| import Mathlib |
| import OAI.MeasureTheory.Falconer.Fourier.DistanceSincSpectralIdentity |
| import OAI.MeasureTheory.Falconer.Model |
| import OAI.MeasureTheory.FalconerCompletion.Measures.CharFunSqIntervalIntegrable |
| import OAI.MeasureTheory.FalconerCompletion.Shells.WeightedDistanceCollisionLower |
| import OAI.MeasureTheory.FalconerCompletion.Shells.WidthKernel |
|
|
| namespace OAI |
|
|
| /-! Dyadic shells and prefix decompositions. -/ |
|
|
| open _root_.OAI.PlanarFalconer |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace GraphLiteralSupport |
| namespace CurrentFalconer.TreeND |
| open Set MeasureTheory PlanarFalconer _root_.OAI.GraphLiteralSupport.Attempt4FullFar |
| open scoped ENNReal |
| variable {d n : ℕ} |
| namespace PositiveDistanceState |
| variable (e : PositiveDistanceState d) |
|
|
| noncomputable def prefixShell (j : ℕ) : ℝ := |
| (2:ℝ)^(-(e.a:ℝ))*(∫ r in Icc ((2:ℝ)^j) ((2:ℝ)^(j+1)), ‖charFun e.weightedDistance r‖^2) |
|
|
| noncomputable def prefixLowIndices : Finset ℕ := by |
| classical |
| exact (Finset.Ico e.a e.t).filter (fun j => e.sig*e.N≤(e.t:ℝ)-(j+1:ℕ)) |
|
|
| noncomputable def prefixHighIndices : Finset ℕ := by |
| classical |
| exact (Finset.Ico e.a e.t).filter (fun j => ¬e.sig*e.N≤(e.t:ℝ)-(j+1:ℕ)) |
|
|
| lemma prefixShell_nonneg (j : ℕ) : 0≤e.prefixShell j := |
| mul_nonneg (by positivity) (integral_nonneg (fun r => sq_nonneg ‖charFun e.weightedDistance r‖)) |
|
|
| lemma prefixShell_le_band [IsFiniteMeasure e.weightedDistance] (j : ℕ) : |
| e.prefixShell j≤(2:ℝ)^(-(e.a:ℝ))* |
| (∫ r in Icc (-((2:ℝ)^(j+1))) ((2:ℝ)^(j+1)), ‖charFun e.weightedDistance r‖^2) := by |
| have hsub : Icc ((2:ℝ)^j) ((2:ℝ)^(j+1))⊆Icc (-((2:ℝ)^(j+1))) ((2:ℝ)^(j+1)) := by |
| intro r hr |
| refine ⟨?_,hr.2⟩ |
| have h₁ : 0<(2:ℝ)^j := by positivity |
| have h₂ : 0<(2:ℝ)^(j+1) := by positivity |
| linarith [hr.1] |
| have hi : Integrable (fun r => ‖charFun e.weightedDistance r‖^2) |
| (volume.restrict (Icc (-((2:ℝ)^(j+1))) ((2:ℝ)^(j+1)))) := by |
| simpa only [one_mul] using charFun_sq_integrable_finite e.weightedDistance |
| (volume.restrict (Icc (-((2:ℝ)^(j+1))) ((2:ℝ)^(j+1)))) 1 |
| exact mul_le_mul_of_nonneg_left |
| (integral_mono_measure (Measure.restrict_mono hsub le_rfl) |
| (Filter.Eventually.of_forall (fun r => sq_nonneg ‖charFun e.weightedDistance r‖)) hi) (by positivity) |
|
|
| lemma prefix_frequency_decomposition [IsFiniteMeasure e.weightedDistance] (hat : e.a≤e.t) : |
| (2:ℝ)^(-(e.a:ℝ))*(∫ r in Icc (-((2:ℝ)^e.t)) ((2:ℝ)^e.t), ‖charFun e.weightedDistance r‖^2)= |
| (2:ℝ)^(-(e.a:ℝ))*(∫ r in Icc (-((2:ℝ)^e.a)) ((2:ℝ)^e.a), ‖charFun e.weightedDistance r‖^2)+ |
| 2*∑ j∈e.prefixLowIndices, e.prefixShell j + 2*∑ j∈e.prefixHighIndices, e.prefixShell j := by |
| classical |
| have hp := Finset.sum_filter_add_sum_filter_not (Finset.Ico e.a e.t) |
| (fun j => e.sig*e.N≤(e.t:ℝ)-(j+1:ℕ)) e.prefixShell |
| change (∑ j∈e.prefixLowIndices, e.prefixShell j)+(∑ j∈e.prefixHighIndices, e.prefixShell j)=_ at hp |
| rw [add_assoc,← mul_add,hp,distance_dyadic_band_decomposition e.weightedDistance hat] |
| simp only [prefixShell,← Finset.mul_sum] |
| ring |
|
|
| end PositiveDistanceState |
|
|
| theorem uniform_actual_prefix_decomposition {sig : ℝ} (hsig : 0<sig) {J : ℕ} (hJ : 0<J) |
| {K : ℝ} (hK : 32*(n+2:ℝ)≤K) : |
| ∃ H : ℝ, 0<H ∧ ∃ N₀ : ℕ, ∀ (e : PositiveDistanceState (n+1)), |
| N₀≤e.N → e.sig=sig → e.J=J → 0<e.q → e.a≤e.t → e.t≤e.N → |
| e.X.length=e.a → e.Y.length=e.a → |
| farSeparatedGridND e.a K (fun j => (pathIndex e.X j:ℤ)) (fun j => (pathIndex e.Y j:ℤ)) → |
| ∀ (B : ℕ) (s lam : ℝ), RegularizedProfileData e.ρ e.N B s lam e.mass → |
| e.coincidence.toReal≤H*(2*((2:ℝ)^e.N)^(4*(n+2:ℝ)*lam)+ |
| 2*(32*binKernelConstant.toReal)* |
| (∑ j∈e.prefixLowIndices, ({e with t:=j+1,i:=e.i+1} : PositiveDistanceState (n+1)).coincidence.toReal)+ |
| 2*∑ j∈e.prefixHighIndices, e.prefixShell j) := by |
| classical |
| obtain ⟨H,hH,hpass⟩ := actual_prefix_lowpass hK |
| obtain ⟨N₀,hband⟩ := uniform_actual_prefix_low_band (n:=n) hsig hJ |
| refine ⟨H,hH,N₀,?_⟩ |
| intro e hN hsig' hJ' hq hat ht hX hY hs B s lam hd |
| let := hd.probability |
| let := e.weightedDistance_isFinite hK hX hY hs |
| have hpass' := hpass e hX hY hs |
| rw [mul_assoc,e.prefix_frequency_decomposition hat] at hpass' |
| apply hpass'.trans |
| apply mul_le_mul_of_nonneg_left _ hH.le |
| have hlow : (∑ j∈e.prefixLowIndices, e.prefixShell j)≤(32*binKernelConstant.toReal)* |
| (∑ j∈e.prefixLowIndices, ({e with t:=j+1,i:=e.i+1} : PositiveDistanceState (n+1)).coincidence.toReal) := by |
| rw [Finset.mul_sum] |
| apply Finset.sum_le_sum |
| intro j hj |
| have hjt : j+1≤e.t := Nat.succ_le_of_lt (Finset.mem_Ico.mp (Finset.mem_filter.mp hj).1).2 |
| exact (e.prefixShell_le_band j).trans |
| (hband e hN hsig' hJ' hq ht K hK hX hY hs (j+1) hjt) |
| have hstop := e.prefix_low_frequency_stopping hK hd (hat.trans ht) hX hY hs |
| nlinarith |
| |
| end CurrentFalconer.TreeND |
| end GraphLiteralSupport |
| end |
| end |
| |
| end OAI |
| |
