| import Mathlib |
| import OAI.MeasureTheory.Falconer.Fourier.BinKernel |
| import OAI.MeasureTheory.Falconer.Fourier.DistanceBandKernelBound |
| import OAI.MeasureTheory.Falconer.Geometry.BinEstimates |
| import OAI.MeasureTheory.Falconer.Model |
| import OAI.MeasureTheory.FalconerCompletion.Estimates.MinimalMaskNestedSupport |
|
|
| 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) |
|
|
| lemma width_kernel (z : ℝ × ℝ) : |
| 1/(1+|z.1-z.2|/e.width)^2=1/(1+(2:ℝ)^e.t*|z.1-z.2|)^2 := by |
| simp only [width,Real.rpow_neg (by norm_num : (0:ℝ)≤2),Real.rpow_natCast, |
| div_eq_mul_inv,inv_inv,mul_comm] |
|
|
| lemma distance_self_kernel [IsFiniteMeasure e.ρ] : |
| e.normalization*(∫⁻ z : ℝ × ℝ, ENNReal.ofReal (1/(1+(2:ℝ)^e.t*|z.1-z.2|)^2) |
| ∂e.distance.prod e.distance)≤binKernelConstant*e.coincidence := by |
| have hw : 0<e.width := by unfold width; positivity |
| have h := real_bin_decaying_self_kernel_bound e.distance hw |
| simp_rw [e.width_kernel] at h |
| have hc : (e.distance.prod e.distance) {z : ℝ × ℝ | |z.1-z.2|≤e.width}≤ |
| (e.distance.prod e.distance) {z : ℝ × ℝ | |z.1-z.2|≤100*(d+1)*e.width} := by |
| apply measure_mono |
| intro z hz |
| change |z.1-z.2|≤100*(d+1)*e.width |
| change |z.1-z.2|≤e.width at hz |
| apply hz.trans |
| have hd : (0:ℝ)≤d := Nat.cast_nonneg d |
| nlinarith |
| have hh := mul_le_mul_right (h.trans (mul_le_mul_right hc binKernelConstant)) e.normalization |
| rw [e.distance_collision] at hh |
| simpa only [coincidence,mul_left_comm] using hh |
|
|
| lemma normalized_weight_factor : |
| ENNReal.ofReal ((2:ℝ)^((e.t:ℝ)-e.a))*ENNReal.ofReal (e.levelMass^(-2:ℤ))* |
| ENNReal.ofReal (e.levelMass^(-2:ℤ))=e.normalization := by |
| rw [← ENNReal.ofReal_mul (by positivity),← ENNReal.ofReal_mul (by positivity)] |
| congr 1 |
| simp only [zpow_neg,zpow_ofNat,div_eq_mul_inv] |
| ring |
|
|
| lemma normalized_dominated_distance_kernel [IsFiniteMeasure e.ρ] |
| (μ : Measure ℝ) [IsFiniteMeasure μ] |
| (hμ : μ≤ENNReal.ofReal (e.levelMass^(-2:ℤ)) • e.distance) : |
| ENNReal.ofReal ((2:ℝ)^((e.t:ℝ)-e.a)) * |
| (∫⁻ z : ℝ × ℝ, ENNReal.ofReal (1/(1+(2:ℝ)^e.t*|z.1-z.2|)^2) ∂μ.prod μ)≤ |
| binKernelConstant*e.coincidence := by |
| let F : ℝ × ℝ → ℝ≥0∞ := fun z => ENNReal.ofReal (1/(1+(2:ℝ)^e.t*|z.1-z.2|)^2) |
| have hp := Measure.prod_mono hμ hμ |
| rw [Measure.prod_smul_left,Measure.prod_smul_right,smul_smul] at hp |
| have h := lintegral_mono' hp (fun z => le_refl (F z)) |
| rw [lintegral_smul_measure,smul_eq_mul] at h |
| have hh := mul_le_mul_right h (ENNReal.ofReal ((2:ℝ)^((e.t:ℝ)-e.a))) |
| rw [← mul_assoc,← mul_assoc,e.normalized_weight_factor] at hh |
| exact hh.trans e.distance_self_kernel |
|
|
| lemma dominated_distance_band [IsFiniteMeasure e.ρ] |
| (μ : Measure ℝ) [IsFiniteMeasure μ] |
| (hμ : μ≤ENNReal.ofReal (e.levelMass^(-2:ℤ)) • e.distance) : |
| (2:ℝ)^(-(e.a:ℝ))*(∫ r in Icc (-((2:ℝ)^e.t)) ((2:ℝ)^e.t), ‖charFun μ r‖^2)≤ |
| (32*binKernelConstant.toReal)*e.coincidence.toReal := by |
| have h := mul_le_mul_right (distance_band_kernel_bound_ennreal μ |
| (by positivity : 0<(2:ℝ)^e.t)) (ENNReal.ofReal ((2:ℝ)^(-(e.a:ℝ)))) |
| have hkernel := mul_le_mul_right (e.normalized_dominated_distance_kernel μ hμ) (32:ℝ≥0∞) |
| have he : (2:ℝ)^(-(e.a:ℝ))*(32*(2:ℝ)^e.t)=32*(2:ℝ)^((e.t:ℝ)-e.a) := by |
| rw [mul_left_comm,← Real.rpow_natCast,← Real.rpow_add (by norm_num : (0:ℝ)<2)] |
| congr 2 |
| ring |
| have hc : ENNReal.ofReal ((2:ℝ)^(-(e.a:ℝ)))*ENNReal.ofReal (32*(2:ℝ)^e.t)= |
| 32*ENNReal.ofReal ((2:ℝ)^((e.t:ℝ)-e.a)) := by |
| rw [← ENNReal.ofReal_mul (by positivity),he,ENNReal.ofReal_mul (by norm_num)] |
| norm_num |
| rw [← mul_assoc,hc,mul_assoc] at h |
| have hh := h.trans hkernel |
| have hr := ENNReal.toReal_mono |
| (ENNReal.mul_ne_top (by norm_num) (ENNReal.mul_ne_top binKernelConstant_ne_top e.coincidence_ne_top)) hh |
| rw [ENNReal.toReal_mul,ENNReal.toReal_ofReal (by positivity), |
| ENNReal.toReal_ofReal (integral_nonneg (fun r => sq_nonneg ‖charFun μ r‖)), |
| ENNReal.toReal_mul,ENNReal.toReal_mul] at hr |
| norm_num only [ENNReal.toReal_ofNat] at hr |
| simpa only [mul_assoc] using hr |
|
|
| end PositiveDistanceState |
|
|
| theorem uniform_actual_prefix_low_band {sig : ℝ} (hsig : 0<sig) {J : ℕ} (hJ : 0<J) : |
| ∃ N₀ : ℕ, ∀ (e : PositiveDistanceState (n+1)) [IsFiniteMeasure e.ρ], |
| N₀≤e.N → e.sig=sig → e.J=J → 0<e.q → e.t≤e.N → |
| ∀ K : ℝ, 32*(n+2:ℝ)≤K → 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)) → |
| ∀ v≤e.t, (2:ℝ)^(-(e.a:ℝ)) * |
| (∫ r in Icc (-((2:ℝ)^v)) ((2:ℝ)^v), ‖charFun e.weightedDistance r‖^2)≤ |
| (32*binKernelConstant.toReal)* |
| ({e with t:=v,i:=e.i+1} : PositiveDistanceState (n+1)).coincidence.toReal := by |
| obtain ⟨N₀,h₀⟩ := PositiveDistanceState.uniform_weightedDistance_nested_domination (n:=n) hsig hJ |
| refine ⟨N₀,?_⟩ |
| intro e he hN hsig' hJ' hq ht K hK hX hY hs v hv |
| let := e.weightedDistance_isFinite hK hX hY hs |
| exact ({e with t:=v,i:=e.i+1} : PositiveDistanceState (n+1)).dominated_distance_band |
| e.weightedDistance (h₀ e hN hsig' hJ' hq ht K hK hX hY hs v hv) |
|
|
| end CurrentFalconer.TreeND |
| end GraphLiteralSupport |
| end |
| end |
|
|
| end OAI |
|
|
