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d4317d7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 | import OAI.MeasureTheory.Falconer.Estimates.EndpointTwoSidedScale
import OAI.MeasureTheory.Falconer.Radial.FiniteRadialRefinement
import OAI.MeasureTheory.Falconer.Radial.RadialScaleWindow
import OAI.MeasureTheory.Falconer.Radial.ReverseSlabsAllRadii
namespace OAI
/-! # The same original pair set can be improved throughout a dyadic scale window -/
namespace Falconer.Regularization
open Set MeasureTheory NormedSpace
open scoped ENNReal BigOperators
theorem endpoint_scale_window_gain (hLW : PublishedFiniteLoomisWhitney)
{h : ℕ} {α t q ζ C D s : ℝ}
(hq : q = (h:ℝ)+1) (ht : (h:ℝ)+3/2 < t) (ht' : t < (h:ℝ)+2)
(hζ : 0 < ζ) (htα : t < α) (hC : 0 ≤ C)
(Q : DyadicCube (h+3)) (hQ : Q.side = 1)
(hD : 1 ≤ D) (hDdim : Real.sqrt (h+3:ℕ) ≤ D) (hs : 0 < s) :
∃ ε gain : ℝ, 0 < ε ∧ 0 < gain ∧ ∀ K : ℕ, 0 < K →
∃ c : ℝ, 0 < c ∧ ∃ N₀ : ℕ, ∀ N ≥ N₀,
∀ (μ ν : Measure (Space (h+3))) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν],
μ Q.carrierᶜ = 0 → ν Q.carrierᶜ = 0 →
(∀ y : Space (h+3), ∀ r > 0, μ (Metric.ball y r) ≤ ENNReal.ofReal (C*r^α)) →
(∀ y : Space (h+3), ∀ r > 0, ν (Metric.ball y r) ≤ ENNReal.ofReal (C*r^α)) →
∀ 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 ∈ A ∪ 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)^(ε/(2*(K:ℝ)))*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)^(ε/(2*(K:ℝ)))*r^ζ)) →
(∀ᵐ x ∂ν, ∀ r : ℝ, 1/(2:ℝ)^N ≤ r → r ≤ 1 →
∀ u : Space (h+3), ‖u‖ = 1 →
μ {z | (z,x) ∈ G ∧ dist (normalize (z-x)) u < r} ≤
ENNReal.ofReal (((2:ℝ)^N)^(ε/(2*(K:ℝ)))*r^q)) →
(∀ᵐ x ∂ν, ∀ u : Space (h+3), ‖u‖ = 1 → ∀ r : ℝ,
1/(2:ℝ)^N ≤ r → r ≤ 1 →
μ {z | (z,x) ∈ G ∧ |inner ℝ u (normalize (z-x))| ≤ r} ≤
ENNReal.ofReal (((2:ℝ)^N)^(ε/(2*(K:ℝ)))*r^ζ)) →
∃ G' : Set (Space (h+3) × Space (h+3)), MeasurableSet G' ∧ G' ⊆ G ∧
(μ.prod ν) (G \ G') ≤ 2*ENNReal.ofReal (((2:ℝ)^N)^(-c)) ∧
∀ j : ℕ, N/K ≤ j → j ≤ N →
(∀ x u : Space (h+3), μ {z | (z,x) ∈ G' ∧ ‖normalize (z-x)-u‖ ≤ 1/(2:ℝ)^j} ≤
ENNReal.ofReal ((1/(2:ℝ)^j)^(q+gain))) ∧
(∀ z u : Space (h+3), ν {x | (z,x) ∈ G' ∧ ‖normalize (x-z)-u‖ ≤ 1/(2:ℝ)^j} ≤
ENNReal.ofReal ((1/(2:ℝ)^j)^(q+gain))) := by
obtain ⟨ε,gain,hε,hgain,Ns,hscale⟩ := endpoint_two_sided_one_scale hLW hq ht ht' hζ
htα hC Q hQ hD hDdim hs
refine ⟨ε,gain,hε,hgain,?_⟩
intro K hK
obtain ⟨c,hc,Nc,hcost⟩ := radial_scale_window_cost K hK hgain
refine ⟨c,hc,max (K*Ns) (max (2*K) Nc),?_⟩
intro N hN μ ν _ _ hμroot hνroot hμfrost hνfrost A P hA hP hμA hνP hsep hpin G hG hcap hslab hcap' hslab'
simp only [max_le_iff] at hN
obtain ⟨hNs,hNK,hNc⟩ := hN
have hstart : Ns ≤ N/K := (Nat.le_div_iff_mul_le hK).mpr (by simpa only [Nat.mul_comm] using hNs)
have hcapSwap : ∀ᵐ z ∂ν, ∀ r : ℝ, 1/(2:ℝ)^N ≤ r → r ≤ 1 →
∀ u : Space (h+3), ‖u‖ = 1 →
μ {x | (z,x) ∈ Prod.swap ⁻¹' G ∧ dist (normalize (x-z)) u < r} ≤
ENNReal.ofReal (((2:ℝ)^N)^(ε/(2*(K:ℝ)))*r^q) := by
simpa only [mem_preimage,Prod.swap_prod_mk] using hcap'
have hslabSwap : ∀ᵐ z ∂ν, ∀ u : Space (h+3), ‖u‖ = 1 → ∀ r : ℝ,
1/(2:ℝ)^N ≤ r → r ≤ 1 →
μ {x | (z,x) ∈ Prod.swap ⁻¹' G ∧ |inner ℝ u (normalize (x-z))| ≤ r} ≤
ENNReal.ofReal (((2:ℝ)^N)^(ε/(2*(K:ℝ)))*r^ζ) := by
simpa only [mem_preimage,Prod.swap_prod_mk] using hslab'
obtain ⟨H,hH,hsub,hdel,hbounds⟩ := finite_radial_refinement μ ν G hG (Finset.Icc (N/K) N)
(fun j => 2*ENNReal.ofReal (((2:ℝ)^j)^(-gain)))
(fun j => ENNReal.ofReal ((1/(2:ℝ)^j)^(q+gain))) (fun j => 1/(2:ℝ)^j) (by
intro j hj
obtain ⟨hjlo,hjhi⟩ := Finset.mem_Icc.mp hj
have hloss := radial_scale_window_loss hK hNK hjlo hε.le
have hc₁ := reverse_caps_all_radii μ μ ν Measure.AbsolutelyContinuous.rfl
(Subset.rfl (s := G)) hjhi hloss hcap
have hc₂ := reverse_caps_all_radii ν ν μ Measure.AbsolutelyContinuous.rfl
(Subset.rfl (s := Prod.swap ⁻¹' G)) hjhi hloss hcapSwap
have hs₁ := reverse_slabs_all_radii μ μ ν Measure.AbsolutelyContinuous.rfl
(Subset.rfl (s := G)) hjhi hloss hslab
have hs₂ := reverse_slabs_all_radii ν ν μ Measure.AbsolutelyContinuous.rfl
(Subset.rfl (s := Prod.swap ⁻¹' G)) hjhi hloss hslabSwap
obtain ⟨H,hH,hsub,hdel,hbounds⟩ := hscale j (hstart.trans hjlo) μ ν
hμroot hνroot hμfrost hνfrost A P hA hP hμA hνP hsep hpin G hG hc₁ hs₁
(by simpa only [mem_preimage,Prod.swap_prod_mk] using hc₂)
(by simpa only [mem_preimage,Prod.swap_prod_mk] using hs₂)
refine ⟨H,hH,hsub,hdel,?_⟩
have he : ((2:ℝ)^j)^(-q-gain) = (1/(2:ℝ)^j)^(q+gain) := by
rw [one_div,Real.inv_rpow (by positivity),← Real.rpow_neg (by positivity)]
congr 1
ring
simpa only [he] using hbounds)
refine ⟨H,hH,hsub,?_,fun j hjlo hjhi => hbounds j (Finset.mem_Icc.mpr ⟨hjlo,hjhi⟩)⟩
apply hdel.trans
rw [← Finset.mul_sum]
exact mul_le_mul' le_rfl (hcost N hNc)
end Falconer.Regularization
end OAI
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