| 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 |
|
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| end OAI |
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