File size: 3,725 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 | import OAI.MeasureTheory.Falconer.Shells.ActualComparableShellRebin
import OAI.MeasureTheory.Falconer.Geometry.SeparatedDistanceMaximum
namespace OAI
namespace PlanarFalconer
open Set MeasureTheory
open scoped ENNReal ContDiff
attribute [local instance] Classical.propDecidable
theorem uniform_actual_comparable_shell_distance {Ψ : ℝ → ℝ}
(hΨ : ContDiff ℝ ∞ Ψ) (hsΨ : ∀ t, Ψ t ≠ 0 → t ∈ Ioo (1/4:ℝ) 4)
{sig : ℝ} (hsig : 0 < sig) (K o : ℕ) {J : ℕ} (hJ : 0 < J) :
∃ H : ℝ, 0 ≤ H ∧ ∃ N₀ : ℕ, ∀ N ≥ N₀,
∀ (Q : Fin 2 → DyadicSquare), (∀ l, (Q l).side = 1) →
∀ (ρ : Fin 2 → Measure Plane) [∀ l, IsFiniteMeasure (ρ l)],
(∀ l, ρ l (Q l).carrierᶜ = 0) →
∀ (m : ℝ → ℝ) (q a₀ t₀ a d n v : ℕ),
0 < q → a₀ ≤ a+n → v ≤ t₀ → v ≤ N →
∀ (S : Fin 2 → Finset PolicyDescriptor) (k : Fin 2 → PolicyDescriptor → ℕ),
(∀ l, remainingPolicyDescriptors (fun t => m t-(t:ℝ)/(N:ℝ)) N q a₀ t₀ a ⊆ S l) →
(∀ l, (S l).card ≤ (N+1)^3+1) → (∀ l, ∑ t ∈ S l, k l t ≤ K) →
∀ L : ℕ, (∀ l t, t ∈ S l → descriptorLength t ≤ L) →
∀ L₀ : ℝ, (v:ℝ)-(a+d:ℕ) = 2*L₀ → (L:ℝ) ≤ L₀ → 10*sig*(N:ℝ) ≤ L₀ →
∀ (lam : ℝ) (i : Fin 2 → ℕ) (hsp : Fin 2 → Plane → ℂ),
(∀ l x, ‖hsp l x‖ ≤ 1) →
∀ A B : CellPath, A.length = a → B.length = a →
let β := fun l => inheritedCircleSymbol (S l) id (k l) (Q l) (ρ l) (m)
N (lam) sig J (i l) (hsp l)
let D : Fin 2 → PositiveDistanceData := fun l =>
⟨Q l,ρ l,m,N,q,a+n,v,lam,sig,J,i l+1,[],[]⟩
‖∑ row ∈ sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B d,
∑ col ∈ sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B d,
if ¬ angularNear (Q 0) (Q 1) (a+d) row col then
∫ x in angularLengthShell (a+n), angularStationarySpatialKernel (β 0) (β 1) o v Ψ x
∂angularFourCellMeasure (Q 0) (Q 1) (ρ 0) (ρ 1) row col
else 0‖ₑ ≤
ENNReal.ofReal H * binKernelConstant *
ENNReal.ofReal ((((2:ℝ)^N)^(-m (a+n)))^4*(2:ℝ)^(-2*((v:ℝ)-(a+n:ℕ)))) *
((sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B n).card:ℝ≥0∞) *
257025537024256 *
(D 0).separatedMaximum (D 1) (sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B n) (a+n) := by
obtain ⟨H,hH,N₀,h₀⟩ := uniform_actual_comparable_shell_rebin hΨ hsΨ hsig K o hJ
refine ⟨H,hH,N₀,?_⟩
intro N hN Q hQ ρ inst hcarry m q a₀ t₀ a d n v hq ha₀ ht hv S k hS hcard hK L hL
L₀ hvh hLL hL₀ lam i hsp hspb A B hA hB
let D : Fin 2 → PositiveDistanceData := fun l =>
⟨Q l,ρ l,m,N,q,a+n,v,lam,sig,J,i l+1,[],[]⟩
have he := h₀ N hN Q hQ ρ hcarry (fun _ => m) q a₀ t₀ a d n v hq ha₀ ht hv
S k hS hcard hK L hL L₀ hvh hLL hL₀ (fun _ => lam) i hsp hspb A B hA hB
have hn : (D 0).normalization = (D 1).normalization := rfl
have hk := PositiveDistanceData.rebinned_positive_kernel_max (D 0) (D 1) rfl hn A B n (a+n)
have hc := PositiveDistanceData.separated_sum_le_maximum (D 0) (D 1) (hQ 0) (hQ 1) (a+n)
(sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B n)
(sourceProductVertices_subset _ _ _ _ _ _ a n hA hB)
dsimp only at he ⊢
rw [mul_assoc] at he
have hh := he.trans (mul_le_mul_right hk (ENNReal.ofReal H))
have hh' := hh.trans (mul_le_mul_right
(mul_le_mul_right hc (binKernelConstant * ENNReal.ofReal
((D 0).levelMass^4*(2:ℝ)^(-2*((v:ℝ)-(a+n:ℕ)))))) (ENNReal.ofReal H))
simpa only [D,PositiveDistanceData.levelMass,mul_assoc,Nat.cast_add] using hh'
end PlanarFalconer
end OAI
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