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import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphFixedAngularBudget
namespace OAI
/-! Graph energies and pair estimates. -/
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace CurrentFalconer.TreeND
open Set MeasureTheory Attempt4FullFar
open scoped ComplexConjugate RealInnerProductSpace ContDiff
variable {d : ℕ}
lemma graph_const_one_jet (j : ℕ) (w : EuclideanSpace ℝ (Fin d)) :
‖iteratedFDeriv ℝ j (fun _ : EuclideanSpace ℝ (Fin d) => (1:ℂ)) w‖ ≤ 1 := by
rcases j with _ | j
· simp
· simp [iteratedFDeriv_succ_const]
theorem graph_direct_nonedge_decay {sig c₀ : ℝ} (hsig : 0 < sig) (hc₀ : 0 < c₀) (T : ℝ) :
∃ m : ℕ, ∃ C : ℝ, 0 ≤ C ∧ ∃ N₀ : ℕ, ∀ N ≥ N₀,
∀ (a p : ℕ) (v L : ℝ), (p:ℝ)-a ≤ v-p+2*sig*N → L ≤ v-p+2*sig*N → 0 ≤ v-p →
let R := (2:ℝ)^N
let δ := min c₀ (((2:ℝ)^((p:ℝ)-a))⁻¹*R^(10*sig))
let τ := (2:ℝ)^(-(p:ℝ))*R^(30*sig)
let S := R^sig*max 1 ((2:ℝ)^L*R^(-50*sig))
∀ (ρ κ : Measure (EuclideanSpace ℝ (Fin d))), ∀ [IsFiniteMeasure ρ] [IsFiniteMeasure κ],
∀ (X Y : CellPath d) (row col : CellPath d × CellPath d)
(M₁ M₂ H K B : ℝ) (Ψ : ℝ → ℝ)
(b c : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ) (k₁ k₂ : Fin d → ℤ),
X ∈ dyadicPaths a → Y ∈ dyadicPaths a →
row ∈ (graphChildren ρ a p X).product (graphChildren κ a p Y) →
col ∈ (graphChildren ρ a p X).product (graphChildren κ a p Y) →
0 ≤ M₁ → 0 ≤ M₂ → 0 ≤ H →
(graphCellLaw ρ (treeCell row.1) M₁).real univ ≤ H →
(graphCellLaw ρ (treeCell col.1) M₁).real univ ≤ H →
(graphCellLaw κ (treeCell row.2) M₂).real univ ≤ H →
(graphCellLaw κ (treeCell col.2) M₂).real univ ≤ H →
k₁ ∈ sphereCapGridND d δ → k₂ ∈ sphereCapGridND d δ →
ContDiff ℝ ∞ Ψ → tsupport Ψ ⊆ Icc (1/4:ℝ) 4 →
(∫ u : ℝ, ‖(Ψ u:ℂ)‖) ≤ K →
(∀ u, ‖iteratedDeriv m (fun u => (Ψ u:ℂ)) u‖ ≤ K) → 0 ≤ K →
GraphSymbolJet b m S B → GraphSymbolJet c m S B →
¬retainedHGraph (eligibleCells ρ a p X (tensorCapND δ k₁) b)
(eligibleCells κ a p Y (tensorCapND δ k₂) c)
(sphereCapCenterND δ k₁) (sphereCapCenterND δ k₂) τ row col →
‖graphDirectCellTerm ρ κ M₁ M₂ b c Ψ v δ k₁ k₂ row col‖ ≤ C*K*B^4*H^4*R^(-T)*(2:ℝ)^v := by
obtain ⟨m,C,hC,N₀,hdec⟩ := graph_cap_four_decay d hsig hc₀
(by positivity : (0:ℝ) ≤ 2*(d+1:ℝ)) T
refine ⟨m,C,hC,N₀,?_⟩
intro N hN a p v L hh hL hv R δ τ S ρ κ _ _ X Y row col M₁ M₂ H K B Ψ b c k₁ k₂
hX hY hr hc hM₁ hM₂ hH hP hP' hQ hQ' hk₁ hk₂ hΨ hsΨ hK hdΨ hK0 hb hcj hne
by_cases hin : row ∈ (eligibleCells ρ a p X (tensorCapND δ k₁) b).product
(eligibleCells κ a p Y (tensorCapND δ k₂) c) ∧
col ∈ (eligibleCells ρ a p X (tensorCapND δ k₁) b).product
(eligibleCells κ a p Y (tensorCapND δ k₂) c)
· have hn : ¬retainedCenterTests (sphereCapCenterND δ k₁) (sphereCapCenterND δ k₂)
(treeCenter row.1) (treeCenter row.2) (treeCenter col.1) (treeCenter col.2) τ :=
fun h => hne ⟨hin.1,hin.2,h⟩
have hn' : ¬(‖(treeCenter row.1-treeCenter col.1)-inner ℝ (treeCenter row.1-treeCenter col.1)
(sphereCapCenterND δ k₁) • sphereCapCenterND δ k₁‖ ≤ τ ∧
‖(treeCenter row.2-treeCenter col.2)-inner ℝ (treeCenter row.2-treeCenter col.2)
(sphereCapCenterND δ k₂) • sphereCapCenterND δ k₂‖ ≤ τ ∧
|inner ℝ (sphereCapCenterND δ k₁) (treeCenter row.1-treeCenter col.1)+
inner ℝ (sphereCapCenterND δ k₂) (treeCenter row.2-treeCenter col.2)| ≤ τ) := by
simpa only [retainedCenterTests,graphPerpND,real_inner_comm] using hn
have hr' := Finset.mem_product.mp hr
have hc' := Finset.mem_product.mp hc
have hgeo : ∀ᵐ z ∂((graphCellLaw ρ (treeCell row.1) M₁).prod (graphCellLaw ρ (treeCell col.1) M₁)).prod
((graphCellLaw κ (treeCell row.2) M₂).prod (graphCellLaw κ (treeCell col.2) M₂)),
‖z.1.1-z.1.2‖ ≤ 2*(d+1:ℝ)*(2:ℝ)^(-(a:ℝ)) ∧
‖z.2.1-z.2.2‖ ≤ 2*(d+1:ℝ)*(2:ℝ)^(-(a:ℝ)) ∧
‖(z.1.1-z.1.2)-(treeCenter row.1-treeCenter col.1)‖ ≤ 2*(d+1:ℝ)*(2:ℝ)^(-(p:ℝ)) ∧
‖(z.2.1-z.2.2)-(treeCenter row.2-treeCenter col.2)‖ ≤ 2*(d+1:ℝ)*(2:ℝ)^(-(p:ℝ)) := by
filter_upwards [graphCellLaw_four_ae ρ ρ κ κ (treeCell row.1) (treeCell col.1) (treeCell row.2) (treeCell col.2)
M₁ M₁ M₂ M₂ (measurableSet_treeCell row.1)
(measurableSet_treeCell col.1) (measurableSet_treeCell row.2) (measurableSet_treeCell col.2)] with z hz
refine ⟨?_,?_,graphChildren_center_error hr'.1 hc'.1 hz.1 hz.2.1,
graphChildren_center_error hr'.2 hc'.2 hz.2.2.1 hz.2.2.2⟩
· exact (graphChildren_direct_size hX hr'.1 hc'.1 hz.1 hz.2.1).trans (by nlinarith [Real.rpow_pos_of_pos (by norm_num : (0:ℝ)<2) (-(a:ℝ))])
· exact (graphChildren_direct_size hY hr'.2 hc'.2 hz.2.2.1 hz.2.2.2).trans (by nlinarith [Real.rpow_pos_of_pos (by norm_num : (0:ℝ)<2) (-(a:ℝ))])
have hsΨ' : tsupport (fun u => (Ψ u:ℂ)) ⊆ Icc (1/4:ℝ) 4 := by
have he : tsupport (fun u => (Ψ u:ℂ)) = tsupport Ψ := by
change closure (Function.support _) = closure (Function.support _)
congr 1
ext u
simp only [Function.mem_support,ne_eq,Complex.ofReal_eq_zero]
rwa [he]
have h := hdec N hN a p v L hh hL hv
(graphCellLaw ρ (treeCell row.1) M₁) (graphCellLaw ρ (treeCell col.1) M₁)
(graphCellLaw κ (treeCell row.2) M₂) (graphCellLaw κ (treeCell col.2) M₂)
(fun u => (Ψ u:ℂ)) K B 1 k₁ k₂ (treeCenter row.1-treeCenter col.1)
(treeCenter row.2-treeCenter col.2) (fun _ => 1) (fun _ => 1) b b c c
hk₁ hk₂ (Complex.ofRealCLM.contDiff.comp hΨ) hsΨ' hK hdΨ hK0 (by norm_num)
contDiff_const contDiff_const (fun j _ w _ => graph_const_one_jet j w)
(fun j _ w _ => graph_const_one_jet j w) hb hb hcj hcj hgeo hn'
simp only [graphCellLaw_transform _ _ hM₁,graphCellLaw_transform _ _ hM₂] at h
rw [←graphDirectCellTerm_eq] at h
simp only [one_pow,mul_one] at h
have hmass : (((graphCellLaw ρ (treeCell row.1) M₁).prod (graphCellLaw ρ (treeCell col.1) M₁)).prod
((graphCellLaw κ (treeCell row.2) M₂).prod (graphCellLaw κ (treeCell col.2) M₂))).real univ ≤ H^4 := by
simp only [Measure.real,←Set.univ_prod_univ,Measure.prod_prod,ENNReal.toReal_mul]
calc
_ ≤ (H*H)*(H*H) := mul_le_mul (mul_le_mul hP hP' measureReal_nonneg hH)
(mul_le_mul hQ hQ' measureReal_nonneg hH) (by positivity) (by positivity)
_ = H^4 := by ring
calc
_ ≤ _ := h
_ ≤ C*K*B^4*R^(-T)*(2:ℝ)^v*H^4 := mul_le_mul_of_nonneg_left hmass (by positivity)
_ = _ := by ring
· rw [graph_direct_ineligible_zero ρ κ hr hc M₁ M₂ hb hcj Ψ v δ k₁ k₂ hin,norm_zero]
positivity
end CurrentFalconer.TreeND
end
end
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace CurrentFalconer.TreeND
open Set MeasureTheory Attempt4FullFar
open scoped ComplexConjugate RealInnerProductSpace ContDiff
variable {d : ℕ}
theorem graph_crossed_nonedge_decay {sig c₀ G : ℝ} (hsig : 0 < sig) (hc₀ : 0 < c₀) (hG : 0 ≤ G) (T : ℝ) :
∃ m : ℕ, ∃ C : ℝ, 0 ≤ C ∧ ∃ N₀ : ℕ, ∀ N ≥ N₀,
∀ (a p : ℕ) (v L : ℝ), (p:ℝ)-a ≤ v-p+2*sig*N → L ≤ v-p+2*sig*N → 0 ≤ v-p →
let R := (2:ℝ)^N
let δ := min c₀ (((2:ℝ)^((p:ℝ)-a))⁻¹*R^(10*sig))
let τ := (2:ℝ)^(-(p:ℝ))*R^(30*sig)
let S := R^sig*max 1 ((2:ℝ)^L*R^(-50*sig))
∀ (ρ κ : Measure (EuclideanSpace ℝ (Fin d))), ∀ [IsFiniteMeasure ρ] [IsFiniteMeasure κ],
∀ (X Y : CellPath d) (row col : CellPath d × CellPath d)
(M₁ M₂ H K B P : ℝ) (Ψ : ℝ → ℝ) (ψ : EuclideanSpace ℝ (Fin d) → ℂ)
(b c : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ) (k₁ k₂ : Fin d → ℤ),
X ∈ dyadicPaths a → Y ∈ dyadicPaths a →
‖treeCenter X-treeCenter Y‖ ≤ G*(2:ℝ)^(-(a:ℝ)) →
row ∈ (graphChildren ρ a p X).product (graphChildren κ a p Y) →
col ∈ (graphChildren ρ a p X).product (graphChildren κ a p Y) →
0 ≤ M₁ → 0 ≤ M₂ → 0 ≤ H →
(graphCellLaw ρ (treeCell row.1) M₁).real univ ≤ H →
(graphCellLaw ρ (treeCell col.1) M₁).real univ ≤ H →
(graphCellLaw κ (treeCell row.2) M₂).real univ ≤ H →
(graphCellLaw κ (treeCell col.2) M₂).real univ ≤ H →
k₁ ∈ sphereCapGridND d δ → k₂ ∈ sphereCapGridND d δ →
ContDiff ℝ ∞ Ψ → tsupport Ψ ⊆ Icc (1/4:ℝ) 4 →
(∫ u : ℝ, ‖(Ψ u:ℂ)‖) ≤ K →
(∀ u, ‖iteratedDeriv m (fun u => (Ψ u:ℂ)) u‖ ≤ K) → 0 ≤ K →
0 ≤ P → ContDiff ℝ ∞ ψ →
(∀ j ≤ m, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j ψ w‖ ≤ P) →
GraphSymbolJet b m S B → GraphSymbolJet c m S B →
¬retainedHFGraph (eligibleCells ρ a p X (tensorCapND δ k₁) b)
(eligibleCells ρ a p X (tensorCapND δ k₂) b)
(eligibleCells κ a p Y (tensorCapND δ k₁) c)
(eligibleCells κ a p Y (tensorCapND δ k₂) c)
(sphereCapCenterND δ k₁) (sphereCapCenterND δ k₂) τ row col →
‖graphCrossedCellTerm ρ κ M₁ M₂ b c Ψ ψ v δ k₁ k₂ row col‖ ≤ C*K*P^2*B^4*H^4*R^(-T)*(2:ℝ)^v := by
obtain ⟨m,C,hC,N₀,hdec⟩ := graph_cap_four_decay d hsig hc₀
(by positivity : (0:ℝ) ≤ 2*(d+1:ℝ)+G) T
refine ⟨m,C,hC,N₀,?_⟩
intro N hN a p v L hh hL hv R δ τ S ρ κ _ _ X Y row col M₁ M₂ H K B P Ψ ψ b c k₁ k₂
hX hY hcenters hr hc hM₁ hM₂ hH hP hP' hQ hQ' hk₁ hk₂ hΨ hsΨ hK hdΨ hK0 hP0 hψ hdψ hb hcj hne
by_cases hin : row ∈ (eligibleCells ρ a p X (tensorCapND δ k₁) b).product
(eligibleCells κ a p Y (tensorCapND δ k₂) c) ∧
col ∈ (eligibleCells ρ a p X (tensorCapND δ k₂) b).product
(eligibleCells κ a p Y (tensorCapND δ k₁) c)
· have hn : ¬retainedCenterTests (sphereCapCenterND δ k₁) (sphereCapCenterND δ k₂)
(treeCenter row.1) (treeCenter row.2) (treeCenter col.2) (treeCenter col.1) τ :=
fun h => hne ⟨hin.1,hin.2,h⟩
have hn' : ¬(‖(treeCenter row.1-treeCenter col.2)-inner ℝ (treeCenter row.1-treeCenter col.2)
(sphereCapCenterND δ k₁) • sphereCapCenterND δ k₁‖ ≤ τ ∧
‖(treeCenter row.2-treeCenter col.1)-inner ℝ (treeCenter row.2-treeCenter col.1)
(sphereCapCenterND δ k₂) • sphereCapCenterND δ k₂‖ ≤ τ ∧
|inner ℝ (sphereCapCenterND δ k₁) (treeCenter row.1-treeCenter col.2)+
inner ℝ (sphereCapCenterND δ k₂) (treeCenter row.2-treeCenter col.1)| ≤ τ) := by
simpa only [retainedCenterTests,graphPerpND,real_inner_comm] using hn
have hr' := Finset.mem_product.mp hr
have hc' := Finset.mem_product.mp hc
have hgeo : ∀ᵐ z ∂((graphCellLaw ρ (treeCell row.1) M₁).prod (graphCellLaw κ (treeCell col.2) M₂)).prod
((graphCellLaw κ (treeCell row.2) M₂).prod (graphCellLaw ρ (treeCell col.1) M₁)),
‖z.1.1-z.1.2‖ ≤ (2*(d+1:ℝ)+G)*(2:ℝ)^(-(a:ℝ)) ∧
‖z.2.1-z.2.2‖ ≤ (2*(d+1:ℝ)+G)*(2:ℝ)^(-(a:ℝ)) ∧
‖(z.1.1-z.1.2)-(treeCenter row.1-treeCenter col.2)‖ ≤ (2*(d+1:ℝ)+G)*(2:ℝ)^(-(p:ℝ)) ∧
‖(z.2.1-z.2.2)-(treeCenter row.2-treeCenter col.1)‖ ≤ (2*(d+1:ℝ)+G)*(2:ℝ)^(-(p:ℝ)) := by
filter_upwards [graphCellLaw_four_ae ρ κ κ ρ (treeCell row.1) (treeCell col.2) (treeCell row.2) (treeCell col.1)
M₁ M₂ M₂ M₁ (measurableSet_treeCell row.1)
(measurableSet_treeCell col.2) (measurableSet_treeCell row.2) (measurableSet_treeCell col.1)] with z hz
have hrev : ‖treeCenter Y-treeCenter X‖ ≤ G*(2:ℝ)^(-(a:ℝ)) := by rwa [norm_sub_rev]
refine ⟨graphChildren_cross_size hX hY hr'.1 hc'.2 hcenters hz.1 hz.2.1,
graphChildren_cross_size hY hX hr'.2 hc'.1 hrev hz.2.2.1 hz.2.2.2,?_,?_⟩
· exact (graphChildren_center_error hr'.1 hc'.2 hz.1 hz.2.1).trans
(by nlinarith [Real.rpow_pos_of_pos (by norm_num : (0:ℝ)<2) (-(p:ℝ))])
· exact (graphChildren_center_error hr'.2 hc'.1 hz.2.2.1 hz.2.2.2).trans
(by nlinarith [Real.rpow_pos_of_pos (by norm_num : (0:ℝ)<2) (-(p:ℝ))])
have hsΨ' : tsupport (fun u => (Ψ u:ℂ)) ⊆ Icc (1/4:ℝ) 4 := by
have he : tsupport (fun u => (Ψ u:ℂ)) = tsupport Ψ := by
change closure (Function.support _) = closure (Function.support _)
congr 1
ext u
simp only [Function.mem_support,ne_eq,Complex.ofReal_eq_zero]
rwa [he]
have h := hdec N hN a p v L hh hL hv
(graphCellLaw ρ (treeCell row.1) M₁) (graphCellLaw κ (treeCell col.2) M₂)
(graphCellLaw κ (treeCell row.2) M₂) (graphCellLaw ρ (treeCell col.1) M₁)
(fun u => (Ψ u:ℂ)) K B P k₁ k₂ (treeCenter row.1-treeCenter col.2)
(treeCenter row.2-treeCenter col.1) ψ (fun w => conj (ψ w)) b c c b
hk₁ hk₂ (Complex.ofRealCLM.contDiff.comp hΨ) hsΨ' hK hdΨ hK0 hP0
hψ (Complex.conjCLE.contDiff.comp hψ) hdψ (by
intro j hj w hw
change ‖iteratedFDeriv ℝ j (Complex.conjLIE ∘ ψ) w‖ ≤ P
rw [Complex.conjLIE.norm_iteratedFDeriv_comp_left ψ w j]
exact hdψ j hj w hw) hb hcj hcj hb hgeo hn'
simp only [graphCellLaw_transform _ _ hM₁,graphCellLaw_transform _ _ hM₂] at h
rw [←graphCrossedCellTerm_eq] at h
have hmass : (((graphCellLaw ρ (treeCell row.1) M₁).prod (graphCellLaw κ (treeCell col.2) M₂)).prod
((graphCellLaw κ (treeCell row.2) M₂).prod (graphCellLaw ρ (treeCell col.1) M₁))).real univ ≤ H^4 := by
simp only [Measure.real,←Set.univ_prod_univ,Measure.prod_prod,ENNReal.toReal_mul]
calc
_ ≤ (H*H)*(H*H) := mul_le_mul (mul_le_mul hP hQ' measureReal_nonneg hH)
(mul_le_mul hQ hP' measureReal_nonneg hH) (by positivity) (by positivity)
_ = H^4 := by ring
calc
_ ≤ _ := h
_ ≤ C*K*P^2*B^4*R^(-T)*(2:ℝ)^v*H^4 := mul_le_mul_of_nonneg_left hmass (by positivity)
_ = _ := by ring
· rw [graph_crossed_ineligible_zero ρ κ hr hc M₁ M₂ hb hcj Ψ ψ v δ k₁ k₂ hin,norm_zero]
positivity
end CurrentFalconer.TreeND
end
end
end OAI
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