File size: 4,268 Bytes
7f37411 | 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 | import OAI.MeasureTheory.Falconer.Fourier.MinimalPairSymbol
import OAI.MeasureTheory.Falconer.Shells.HighShellNormalization
import OAI.MeasureTheory.Falconer.Shells.HighShellCone
import OAI.MeasureTheory.Falconer.Angular.InverseCircleArcLength
namespace OAI
namespace PlanarFalconer
open Set MeasureTheory
open scoped ContDiff Topology ComplexConjugate
namespace PositiveDistanceData
theorem uniform_actual_high_shell_inverse {sig T : ℝ} (hsig : 0 < sig) :
∃ n N₀ : ℕ, ∀ d : PositiveDistanceData, N₀ ≤ d.N → d.sig = sig →
0 < d.Q.side → d.StronglySeparated → 0 < d.q →
d.sig*d.N ≤ (d.q:ℝ) → d.a+100*d.q < d.t → d.t ≤ d.N →
∀ v r : ℝ, (d.t:ℝ)-d.sig*d.N ≤ v → (2:ℝ)^(v-1) ≤ r →
∀ x ∈ (d.Q.descendant d.X).carrier, ∀ y ∈ (d.Q.descendant d.Y).carrier,
‖Complex.exp (Complex.I*((r*dist x y:ℝ):ℂ))*(d.minimalMask (x,y):ℂ) -
sourceInverseCircleIntegral n r (x-y)
(sourceConeCircle (sourceParentAxis (d.Q.descendant d.X) (d.Q.descendant d.Y)))
(d.pairMaskSymbol (x,y))‖ ≤ ((2:ℝ)^d.N)^(-T) := by
obtain ⟨n,hn⟩ := rotated_inverse_circle_power (T := T) (c := 90) (by norm_num) hsig
obtain ⟨P,hP,hPb⟩ := sourceConeCircle_uniform_derivatives (4*(n+1)+2)
obtain ⟨C,hC,N₁,h₁⟩ := uniform_actual_pair_derivatives hsig 0 (4*(n+1)+2)
obtain ⟨N₂,h₂⟩ := hn P C hP hC
refine ⟨n,max N₁ N₂,?_⟩
intro d hN hdsig hQ hs hq hqscale hw ht v r hv hr x hx y hy
subst sig
let S := policyDescriptorList (fun n => d.mass n-(n:ℝ)/(d.N:ℝ)) d.N d.q d.a d.t
let L := (d.t-d.a)/2
have hLN : L ≤ d.N := by dsimp [L]; omega
have hcard : S.card ≤ (d.N+1)^3+1 := (policyDescriptorList_card _ hq).trans (by omega)
have hL : ∀ b ∈ S, descriptorLength b ≤ L := fun b hb => policyDescriptor_length_le_half _ hq hb
have hder := h₁ d.N ((le_max_left _ _).trans hN) S id (fun _ => 0) hcard
(by simp) L hL d.Q hQ d.ρ d.mass d.lam d.J d.i (fun _ => 1) x y (by simp) (by simp)
let f := fun θ => d.pairMaskSymbol (x,y) (sourceUnitDirection θ)
have he : f = fun θ =>
inheritedCircleSymbol S id (fun _ => 0) d.Q d.ρ d.mass d.N d.lam d.sig d.J d.i (fun _ => 1) x (sourceUnitDirection θ) *
conj (inheritedCircleSymbol S id (fun _ => 0) d.Q d.ρ d.mass d.N d.lam d.sig d.J d.i (fun _ => 1) y (sourceUnitDirection θ)) :=
funext (fun θ => d.pairMaskSymbol_inherited (x,y) (sourceUnitDirection θ))
have hf : ContDiff ℝ ∞ f := by rw [he]; exact hder.1
have hb : ∀ j ≤ 4*(n+1)+2, ∀ θ, ‖iteratedDeriv j f θ‖ ≤ C*sourceSymbolScale d.N d.sig L^j := by
rw [he]; exact hder.2
let axis := sourceParentAxis (d.Q.descendant d.X) (d.Q.descendant d.Y)
obtain ⟨φ,hφ,hdir,hgeom⟩ := strong_separation_parent_angle _ _
(d.Q.descendant_side_pos hQ _) (d.Q.descendant_side_pos hQ _)
hs.1 hs.2.1 hs.2.2.1 hx hy
have hdir' : sourceUnitDirection (φ+axis) = sourceRadialUnit y x := by
have hh := congrArg (sourceCircleRotate axis) hdir
simpa only [axis,sourceCircleRotate_comp,sourceCircleRotate_add,neg_add_cancel,
sourceCircleRotate_zero] using hh
have hval : f (φ+axis) = (d.minimalMask (x,y):ℂ) := by
change d.pairMaskSymbol (x,y) (sourceUnitDirection (φ+axis)) = _
rw [hdir',pairMaskSymbol_radial]
have hslack := d.actual_high_shell_stationary_slack hQ hs hsig.le hqscale hw hv hr (z := (x,y)) hx hy
have hψ := sourceConeCircle_rotated_contDiff axis
obtain ⟨hg₁,hg₂⟩ := sourceConeCircle_inverse_germs axis hφ
have hpψ : Function.Periodic (fun θ => sourceConeCircle axis (sourceUnitDirection θ)) (2*Real.pi) :=
fun θ => congrArg (sourceConeCircle axis) (sourceUnitDirection_periodic θ)
have hpf : Function.Periodic f (2*Real.pi) :=
fun θ => congrArg (d.pairMaskSymbol (x,y)) (sourceUnitDirection_periodic θ)
have hh := h₂ d.N ((le_max_right _ _).trans hN) hψ hf hpψ hpf (φ+axis)
hg₁ hg₂ (sourceSymbolScale_one_le d.N hsig.le L) hslack.1
(sourceSymbolScale_upper d.N hsig.le hLN) hslack.2 (hPb axis) hb
rw [hval] at hh
rw [sourceInverseCircleIntegral_eq n r (φ+axis) hgeom hψ.continuous hf]
simpa only [dist_eq_norm] using hh
end PositiveDistanceData
end PlanarFalconer
end OAI
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