| import OAI.MeasureTheory.Falconer.Recursion.ProfileEnvelopes |
| import OAI.MeasureTheory.Falconer.Estimates.RegularizationSelection |
| import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics |
|
|
| namespace OAI |
|
|
| /-! |
| |
|
|
| The enclosing-ball constant is 2*sqrt d, and the numerical choice of block |
| length is independent of dimension. |
| -/ |
|
|
| namespace Falconer.Regularization |
| open Set MeasureTheory Metric |
| open scoped ENNReal |
| variable {d : ℕ} |
|
|
| theorem dyadic_rpow_scale {N : ℕ} (hN : 0 < N) (j : ℕ) (s : ℝ) : |
| ((2:ℝ)^N)^(-s*(j:ℝ)/(N:ℝ)) = (((2:ℝ)^j)^s)⁻¹ := by |
| rw [← Real.rpow_natCast_mul (by norm_num : (0:ℝ) ≤ 2)] |
| have hN' : (N:ℝ) ≠ 0 := by exact_mod_cast hN.ne' |
| have he : (N:ℝ)*(-s*(j:ℝ)/(N:ℝ)) = -((j:ℝ)*s) := by field_simp |
| rw [he,Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),Real.rpow_natCast_mul (by norm_num : (0:ℝ) ≤ 2)] |
|
|
| theorem dyadic_cell_frostman_bound (hd : 0 < d) (Q : DyadicCube d) (hQ : 0 < Q.side) |
| (μ : Measure (Space d)) {s C : ℝ} |
| (hball : ∀ x : Space d, ∀ r > 0, μ (ball x r) ≤ ENNReal.ofReal (C*r^s)) |
| (p : CellPath d) : |
| μ (Q.descendant p).carrier ≤ ENNReal.ofReal (C*(2*Real.sqrt d*(Q.descendant p).side)^s) := by |
| have hside := Q.descendant_side_pos hQ p |
| have hsqrt : 0 < Real.sqrt (d : ℝ) := Real.sqrt_pos.mpr (by exact_mod_cast hd) |
| apply (measure_mono (show (Q.descendant p).carrier ⊆ |
| ball (Q.descendant p).center (2*Real.sqrt d*(Q.descendant p).side) from ?_)).trans |
| (hball _ _ (by positivity)) |
| intro x hx |
| have hh := (Q.descendant p).dist_le hside.le hx ((Q.descendant p).center_mem hside) |
| exact mem_ball.mpr (by nlinarith [mul_pos hsqrt hside]) |
|
|
| theorem dyadic_cell_frostman_profile_bound (hd : 0 < d) (Q : DyadicCube d) (hQ : 0 < Q.side) |
| (μ : Measure (Space d)) [IsFiniteMeasure μ] {s C : ℝ} (hC : 0 ≤ C) |
| (hball : ∀ x : Space d, ∀ r > 0, μ (ball x r) ≤ ENNReal.ofReal (C*r^s)) |
| {N : ℕ} (hN : 0 < N) (p : CellPath d) : |
| (μ (Q.descendant p).carrier).toReal ≤ |
| (C*(2*Real.sqrt d*Q.side)^s)*((2:ℝ)^N)^(-s*(p.length:ℝ)/(N:ℝ)) := by |
| have hh := ENNReal.toReal_mono ENNReal.ofReal_ne_top (dyadic_cell_frostman_bound hd Q hQ μ hball p) |
| have hside := Q.descendant_side_pos hQ p |
| rw [ENNReal.toReal_ofReal (by positivity : 0 ≤ C*(2*Real.sqrt d*(Q.descendant p).side)^s)] at hh |
| apply hh.trans_eq |
| rw [Q.descendant_side,dyadic_rpow_scale hN] |
| rw [show 2*Real.sqrt d*(Q.side/(2:ℝ)^p.length) = |
| (2*Real.sqrt d*Q.side)/(2:ℝ)^p.length by ring, |
| Real.div_rpow (by positivity) (by positivity)] |
| ring |
|
|
| theorem fixed_constant_absorbed {lam : ℝ} (hlam : 0 < lam) (C : ℝ) : |
| ∃ N₀ : ℕ, ∀ N ≥ N₀, C ≤ ((2:ℝ)^N)^lam := by |
| have ht : Filter.Tendsto (fun N : ℕ => ((2:ℝ)^N)^lam) Filter.atTop Filter.atTop := |
| (tendsto_rpow_atTop hlam).comp (tendsto_pow_atTop_atTop_of_one_lt (by norm_num)) |
| exact Filter.eventually_atTop.mp (ht.eventually (Filter.eventually_ge_atTop C)) |
|
|
| theorem retained_label_real_mass_lower (Q : DyadicCube d) (μ : Measure (Space d)) |
| [IsFiniteMeasure μ] {N B : ℕ} {p : CellPath d} {lam : ℝ} |
| (hp : regularizationLabel Q μ N B p ∈ regularizationIndex Q μ N B |
| (ENNReal.ofReal (((2:ℝ)^N)^(-lam)))) : |
| ((2:ℝ)^N)^(-lam) ≤ (classMeasure Q μ (regularizationLeaves Q μ N) |
| (regularizationLabel Q μ N B) (regularizationLabel Q μ N B p) univ).toReal := by |
| have hm := regularizationSet_mass Q μ N B _ hp |
| have hh := ENNReal.toReal_mono (measure_ne_top μ _) hm |
| simpa only [classMeasure,Measure.restrict_apply_univ,regularizationSet, |
| ENNReal.toReal_ofReal (by positivity : 0 ≤ ((2:ℝ)^N)^(-lam))] using hh |
|
|
| theorem sourceProfile_frostman_envelope (hd : 0 < d) (Q : DyadicCube d) (hQ : 0 < Q.side) |
| (μ : Measure (Space d)) [IsProbabilityMeasure μ] {s C lam : ℝ} (hC : 0 ≤ C) |
| (hlam : 0 < lam) (hball : ∀ x : Space d, ∀ r > 0, |
| μ (ball x r) ≤ ENNReal.ofReal (C*r^s)) : |
| ∃ H N₀ : ℕ, 0 < H ∧ ∀ N ≥ N₀, ∀ p ∈ regularizationLeaves Q μ N, |
| regularizationLabel Q μ N (N/H) p ∈ regularizationIndex Q μ N (N/H) |
| (ENNReal.ofReal (((2:ℝ)^N)^(-lam))) → |
| ∀ j ≤ N, s*(j:ℝ)/(N:ℝ)-((d:ℝ)+3)*lam ≤ sourceProfile Q μ N (N/H) p j := by |
| obtain ⟨H,N₁,hH,hscales⟩ := PlanarFalconer.exists_regularization_block_control hlam |
| obtain ⟨N₂,hconst⟩ := fixed_constant_absorbed hlam (C*(2*Real.sqrt d*Q.side)^s) |
| refine ⟨H,max N₁ N₂,hH,?_⟩ |
| intro N hN p hp hret j hj |
| have hs := hscales N ((le_max_left _ _).trans hN) |
| apply sourceProfile_frostman_lower Q hQ μ hs.1 hs.2.1 hs.2.2.2.1 hs.2.2.2.2 hp |
| (retained_label_real_mass_lower Q μ hret) _ hj |
| intro k hk P hP |
| have hh := dyadic_cell_frostman_profile_bound hd Q hQ μ hC hball hs.1 P |
| rw [hP] at hh |
| exact hh.trans (mul_le_mul_of_nonneg_right (hconst N ((le_max_right _ _).trans hN)) (by positivity)) |
|
|
| end Falconer.Regularization |
|
|
| end OAI |
|
|
