| import OAI.MeasureTheory.Falconer.Angular.GnomonicTripleKernel |
| import OAI.MeasureTheory.Falconer.Radial.RadialAnglePotential |
|
|
| namespace OAI |
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| /-! # The actual Gaussian bad-triple weight is integrable on every retained anchor -/ |
|
|
| namespace Falconer |
| open Set MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
| variable {n h : ℕ} |
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| noncomputable def gaussianChartTripleWeight (h : ℕ) (r : ℝ) (p q z : Space n) : ℝ≥0∞ := |
| (ENNReal.ofReal (r^(-(h:ℝ))))^3 * |
| gaussianRows (n+1) h {A | ∀ i, |
| |inner ℝ (gnomonicChart p : Space (n+1)) (A i)| ≤ r ∧ |
| |inner ℝ (gnomonicChart q : Space (n+1)) (A i)| ≤ r ∧ |
| |inner ℝ (gnomonicChart z : Space (n+1)) (A i)| ≤ r} |
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| theorem anchored_bad_triple_potential (μ : Measure (Space n)) |
| (G : Set (Space n × Space n)) (E : Set (Space n)) (hE : MeasurableSet E) |
| (p q : Space n) (hp : ‖p‖ ≤ 1) (hq : ‖q‖ ≤ 1) |
| {b W K ℓ : ℝ} (hb : 0 < b) (hb1 : b ≤ 1) (hpq : b ≤ ‖q-p‖) |
| (hK : 0 ≤ K) (hℓ : (h:ℝ) < ℓ) (g k : ℕ) |
| (hW : 8*W/b^2 ≤ 1/(2:ℝ)^g) |
| (hretained : ∀ z ∈ E, (p,z) ∈ G ∧ ‖z‖ ≤ 1 ∧ b ≤ ‖z-p‖ ∧ |
| Real.sqrt (tripleGram (gnomonicChart p : Space (n+1)) |
| (gnomonicChart q : Space (n+1)) (gnomonicChart z : Space (n+1))) < W) |
| (hdiag : ∀ z, (p,z) ∈ G → z ≠ p) |
| (hcap : ∀ j : ℕ, j ≤ g+k → ∀ u : Space n, ‖u‖ = 1 → |
| μ {z | (p,z) ∈ G ∧ dist (NormedSpace.normalize (z-p)) u < 3*(1/(2:ℝ)^j)} ≤ |
| ENNReal.ofReal (K*(3*(1/(2:ℝ)^j))^ℓ)) : |
| (∫⁻ z in E, gaussianChartTripleWeight h (1/(2:ℝ)^(g+k)) p q z ∂μ) ≤ |
| ENNReal.ofReal ((gaussianTripleBandConstant/(b/4)^2)^h) * |
| ENNReal.ofReal ((2:ℝ)^(h:ℝ)*((2*K*(3:ℝ)^ℓ)*(((2:ℝ)^g)^((h:ℝ)-ℓ))* |
| (1+(2:ℝ)^(h:ℝ)/(1-(2:ℝ)^((h:ℝ)-ℓ))))) := by |
| let r : ℝ := 1/(2:ℝ)^(g+k) |
| let F : Space n → ℝ≥0∞ := fun z => ENNReal.ofReal ((1/max (displacementAngle p q z) r)^(h:ℝ)) |
| let C : ℝ≥0∞ := ENNReal.ofReal ((gaussianTripleBandConstant/(b/4)^2)^h) |
| have hr : 0 < r := by dsimp [r]; positivity |
| have hqne : q ≠ p := by |
| intro he |
| rw [he,sub_self,norm_zero] at hpq |
| linarith |
| have hbound (z : Space n) (hz : z ∈ E) : gaussianChartTripleWeight h r p q z ≤ C*F z := by |
| obtain ⟨_,hz1,hpz,_⟩ := hretained z hz |
| apply (gnomonic_triple_kernel p q z hp hq hz1 hb hb1 hr hpq hpz).trans_eq |
| dsimp [C,F] |
| have hCpos : 0 ≤ (gaussianTripleBandConstant/(b/4)^2)^h := |
| pow_nonneg (div_nonneg gaussianTripleBandConstant_pos.le (sq_nonneg _)) h |
| rw [← ENNReal.ofReal_mul hCpos,Real.rpow_natCast] |
| congr 1 |
| rw [← mul_pow] |
| congr 1 |
| ring |
| have hsub : E ⊆ {z | displacementAngle p q z < 1/(2:ℝ)^g} ∩ {z | (p,z) ∈ G} := by |
| intro z hz |
| obtain ⟨hG,hz1,hpz,hw⟩ := hretained z hz |
| exact ⟨(displacementAngle_lt_of_tripleGram hp hq hz1 hb hpq hpz hw).trans_le hW,hG⟩ |
| have hm : μ.restrict E ≤ (μ.restrict {z | (p,z) ∈ G}).restrict |
| {z | displacementAngle p q z < 1/(2:ℝ)^g} := by |
| rw [Measure.restrict_restrict (measurableSet_lt (displacementAngle_measurable p q) measurable_const)] |
| exact Measure.restrict_mono hsub le_rfl |
| calc |
| _ ≤ ∫⁻ z in E, C*F z ∂μ := lintegral_mono_ae (by |
| filter_upwards [ae_restrict_mem hE] with z hz |
| exact hbound z hz) |
| _ = C * ∫⁻ z in E, F z ∂μ := lintegral_const_mul' C _ ENNReal.ofReal_ne_top |
| _ ≤ C * ∫⁻ z in {z | displacementAngle p q z < 1/(2:ℝ)^g}, F z |
| ∂μ.restrict {z | (p,z) ∈ G} := mul_le_mul' le_rfl (lintegral_mono' hm le_rfl) |
| _ ≤ _ := mul_le_mul' le_rfl |
| (radial_angle_potential μ G p q hqne hdiag hK (Nat.cast_nonneg _) hℓ g k hcap) |
|
|
| end Falconer |
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| end OAI |
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