| import OAI.MeasureTheory.Falconer.Incidence.FiniteBallBounds | |
| import OAI.MeasureTheory.Falconer.Recursion.TruncatedScalarPotential | |
| import Mathlib.MeasureTheory.Integral.Lebesgue.Countable | |
| namespace OAI | |
| /-! The finite point potential needed by the elementary planar incidence bound. -/ | |
| namespace Falconer | |
| open Set MeasureTheory | |
| open scoped ENNReal BigOperators | |
| attribute [local instance] Classical.propDecidable | |
| noncomputable def planarCountingMeasure (P : Finset (Space 2)) : Measure (Space 2) := | |
| ∑ x ∈ P, Measure.dirac x | |
| theorem planarCountingMeasure_apply (P : Finset (Space 2)) (A : Set (Space 2)) : | |
| planarCountingMeasure P A = (P.filter (fun x => x ∈ A)).card := by | |
| rw [planarCountingMeasure,Measure.finsetSum_apply] | |
| simp_rw [Measure.dirac_apply,Set.indicator_apply,Pi.one_apply] | |
| rw [← Finset.sum_filter] | |
| simp | |
| theorem planarCountingMeasure_lintegral (P : Finset (Space 2)) (f : Space 2 → ℝ≥0∞) : | |
| (∫⁻ x, f x ∂planarCountingMeasure P) = ∑ x ∈ P, f x := by | |
| rw [planarCountingMeasure,lintegral_finsetSum_measure] | |
| simp only [lintegral_dirac] | |
| theorem finite_dyadic_truncated_potential (P : Finset (Space 2)) (x : Space 2) | |
| (N : ℕ) {K b f : ℝ} (hK : 0 ≤ K) (hb : 0 ≤ b) (hbf : b < f) | |
| (hball : NormalizedFiniteBallBound (1/(2:ℝ)^N) f K P) : | |
| (∑ y ∈ P, ENNReal.ofReal ((1/max (dist x y) (1/(2:ℝ)^N))^b)) ≤ | |
| P.card + ENNReal.ofReal ((2:ℝ)^b*(K*(P.card : ℝ))*(1+(2:ℝ)^b/(1-(2:ℝ)^(b-f)))) := by | |
| let μ := planarCountingMeasure P | |
| have hsmall (j : ℕ) (hj : j ≤ 0+N) : | |
| μ {y | dist x y < 1/(2:ℝ)^j} ≤ | |
| ENNReal.ofReal ((K*(P.card : ℝ))*(1/(2:ℝ)^j)^f) := by | |
| rw [planarCountingMeasure_apply] | |
| have hr0 : 1/(2:ℝ)^N ≤ 1/(2:ℝ)^j := by | |
| apply one_div_le_one_div_of_le (by positivity) | |
| exact pow_le_pow_right₀ (by norm_num) (by simpa using hj) | |
| have hr1 : 1/(2:ℝ)^j ≤ 1 := by | |
| apply (div_le_one (by positivity)).2 | |
| exact one_le_pow₀ (by norm_num) | |
| have hs : P.filter (fun y => dist x y < 1/(2:ℝ)^j) ⊆ | |
| P.filter (fun y => dist y x ≤ 1/(2:ℝ)^j) := by | |
| intro y hy | |
| obtain ⟨hyP,hy⟩ := Finset.mem_filter.mp hy | |
| exact Finset.mem_filter.mpr ⟨hyP,by simpa only [dist_comm] using hy.le⟩ | |
| have hc := (show ((P.filter (fun y => dist x y < 1/(2:ℝ)^j)).card : ℝ) ≤ | |
| (P.filter (fun y => dist y x ≤ 1/(2:ℝ)^j)).card from by exact_mod_cast Finset.card_le_card hs) | |
| have hh := hc.trans (hball x _ hr0 hr1) | |
| rw [show (K*(P.card : ℝ))*(1/(2:ℝ)^j)^f = K*(1/(2:ℝ)^j)^f*(P.card : ℝ) by ring] | |
| exact_mod_cast ENNReal.ofReal_le_ofReal hh | |
| have hlocal := truncated_scalar_potential μ (fun y => dist x y) (by fun_prop) | |
| (fun _ => dist_nonneg) (mul_nonneg hK (Nat.cast_nonneg _)) hb hbf 0 N hsmall | |
| simp only [pow_zero,one_div_one,zero_add,Real.one_rpow,mul_one] at hlocal | |
| let A := {y : Space 2 | dist x y < 1} | |
| let F := fun y : Space 2 => ENNReal.ofReal ((1/max (dist x y) (1/(2:ℝ)^N))^b) | |
| have hfar : (∫⁻ y in Aᶜ, F y ∂μ) ≤ P.card := by | |
| calc | |
| _ ≤ ∫⁻ _ in Aᶜ, (1 : ℝ≥0∞) ∂μ := by | |
| apply setLIntegral_mono' (by measurability) | |
| intro y hy | |
| change ¬ dist x y < 1 at hy | |
| have hd : 1 ≤ max (dist x y) (1/(2:ℝ)^N) := | |
| (le_of_not_gt hy).trans (le_max_left _ _) | |
| have hinv : 1/max (dist x y) (1/(2:ℝ)^N) ≤ 1 := | |
| (div_le_one (zero_lt_one.trans_le hd)).2 hd | |
| exact ENNReal.ofReal_le_one.mpr ((Real.rpow_le_rpow (by positivity) hinv hb).trans_eq (Real.one_rpow _)) | |
| _ ≤ μ univ := by | |
| simp only [lintegral_one,Measure.restrict_apply_univ] | |
| exact measure_mono (subset_univ _) | |
| _ = P.card := by rw [planarCountingMeasure_apply]; simp | |
| rw [← planarCountingMeasure_lintegral] | |
| change (∫⁻ y, F y ∂μ) ≤ _ | |
| rw [← lintegral_add_compl F (show MeasurableSet A from by dsimp [A]; measurability)] | |
| simpa only [mul_assoc,add_comm] using add_le_add hlocal hfar | |
| noncomputable def planarPotentialConstant (b f : ℝ) : ℝ := | |
| 1+(2:ℝ)^b*(1+(2:ℝ)^b/(1-(2:ℝ)^(b-f))) | |
| theorem planarPotentialConstant_pos {b f : ℝ} (hbf : b < f) : | |
| 0 < planarPotentialConstant b f := by | |
| have hd : 0 < 1-(2:ℝ)^(b-f) := sub_pos.mpr | |
| (Real.rpow_lt_one_of_one_lt_of_neg (by norm_num) (sub_neg.mpr hbf)) | |
| unfold planarPotentialConstant | |
| positivity | |
| theorem finite_dyadic_truncated_potential_real (P : Finset (Space 2)) (x : Space 2) | |
| (N : ℕ) {K b f : ℝ} (hK : 1 ≤ K) (hb : 0 ≤ b) (hbf : b < f) | |
| (hball : NormalizedFiniteBallBound (1/(2:ℝ)^N) f K P) : | |
| (∑ y ∈ P, (1/max (dist x y) (1/(2:ℝ)^N))^b) ≤ | |
| K*planarPotentialConstant b f*(P.card : ℝ) := by | |
| have hK0 : 0 ≤ K := zero_le_one.trans hK | |
| have hd : 0 < 1-(2:ℝ)^(b-f) := sub_pos.mpr | |
| (Real.rpow_lt_one_of_one_lt_of_neg (by norm_num) (sub_neg.mpr hbf)) | |
| have hR : 0 ≤ (P.card : ℝ)+(2:ℝ)^b*(K*(P.card : ℝ))*(1+(2:ℝ)^b/(1-(2:ℝ)^(b-f))) := by | |
| positivity | |
| have hh := finite_dyadic_truncated_potential P x N hK0 hb hbf hball | |
| rw [← ENNReal.ofReal_sum_of_nonneg (fun y _ => by positivity), | |
| ← ENNReal.ofReal_natCast P.card, ← ENNReal.ofReal_add (Nat.cast_nonneg _) (by positivity)] at hh | |
| have hh' := (ENNReal.ofReal_le_ofReal_iff hR).mp hh | |
| apply hh'.trans | |
| unfold planarPotentialConstant | |
| have hC : 0 ≤ (2:ℝ)^b*(1+(2:ℝ)^b/(1-(2:ℝ)^(b-f))) := by positivity | |
| nlinarith [Nat.cast_nonneg (α := ℝ) P.card] | |
| end Falconer | |
| end OAI | |