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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 | 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
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