| import OAI.MeasureTheory.Falconer.Radial.EndpointSourceMiddle |
| import OAI.MeasureTheory.Falconer.Radial.SourceClassOuterRadial |
| import OAI.MeasureTheory.Falconer.Recursion.SourceClassIntervals |
| import OAI.MeasureTheory.Falconer.Estimates.RelativePairCore |
| import OAI.MeasureTheory.Falconer.Angular.ReverseCapsAllRadii |
| import OAI.MeasureTheory.Falconer.Geometry.UniformMiddleGeometry |
| import OAI.MeasureTheory.Falconer.Recursion.ThreeIntervalDyadicGain |
| import OAI.MeasureTheory.Falconer.Recursion.ThreeIntervalCost |
| import OAI.MeasureTheory.Falconer.Recursion.ThreeIntervalShift |
| import OAI.MeasureTheory.Falconer.Geometry.OneScaleGeometry |
|
|
| namespace OAI |
|
|
| /-! # Endpoint one-scale gain for an actual source class with a contact interval |
|
|
| All three local estimates are applied to the original conditional source laws. |
| The incoming relation and pin law are never renormalized. |
| -/ |
|
|
| namespace Falconer.Regularization |
| open Set MeasureTheory NormedSpace |
| open scoped ENNReal BigOperators |
| attribute [local instance] Classical.propDecidable |
|
|
| theorem endpoint_contact_gain (hLW : PublishedFiniteLoomisWhitney) |
| {h : ℕ} {t q ζ : ℝ} (hq : q = (h:ℝ)+1) |
| (ht : (h:ℝ)+3/2 < t) (ht' : t < (h:ℝ)+2) (hζ : 0 < ζ) |
| (Q : DyadicCube (h+3)) (hQ : Q.side = 1) |
| {D s : ℝ} (hD : 1 ≤ D) (hDdim : Real.sqrt (h+3:ℕ) ≤ D) (hs : 0 < s) : |
| ∃ ε gain : ℝ, 0 < ε ∧ 0 < gain ∧ |
| ∀ τ γ : ℝ, 0 < τ → 0 < γ → |
| (2*((h+3:ℕ):ℝ)+4)*τ ≤ ε/2 → |
| (20*(2*((h+3:ℕ):ℝ)+4)+4)*τ ≤ gain/8 → 2*τ ≤ gain → |
| ∃ N₀ : ℕ, ∀ N ≥ N₀, |
| ∀ (μ ν : Measure (Space (h+3))) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν], |
| ∀ B : ℕ, 0 < B → ∀ lam : ℝ, 0 ≤ lam → lam ≤ τ*γ → |
| (B:ℝ) ≤ lam*(N:ℝ) → (2:ℝ)^(N/B+2) ≤ ((2:ℝ)^N)^lam → |
| ∀ p₀ ∈ regularizationLeaves Q μ N, |
| RobustProfileContact (sourceUnitProfile Q μ N B p₀) q t τ γ → |
| ∀ A P : Set (Space (h+3)), MeasurableSet A → MeasurableSet P → |
| μ Aᶜ = 0 → ν Pᶜ = 0 → |
| (∀ z ∈ A, ∀ x ∈ P, s ≤ ‖z-x‖) → |
| (∀ y ∈ Q.carrier, ∀ x ∈ P, ‖y-x‖ ≤ D) → |
| ∀ G : Set (Space (h+3) × Space (h+3)), MeasurableSet G → |
| (∀ᵐ z ∂μ, ∀ r : ℝ, 1/(2:ℝ)^N ≤ r → r ≤ 1 → |
| ∀ u : Space (h+3), ‖u‖ = 1 → |
| ν {x | (z,x) ∈ G ∧ dist (normalize (x-z)) u < r} ≤ |
| ENNReal.ofReal (((2:ℝ)^N)^(τ*γ)*r^q)) → |
| (∀ᵐ z ∂μ, ∀ u : Space (h+3), ‖u‖ = 1 → ∀ r : ℝ, |
| 1/(2:ℝ)^N ≤ r → r ≤ 1 → |
| ν {x | (z,x) ∈ G ∧ |inner ℝ u (normalize (x-z))| ≤ r} ≤ |
| ENNReal.ofReal (((2:ℝ)^N)^(τ*γ)*r^ζ)) → |
| ∃ G' : Set (Space (h+3) × Space (h+3)), MeasurableSet G' ∧ G' ⊆ G ∧ |
| ((sourceClass Q μ N B p₀).prod ν) (G \ G') ≤ |
| ENNReal.ofReal (((2:ℝ)^N)^(-(τ*γ/2))) ∧ |
| ∀ x u : Space (h+3), sourceClass Q μ N B p₀ |
| {z | (z,x) ∈ G' ∧ ‖normalize (z-x)-u‖ ≤ 1/(2:ℝ)^N} ≤ |
| ENNReal.ofReal (((2:ℝ)^N)^(-q-gain*γ/8)) := by |
| obtain ⟨ε,gain,hε,hgain,M₀,hmiddle⟩ := |
| endpoint_source_class_middle_pruning hLW ht ht' hζ |
| refine ⟨ε,gain,hε,hgain,?_⟩ |
| intro τ γ hτ hγ hτε hτgain hτgain' |
| let L : ℝ := 2*((h+3:ℕ):ℝ)+4 |
| let σ : ℝ := cellContraction (h+3) |
| let C : ℝ := 4*Real.sqrt (h+3:ℕ)/s+2 |
| have hL : 1 ≤ L := by dsimp [L]; have hh := Nat.cast_nonneg (α := ℝ) (h+3); linarith only [hh] |
| have hq0 : 0 ≤ q := by rw [hq]; positivity |
| have ht0 : 0 ≤ t := by have hh := Nat.cast_nonneg (α := ℝ) h; linarith only [hh,ht] |
| have hside : 0 < Q.side := by rw [hQ]; norm_num |
| have hσ : 0 < σ := cellContraction_pos _ |
| have hη₀ : 0 < L*τ*γ := by positivity |
| obtain ⟨Jo,No,houter⟩ := source_class_outer_radial_pruning Q hQ hD hDdim hη₀ hq0 C |
| obtain ⟨Jm,Nm,hgeom⟩ := uniform_middle_geometry hσ (by positivity : 0 < s/2) |
| (by positivity : 0 < γ/2) D C M₀ |
| obtain ⟨Na,hmargin⟩ := eventually_integer_margin (by positivity : 0 < γ/2) Jo |
| obtain ⟨Ns,hsmall⟩ := eventually_cell_diameter_small (h+3) (by positivity : 0 < γ/2) hs |
| let K : ℝ := (((2*Nat.ceil (2+4*Real.sqrt (h+3:ℕ))+1)^(h+3):ℕ):ℝ) |
| obtain ⟨Nf,hfirst⟩ := eventually_contact_outer_coefficient (h+3) hτ hγ (K*(2:ℝ)^q) 0 |
| obtain ⟨Nl,hlast⟩ := eventually_contact_outer_coefficient (h+3) hτ hγ |
| (K*(2:ℝ)^q) (((h+3:ℕ):ℝ)+q) |
| obtain ⟨Nt,htcoef⟩ := eventually_contact_frostman_coefficient (h+3) hτ hγ hτε |
| (K*(2:ℝ)^t*σ^(-t)) (((h+3:ℕ):ℝ)+t) |
| obtain ⟨Nc,hconst⟩ := fixed_constant_absorbed (mul_pos hτ hγ) |
| (((2:ℝ)^Jm)^(q+gain/2)) |
| obtain ⟨Nd,hcost⟩ := eventually_three_interval_cost hτ hγ |
| (by linarith : 1 ≤ 2*L) hτgain' |
| obtain ⟨Nlarge,hlarge⟩ := exists_nat_gt (4/γ) |
| let N₀ := max 1 (max No (max Nm (max Na (max Ns (max Nf (max Nl (max Nt |
| (max Nc (max Nd Nlarge))))))))) |
| refine ⟨N₀,?_⟩ |
| intro N hN μ ν _ _ B hB lam hlam hlamτ hblock hblock' |
| p₀ hp₀ hcontact A P hA hP hAfull hPfull hsep hpin G hG hcap hslab |
| change max 1 (max No (max Nm (max Na (max Ns (max Nf (max Nl (max Nt |
| (max Nc (max Nd Nlarge))))))))) ≤ N at hN |
| simp only [max_le_iff] at hN |
| rcases hN with ⟨h1,hNo,hNm,hNa,hNs,hNf,hNl,hNt,hNc,hNd,hNL⟩ |
| have hNp : 0 < N := by omega |
| have hNN : (Nlarge:ℝ) ≤ N := by exact_mod_cast hNL |
| have hlarge' : 4 ≤ γ*(N:ℝ) := by |
| have hh := (div_lt_iff₀ hγ).mp hlarge |
| nlinarith only [hh,hNN,hγ] |
| obtain ⟨α,β,a,b,hγΔ,hΔ1,hab,hbN,hgap,_,ha,_,hNb,hfreeze,hi₀,hi₁,hi₂⟩ := |
| source_class_contact_intervals Q μ hNp hB p₀ hq0 ht0 hγ hlarge' hcontact |
| let Δ := β-α |
| let η := L*τ*Δ |
| let ξ := τ*Δ |
| let M := (b-a)-Jm |
| have hΔ : 0 < Δ := hγ.trans_le hγΔ |
| have hη : L*τ*γ ≤ η := mul_le_mul_of_nonneg_left hγΔ (by positivity) |
| obtain ⟨hJm,hM,hNM,hfr,hwi⟩ := hgeom N hNm a b Δ hab |
| (by dsimp [Δ]; linarith only [hγΔ,hγ]) hgap hfreeze |
| have hJoA : Jo ≤ a := hmargin N hNa a ha |
| have hJoB : b+Jo ≤ N := by |
| have hh := hmargin N hNa (N-b) (by linarith only [hNb,hγ]) |
| omega |
| have hsmallA : Real.sqrt (h+3:ℕ)*(Q.side/(2:ℝ)^a) ≤ s/2 := by |
| rw [hQ,mul_one_div] |
| exact hsmall N hNs a ha |
| have hsmallAll {n : ℕ} (han : a ≤ n) : |
| Real.sqrt (h+3:ℕ)*(Q.side/(2:ℝ)^n) ≤ s/2 := by |
| apply le_trans _ hsmallA |
| exact mul_le_mul_of_nonneg_left |
| (div_le_div_of_nonneg_left hside.le (by positivity) |
| (pow_le_pow_right₀ (by norm_num) han)) (Real.sqrt_nonneg _) |
| have hcoef₀ : K*(2:ℝ)^q*((2:ℝ)^N)^(τ*Δ+2*(((h+3:ℕ):ℝ)+1)*lam) ≤ |
| ((2:ℝ)^N)^η := by |
| simpa only [zero_div,add_zero] using hfirst N hNf hNp Δ lam hγΔ hlamτ |
| have hcoef₂ := hlast N hNl hNp Δ lam hγΔ hlamτ |
| have hcoef₁ : K*(2:ℝ)^t*((2:ℝ)^N)^ |
| ((τ*Δ+(((h+3:ℕ):ℝ)+t)/(N:ℝ))+2*(((h+3:ℕ):ℝ)+1)*lam)*σ^(-t) ≤ |
| (1/(2:ℝ)^M)^(-ε) := by |
| convert htcoef N hNt hNp M Δ lam hγΔ hlamτ hNM using 1; ring |
| have hreverseOuter : ((2:ℝ)^N)^(τ*γ) ≤ ((2:ℝ)^N)^η := by |
| apply Real.rpow_le_rpow_of_exponent_le (one_le_pow₀ (by norm_num)) |
| have hh := mul_le_mul_of_nonneg_right hL (mul_nonneg hτ.le hγ.le) |
| dsimp [η] |
| nlinarith only [hh,hη] |
| have hreverseMiddle : ((2:ℝ)^N)^(τ*γ) ≤ (1/(2:ℝ)^M)^(-ε) := by |
| rw [inverse_dyadic_neg_power] |
| apply dyadic_relative_scale_bound |
| have hh := contact_loss_relative_budget hτ.le hγΔ hlamτ hτε hNM |
| have hx : 0 ≤ (N:ℝ)*(τ*Δ+2*(((h+3:ℕ):ℝ)+1)*lam) := by positivity |
| nlinarith only [hh,hx] |
| let ρ := sourceClass Q μ N B p₀ |
| have : IsProbabilityMeasure ρ := regularization_class_probability Q hside μ N B hp₀ |
| have hρA : ρ Aᶜ = 0 := sourceClass_absolutelyContinuous Q μ N B p₀ hAfull |
| obtain ⟨G₀,hG₀,hG₀G,hG₀core,hnull⟩ := |
| source_class_pair_core Q hside μ ν N B p₀ hp₀ A P hA hP hAfull hPfull G hG |
| have hcore : G₀ ⊆ Q.occupiedTreeCore ρ N ×ˢ P := fun _ hz => ⟨(hG₀core hz).1.2,(hG₀core hz).2⟩ |
| have hdiag {z x : Space (h+3)} (hz : (z,x) ∈ G₀) : z ≠ x := by |
| have hh := hsep z (hG₀core hz).1.1 x (hG₀core hz).2 |
| intro he |
| simp only [he,sub_self,norm_zero] at hh |
| exact (not_le_of_gt hs) hh |
| let cost : ℕ → ℝ≥0∞ := fun j => if j = 1 then 2*ENNReal.ofReal ((1/(2:ℝ)^M)^gain) |
| else ENNReal.ofReal (((2:ℝ)^N)^(-(2*η))) |
| have hlocal : ∀ j < 3, ∀ p ∈ Q.occupiedPaths ρ (threeIntervalDepth a b N j), |
| ∃ H : Set (Space (h+3) × Space (h+3)), MeasurableSet H ∧ |
| ((ProbabilityTheory.cond ρ (Q.descendant p).carrier).prod ν) |
| ((G₀ ∩ ((Q.descendant p).carrier ×ˢ univ)) \ H) ≤ cost j ∧ |
| ∀ z ∈ H, radialCollisionMass (ProbabilityTheory.cond ρ (Q.descendant p).carrier) |
| (C/(2:ℝ)^(threeIntervalDepth a b N (j+1))) z ≤ |
| threeIntervalFactor a b N Jm q (gain/2) η j := by |
| intro j hj p hp |
| have hpPos : 0 < ρ (Q.descendant p).carrier := (Finset.mem_filter.mp hp).2 |
| have hpLen : p.length = threeIntervalDepth a b N j := |
| (mem_cubePaths _ _ _).mp (Finset.mem_filter.mp hp).1 |
| let V := G₀ ∩ ((Q.descendant p).carrier ×ˢ univ) |
| have hV : MeasurableSet V := hG₀.inter ((Q.descendant p).measurable.prod MeasurableSet.univ) |
| have hVG : V ⊆ G := inter_subset_left.trans hG₀G |
| have hVdiag : ∀ z x, (z,x) ∈ V → x ≠ z := fun _ _ hz => (hdiag hz.1).symm |
| have hVpin : ∀ z x, (z,x) ∈ V → ∀ y ∈ (Q.descendant p).carrier, ‖y-x‖ ≤ D := |
| fun _ _ hz y hy => hpin y (Q.descendant_subset p hy) _ (hG₀core hz.1).2 |
| have hac := sourceClass_conditional_absolutelyContinuous Q μ N B p₀ p |
| interval_cases j |
| · have hlen : p.length = 0 := by simpa [threeIntervalDepth] using hpLen |
| obtain ⟨H,hH,_,hd,hk⟩ := houter N hNo η hη 0 a (by omega) (hab.trans hbN) |
| μ ν B hNp hB lam (τ*Δ) hblock hblock' p₀ hp₀ p hlen hpPos |
| (by intro n _ hn; simpa only [Nat.sub_zero,sourceProfile,Nat.cast_zero,regularizedProfile_zero,sub_zero] using hi₀ n hn) |
| hcoef₀ V hV hVdiag hVpin |
| (reverse_caps_all_radii μ _ ν hac hVG (hab.trans hbN) hreverseOuter hcap) |
| refine ⟨H,hH,?_,?_⟩ |
| · simpa [cost] using hd |
| · simpa [threeIntervalDepth,threeIntervalFactor] using hk |
| · have hlen : p.length = a := by simpa [threeIntervalDepth] using hpLen |
| have hp' : p ∈ Q.occupiedPaths ρ a := by simpa [threeIntervalDepth] using hp |
| let w₀ : Sphere (h+3) := ⟨EuclideanSpace.basisFun (Fin (h+3)) ℝ 0,by |
| simpa only [Metric.mem_sphere,dist_zero_right] using |
| (EuclideanSpace.basisFun (Fin (h+3)) ℝ).norm_eq_one 0⟩ |
| obtain ⟨H,hH,_,hd,hk⟩ := hmiddle M hM Q hQ μ ν (by simp) N B hNp hB lam |
| (τ*Δ+(((h+3:ℕ):ℝ)+t)/(N:ℝ)) σ (by positivity) hlam hσ |
| (cellContraction_le_one _) (cellContraction_sqrt _) hblock hblock' p₀ hp₀ |
| a b hab hbN (Nat.sub_le _ _) p hlen hpPos |
| (by intro n hn hn'; have hh := hi₁ n hn hn'; linarith only [hh]) hcoef₁ w₀ V hV |
| (fun z x hz => ⟨hdiag hz.1,hz.2.1⟩) |
| (by |
| have hmN : M ≤ N := (Nat.sub_le _ _).trans ((Nat.sub_le _ _).trans hbN) |
| have hcm := reverse_caps_transfer μ _ ν hac hVG hmN hreverseMiddle hcap |
| have hsm := reverse_slabs_transfer μ _ ν hac hVG hmN hreverseMiddle hslab |
| filter_upwards [hcm,hsm] with z hz hs' |
| exact ⟨by simpa only [hq] using hz,hs'⟩) |
| (s/2) D (C/(2:ℝ)^b) (by positivity) hfr hwi (by |
| intro x hx |
| obtain ⟨z,hz⟩ := hx |
| refine ⟨occupied_cell_center_separation Q hside ρ A P hρA hsep hsmallA hp' |
| x (hG₀core hz.1).2,?_⟩ |
| exact ProbabilityTheory.ae_cond_of_forall_mem (Q.descendant p).measurable |
| (fun y hy => hVpin z x hz y hy)) |
| refine ⟨H,hH,?_,?_⟩ |
| · simpa [cost] using hd |
| · simpa [threeIntervalDepth,threeIntervalFactor,M,hq] using hk |
| · have hlen : p.length = b := by simpa [threeIntervalDepth] using hpLen |
| obtain ⟨H,hH,_,hd,hk⟩ := houter N hNo η hη b N hJoB le_rfl |
| μ ν B hNp hB lam (τ*Δ+(((h+3:ℕ):ℝ)+q)/(N:ℝ)) hblock hblock' p₀ hp₀ p hlen hpPos |
| (by intro n hn hn'; have hh := hi₂ n hn hn'; linarith only [hh]) |
| hcoef₂ V hV hVdiag hVpin (reverse_caps_all_radii μ _ ν hac hVG le_rfl hreverseOuter hcap) |
| refine ⟨H,hH,?_,?_⟩ |
| · simpa [cost] using hd |
| · simpa [threeIntervalDepth,threeIntervalFactor] using hk |
| have hβ : (Δ/2)*(N:ℝ) ≤ (b-a:ℕ) := by |
| have hh : ((b-a)-Jm:ℕ) ≤ b-a := Nat.sub_le _ _ |
| have hh' : (M:ℝ) ≤ (b-a:ℕ) := by exact_mod_cast hh |
| nlinarith only [hh',hNM] |
| have hconst' : ((2:ℝ)^Jm)^(q+gain/2) ≤ ((2:ℝ)^N)^ξ := |
| (hconst N hNc).trans (Real.rpow_le_rpow_of_exponent_le |
| (one_le_pow₀ (by norm_num)) (mul_le_mul_of_nonneg_left hγΔ hτ.le)) |
| have hbudget : 20*η+4*ξ+gain*γ/8 ≤ (gain/2)*(Δ/2) := |
| three_interval_relative_budget hΔ.le hγΔ hgain.le hτgain |
| obtain ⟨G',hG',hsub,hdel,hcaps⟩ := three_interval_dyadic_gain ρ ν Q hside hab hbN hJm |
| A P hρA hs hsep G₀ hG₀ hcore |
| (fun j => C/(2:ℝ)^(threeIntervalDepth a b N (j+1))) cost |
| (by positivity : 0 ≤ gain/2) hβ hconst' hbudget |
| (fun j hj => hsmallAll (threeIntervalDepth_child hab hbN hj)) |
| (fun j hj => by |
| rw [hQ] |
| exact dyadic_collision_width (h+3) hs le_rfl |
| (threeIntervalDepth_le hab hbN (by omega))) hlocal |
| refine ⟨G',hG',hsub.trans hG₀G,?_,hcaps⟩ |
| have hz : (ρ.prod ν) (G \ G₀) ≤ 0 := hnull.le |
| have htotal := relative_pruning_cost_trans (ρ.prod ν) G G₀ G' hz hdel |
| rw [zero_add] at htotal |
| apply htotal.trans |
| have hc := hcost N hNd M Δ hγΔ hNM |
| have he : (∑ j ∈ Finset.range 3, cost j)+4*ENNReal.ofReal (((2:ℝ)^N)^(-ξ)) = |
| ENNReal.ofReal (2*((2:ℝ)^N)^(-(2*η))+2*(1/(2:ℝ)^M)^gain+ |
| 4*((2:ℝ)^N)^(-ξ)) := by |
| rw [ENNReal.ofReal_add (by positivity) (by positivity), |
| ENNReal.ofReal_add (by positivity) (by positivity)] |
| simp only [Finset.sum_range_succ,Finset.sum_range_zero,cost,show (0:ℕ) ≠ 1 by omega, |
| show (2:ℕ) ≠ 1 by omega,ite_false,ite_true,zero_add, |
| ENNReal.ofReal_mul (by norm_num : (0:ℝ) ≤ 2), |
| ENNReal.ofReal_mul (by norm_num : (0:ℝ) ≤ 4),ENNReal.ofReal_ofNat] |
| ring |
| rw [he] |
| exact ENNReal.ofReal_le_ofReal hc |
|
|
| end Falconer.Regularization |
|
|
| end OAI |
|
|
