File size: 3,761 Bytes
d4317d7 | 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 | import OAI.MeasureTheory.Falconer.Angular.GnomonicTripleKernel
import OAI.MeasureTheory.Falconer.Radial.RadialAnglePotential
namespace OAI
/-! # 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}
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}
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
end OAI
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