File size: 4,867 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 77 78 79 80 81 82 83 84 | import OAI.MeasureTheory.Falconer.Shells.ActualComparableShellDistance
import OAI.MeasureTheory.Falconer.Measures.ComparableNormalization
namespace OAI
namespace PlanarFalconer
open Set MeasureTheory
open scoped ENNReal ContDiff
attribute [local instance] Classical.propDecidable
theorem uniform_actual_comparable_shell_normalized {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 →
∀ (B₀ : ℕ) (δ lam : ℝ),
(∀ l, RegularizedProfileData (Q l) (ρ l) N B₀ δ lam m) → a+n ≤ N →
∀ (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,[],[]⟩
‖((2:ℝ)^(2*((v:ℝ)-a))/(((2:ℝ)^N)^(-m a))^4) •
(∑ 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)‖ ≤
H * ((2:ℝ)^N)^(-2*(m (a+n)-(a+n:ℕ)/N-(m a-(a:ℝ)/N))+12*lam) *
((D 0).separatedMaximum (D 1) (sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B n) (a+n)).toReal := by
obtain ⟨H,hH,N₀,h₀⟩ := uniform_actual_comparable_shell_distance hΨ hsΨ hsig K o hJ
let C : ℝ≥0∞ := ENNReal.ofReal H*binKernelConstant*257025537024256*4096
have hC : C ≠ ⊤ := by
dsimp only [C]
exact ENNReal.mul_ne_top
(ENNReal.mul_ne_top (ENNReal.mul_ne_top ENNReal.ofReal_ne_top binKernelConstant_ne_top) (by norm_num))
(by norm_num)
refine ⟨C.toReal,ENNReal.toReal_nonneg,max 1 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 B₀ δ lam hd hp 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 hNp : 0 < N := lt_of_lt_of_le (by norm_num : 0 < 1) ((le_max_left _ _).trans hN)
have he := h₀ N ((le_max_right _ _).trans hN) Q hQ ρ 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
have hc := sourceProductVertices_profile_count (Q 0) (Q 1) (hQ 0) (hQ 1) (ρ 0) (ρ 1)
N B₀ δ lam m m (hd 0) (hd 1) A B a n hA hB hp
have hcc : ((sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B n).card:ℝ) ≤
4096*((2:ℝ)^N)^(2*(m (a+n)-m a)+12*lam) := by
convert hc using 1
congr 2
ring
have hn := comparable_normalized_bound N hNp m a (a+n:ℕ) v lam
(sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B n).card
(ENNReal.ofReal H*binKernelConstant)
((D 0).separatedMaximum (D 1) (sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B n) (a+n))
_ (by simpa only [Nat.cast_add] using hcc) (by simpa only [Nat.cast_add] using he)
have hmax := PositiveDistanceData.separatedMaximum_ne_top (D 0) (D 1)
(sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B n) (a+n)
have hfin : C * ENNReal.ofReal (((2:ℝ)^N)^(-2*(m (a+n)-(a+n:ℕ)/N-(m a-(a:ℝ)/N))+12*lam)) *
(D 0).separatedMaximum (D 1) (sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B n) (a+n) ≠ ⊤ :=
ENNReal.mul_ne_top (ENNReal.mul_ne_top hC ENNReal.ofReal_ne_top) hmax
simp only [Nat.cast_add] at hfin hn
have ht := ENNReal.toReal_mono hfin hn
have hrpow : 0 ≤ ((2:ℝ)^N)^(-2*(m ((a:ℝ)+n)-((a:ℝ)+n)/N-(m a-(a:ℝ)/N))+12*lam) := by positivity
simpa only [← ofReal_norm,ENNReal.toReal_ofReal (norm_nonneg _),ENNReal.toReal_mul,
ENNReal.toReal_ofReal hrpow,C,D,Nat.cast_add] using ht
end PlanarFalconer
end OAI
|