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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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 | import OAI.MeasureTheory.Falconer.Radial.RadialZoomBounds
import OAI.MeasureTheory.Falconer.Geometry.FiniteScaleBall
import OAI.MeasureTheory.Falconer.Recursion.ProfileIncrementFrostman
namespace OAI
/-! # The actual cell normalization used by the middle radial interval -/
namespace Falconer.Regularization
open Set MeasureTheory
open scoped ENNReal
variable {d : ℕ}
theorem cell_normalization_map (ρ : Measure (Space d)) (p : ℕ) (c : Space d)
{b : ℝ} (hb : 0 < b) :
Measure.map (radialZoom (b*(2:ℝ)^p) (by positivity) c) ρ =
Measure.map (radialZoom b hb.ne' 0) (Measure.map (cellScaleMap p c) ρ) := by
rw [Measure.map_map (radialZoom b hb.ne' 0).measurable (cellScaleMap_continuous p c).measurable]
congr 1
ext x
simp only [Function.comp_apply,radialZoom_apply,cellScaleMap,sub_zero,smul_smul]
theorem cell_normalization_support (Q : DyadicCube d) (hQ : Q.side = 1)
(ρ : Measure (Space d)) (P : CellPath d) {b : ℝ} (hb : 0 < b) :
∀ᵐ z ∂Measure.map (radialZoom (b*(2:ℝ)^P.length) (by positivity) (Q.descendant P).center)
(ProbabilityTheory.cond ρ (Q.descendant P).carrier), ‖z‖ ≤ b*Real.sqrt d := by
rw [cell_normalization_map _ _ _ hb]
apply radialZoom_ae_bound _ hb 0
simpa only [sub_zero] using rescaled_conditional_support Q hQ ρ P
theorem cell_normalization_quarter_point (Q : DyadicCube d) (hQ : Q.side = 1)
(P : CellPath d) {b : ℝ} (hb : 0 < b) (hsmall : b*Real.sqrt d ≤ 1/4)
{z : Space d} (hz : z ∈ (Q.descendant P).carrier) :
‖z-(Q.descendant P).center‖ ≤ 1/(4*(b*(2:ℝ)^P.length)) := by
have hh := rescaled_cell_norm_bound Q hQ P hz
rw [cellScaleMap,norm_smul,Real.norm_eq_abs,abs_of_pos (by positivity : 0 < (2:ℝ)^P.length)] at hh
have hm := (mul_le_mul_of_nonneg_left hh hb.le).trans hsmall
apply (le_div_iff₀ (by positivity : 0 < 4*(b*(2:ℝ)^P.length))).mpr
nlinarith
theorem conditional_frostman_coefficient_ge_one (d N : ℕ) {s E lam : ℝ}
(hs : 0 ≤ s) (hE : 0 ≤ E) (hlam : 0 ≤ lam) :
1 ≤ (((2*Nat.ceil (2+4*Real.sqrt d)+1)^d:ℕ):ℝ)*(2:ℝ)^s*
((2:ℝ)^N)^(E+2*((d:ℝ)+1)*lam) := by
have hdim : (1:ℝ) ≤ (((2*Nat.ceil (2+4*Real.sqrt d)+1)^d:ℕ):ℝ) := by
exact_mod_cast (one_le_pow₀ (by omega : 1 ≤ 2*Nat.ceil (2+4*Real.sqrt d)+1) :
1 ≤ (2*Nat.ceil (2+4*Real.sqrt d)+1)^d)
have htwo : 1 ≤ (2:ℝ)^s := Real.one_le_rpow (by norm_num) hs
have hR : 1 ≤ ((2:ℝ)^N)^(E+2*((d:ℝ)+1)*lam) :=
Real.one_le_rpow (one_le_pow₀ (by norm_num)) (by positivity)
exact one_le_mul_of_one_le_of_one_le (one_le_mul_of_one_le_of_one_le hdim htwo) hR
theorem sourceClass_normalized_conditional_frostman (Q : DyadicCube d) (hQ : Q.side = 1)
(μ : Measure (Space d)) [IsProbabilityMeasure μ] {N B : ℕ} (hN : 0 < N) (hB : 0 < B)
{lam s E b : ℝ} (hs : 0 ≤ s) (hE : 0 ≤ E) (hlam : 0 ≤ lam)
(hb : 0 < b) (hb1 : b ≤ 1) (hwidth : (B:ℝ) ≤ lam*(N:ℝ))
(hlarge : (2:ℝ)^(N/B+2) ≤ ((2:ℝ)^N)^lam)
{p₀ : CellPath d} (hp₀ : p₀ ∈ regularizationLeaves Q μ N)
{p t : ℕ} (hpt : p ≤ t) (htN : t ≤ N) {P : CellPath d} (hP : P.length = p)
(hPpos : 0 < sourceClass Q μ N B p₀ (Q.descendant P).carrier)
(hinc : ∀ n, p ≤ n → n ≤ t →
s*(n-p:ℕ)/N-E ≤ sourceProfile Q μ N B p₀ n-sourceProfile Q μ N B p₀ p) :
∀ x : Space d, ∀ r : ℝ, 0 < r → 1/(2:ℝ)^(t-p) ≤ r →
Measure.map (radialZoom (b*(2:ℝ)^p) (by positivity) (Q.descendant P).center)
(ProbabilityTheory.cond (sourceClass Q μ N B p₀) (Q.descendant P).carrier)
(Metric.ball x r) ≤
ENNReal.ofReal (((((2*Nat.ceil (2+4*Real.sqrt d)+1)^d:ℕ):ℝ)*(2:ℝ)^s*
((2:ℝ)^N)^(E+2*((d:ℝ)+1)*lam))*b^(-s)*r^s) := by
have hside : 0 < Q.side := by rw [hQ]; norm_num
have : IsProbabilityMeasure (ProbabilityTheory.cond (sourceClass Q μ N B p₀)
(Q.descendant P).carrier) := sourceClass_conditional_probability Q hside μ N B hp₀ P hPpos
intro x r hr hδ
rw [cell_normalization_map _ _ _ hb]
apply radialZoom_frostman_bound _ hb hb1
(conditional_frostman_coefficient_ge_one d N hs hE hlam) hs (by simp) 0
(sourceClass_conditional_frostman_of_increment Q hQ μ hN hB hs hwidth hlarge
hp₀ hpt htN hP hPpos hinc) x r hr hδ
theorem sourceClass_scaled_dyadic_ball (Q : DyadicCube d) (hQ : Q.side = 1)
(μ : Measure (Space d)) [IsProbabilityMeasure μ] {N B : ℕ} (hN : 0 < N) (hB : 0 < B)
{lam s E b K : ℝ} (hs : 0 ≤ s) (hE : 0 ≤ E) (hlam : 0 ≤ lam)
(hb : 0 < b) (hb1 : b ≤ 1) (hwidth : (B:ℝ) ≤ lam*(N:ℝ))
(hlarge : (2:ℝ)^(N/B+2) ≤ ((2:ℝ)^N)^lam)
{p₀ : CellPath d} (hp₀ : p₀ ∈ regularizationLeaves Q μ N)
{p t M : ℕ} (hpt : p ≤ t) (htN : t ≤ N) (hM : M ≤ t-p)
{P : CellPath d} (hP : P.length = p)
(hPpos : 0 < sourceClass Q μ N B p₀ (Q.descendant P).carrier)
(hinc : ∀ n, p ≤ n → n ≤ t →
s*(n-p:ℕ)/N-E ≤ sourceProfile Q μ N B p₀ n-sourceProfile Q μ N B p₀ p)
(hcoef : (((((2*Nat.ceil (2+4*Real.sqrt d)+1)^d:ℕ):ℝ)*(2:ℝ)^s*
((2:ℝ)^N)^(E+2*((d:ℝ)+1)*lam))*b^(-s)) ≤ K) :
∀ x : Space d, ∀ j : ℕ, j ≤ M →
ProbabilityTheory.cond (sourceClass Q μ N B p₀) (Q.descendant P).carrier
(Metric.ball x ((1/(2:ℝ)^j)/(b*(2:ℝ)^p))) ≤
ENNReal.ofReal (K*(1/(2:ℝ)^j)^s) := by
intro x j hj
have ha : 0 < b*(2:ℝ)^p := by positivity
have hr : 1/(2:ℝ)^(t-p) ≤ 1/(2:ℝ)^j :=
one_div_le_one_div_of_le (by positivity) (pow_le_pow_right₀ (by norm_num) (hj.trans hM))
have hh := sourceClass_normalized_conditional_frostman Q hQ μ hN hB hs hE hlam hb hb1
hwidth hlarge hp₀ hpt htN hP hPpos hinc
(radialZoom (b*(2:ℝ)^p) ha.ne' (Q.descendant P).center x)
(1/(2:ℝ)^j) (by positivity) hr
rw [radialZoom_ball_apply _ ha] at hh
have hc : (Q.descendant P).center+(b*(2:ℝ)^p)⁻¹ •
radialZoom (b*(2:ℝ)^p) ha.ne' (Q.descendant P).center x = x := by
simp only [radialZoom_apply,smul_smul,inv_mul_cancel₀ ha.ne',one_smul]
abel
rw [hc] at hh
apply hh.trans
exact ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_right hcoef (by positivity))
end Falconer.Regularization
end OAI
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