, ∀ y ∈ Y l, δ ≤ dist x y ∧ dist x y ≤ K) → | |
| (∀ l ∈ I, ∀ x ∈ X l, ∀ y ∈ Y l, | |
| ψ =ᶠ[𝓝 ((dist x y)⁻¹ • (x-y))] (fun _ => 1) ∧ | |
| ψ =ᶠ[𝓝 (-((dist x y)⁻¹ • (x-y)))] (fun _ => 0)) → | |
| (Set.Pairwise (↑I : Set ℕ) (fun l m => Disjoint (X l ×ˢ Y l) (X m ×ˢ Y m))) → | |
| (I.card : ℝ) ≤ A*((2:ℝ)^N)^(2*(n+1:ℝ)*omega) → | |
| (∫ r in {r : ℝ | ((2:ℝ)^N)/2 ≤ |r| ∧ |r| ≤ 2*((2:ℝ)^N)}, | |
| ‖charFun (assemblyTau F G kap Ctest beta I X Y) r‖^2) ≤ | |
| 2*A^2*((2:ℝ)^N)^(-gamma) | |
| := by | |
| obtain ⟨Cd,hCd,hshell⟩ := actual_two_profile_component_shell hn Ctest | |
| obtain ⟨beta,hbeta,hbetasm,kap,hkap,hkap1,sig,hsig,hsigkap,hsmall,omega,homega,homegasm,hexp⟩ := | |
| assembly_small_parameters n hchi hCd | |
| refine ⟨beta,hbeta,hbetasm,kap,hkap,hkap1,sig,hsig,hsigkap,hsmall,omega,beta/2,homega,homegasm, | |
| by positivity,?_⟩ | |
| intro Ψ hΨ hsupp hΨnon hΨone ψ hψ hψone δ K A hδ hK hA | |
| obtain ⟨N₀,hN₀⟩ := hshell chi beta sig kap (beta/8) hchi hbeta hbetasm hkap hkap1 hsig | |
| hsigkap hsmall (by positivity) Ψ hΨ hsupp hΨnon hΨone ψ hψ hψone δ K hδ hK | |
| refine ⟨N₀,?_⟩ | |
| intro N hN ι κ _ _ μ ν _ _ B F G I X Y hXY hroot hsep hout hdis hcard | |
| let R : ℝ := (2:ℝ)^N | |
| let E : ℝ := -beta+Cd*(kap+sig/kap)+beta/8 | |
| have hR : 1≤R := one_le_pow₀ (by norm_num) | |
| have hRp : 0<R := by dsimp[R];positivity | |
| have hcomp (i : ι) (j : κ) (l : ℕ) (hl : l∈I) : | |
| assemblyPositiveEnergy (assemblyComponentMeasure (F.env kap Ctest i) (G.env kap Ctest j) | |
| (assemblyPairEntry F G kap Ctest beta i j) (X l) (Y l)) R ≤ R^E := by | |
| let := (F.profile i).probability | |
| let := (G.profile j).probability | |
| have hm : ∀ᵐ z ∂((assemblyClassLaw μ (F.domain i)).restrict (X l)).prod | |
| ((assemblyClassLaw ν (G.domain j)).restrict (Y l)), z.1∈X l ∧ z.2∈Y l := | |
| (Measure.quasiMeasurePreserving_fst.ae (ae_restrict_mem (hXY l hl).1)).and | |
| (Measure.quasiMeasurePreserving_snd.ae (ae_restrict_mem (hXY l hl).2)) | |
| have ho : ∀ᵐ z ∂((assemblyClassLaw μ (F.domain i)).restrict (X l)).prod | |
| ((assemblyClassLaw ν (G.domain j)).restrict (Y l)), | |
| ψ =ᶠ[𝓝 ((dist z.1 z.2)⁻¹ • (z.1-z.2))] (fun _ => 1) ∧ | |
| ψ =ᶠ[𝓝 (-((dist z.1 z.2)⁻¹ • (z.1-z.2)))] (fun _ => 0) := by | |
| filter_upwards [hm] with z hz | |
| exact hout l hl z.1 hz.1 z.2 hz.2 | |
| have hh := hN₀ N hN (assemblyClassLaw μ (F.domain i)) (assemblyClassLaw ν (G.domain j)) | |
| B (F.mass i) (G.mass j) (F.profile i) (G.profile j) (F.law_root i) (G.law_root j) | |
| (X l) (Y l) (fun _ => 1) (fun _ => 1) (hXY l hl).1 (hXY l hl).2 | |
| (hroot l hl).1 (hroot l hl).2 measurable_const measurable_const | |
| (by intro x;simp) (by intro x;simp) (hsep l hl) ho | |
| have heq (r : ℝ) := assemblyComponentMeasure_fourier (F.env kap Ctest i) | |
| (G.env kap Ctest j) (assemblyPairEntry F G kap Ctest beta i j) (X l) (Y l) | |
| (F.profile i).zero (G.profile j).zero r | |
| unfold assemblyPositiveEnergy | |
| simp_rw [heq] | |
| exact hh.2 | |
| obtain ⟨hfinite,hpos⟩ := assemblyTau_positive_energy F G kap Ctest beta I X Y hXY hdis hδ | |
| (fun l hl x hx y hy => (hsep l hl x hx y hy).1) R (R^E) (Real.rpow_nonneg hRp.le E) hcomp | |
| let := hfinite | |
| rw [assemblyEnergy_two_sided _ hRp] | |
| have hcard2 : (I.card:ℝ)^2 ≤ A^2*R^(4*(n+1:ℝ)*omega) := by | |
| change (I.card:ℝ) ≤ A*R^(2*(n+1:ℝ)*omega) at hcard | |
| have hh := pow_le_pow_left₀ (Nat.cast_nonneg I.card) hcard 2 | |
| have hp : (R^(2*(n+1:ℝ)*omega))^2 = R^(4*(n+1:ℝ)*omega) := by | |
| rw [← Real.rpow_mul_natCast hRp.le] | |
| congr 1 | |
| push_cast | |
| ring | |
| simpa only [mul_pow,hp] using hh | |
| have hbound : (I.card:ℝ)^2*R^E ≤ A^2*R^(-(beta/2)) := by | |
| calc | |
| _ ≤ (A^2*R^(4*(n+1:ℝ)*omega))*R^E := | |
| mul_le_mul_of_nonneg_right hcard2 (Real.rpow_nonneg hRp.le E) | |
| _ = A^2*R^(E+4*(n+1:ℝ)*omega) := by | |
| rw [mul_assoc,← Real.rpow_add hRp] | |
| congr 2 | |
| ring | |
| _ ≤ _ := by | |
| simpa only [neg_div] using | |
| mul_le_mul_of_nonneg_left (Real.rpow_le_rpow_of_exponent_le hR hexp) (sq_nonneg A) | |
| have hh := mul_le_mul_of_nonneg_left (hpos.trans hbound) (by norm_num : (0:ℝ)≤2) | |
| simpa only [mul_assoc] using hh | |
| end GraphLiteralSupport.CurrentFalconer.TreeND | |
| end | |
| end | |
| end OAI | |