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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 | import OAI.MeasureTheory.Falconer.Measures.DyadicMeasure
import Mathlib.MeasureTheory.Measure.Prod
import Mathlib.Data.Finset.Union
namespace OAI
namespace PlanarFalconer
open Set Metric MeasureTheory
open scoped ENNReal NNReal
def dyadicPaths : ℕ → Finset (List (Bool × Bool))
| 0 => {[]}
| n+1 => Finset.univ.biUnion fun a : Bool × Bool =>
(dyadicPaths n).image (List.cons a)
@[simp] theorem mem_dyadicPaths (n : ℕ) (p : List (Bool × Bool)) :
p ∈ dyadicPaths n ↔ p.length = n := by
induction n generalizing p with
| zero => simp [dyadicPaths]
| succ n ih =>
cases p with
| nil => simp [dyadicPaths]
| cons a p => simp [dyadicPaths, ih]
theorem DyadicSquare.descendant_side (Q : DyadicSquare) (p : List (Bool × Bool)) :
(Q.descendant p).side = Q.side / (2 : ℝ)^p.length := by
induction p generalizing Q with
| nil => simp [descendant]
| cons a p ih =>
simp only [descendant, ih, child, List.length_cons, pow_succ]
ring
theorem DyadicSquare.descendants_disjoint (Q : DyadicSquare) (hQ : 0 < Q.side)
{p q : List (Bool × Bool)} (hlen : p.length = q.length) (hne : p ≠ q) :
Disjoint (Q.descendant p).carrier (Q.descendant q).carrier := by
induction p generalizing Q q with
| nil =>
have : q = [] := by cases q <;> simp_all
exact (hne this.symm).elim
| cons a p ih =>
cases q with
| nil => simp at hlen
| cons b q =>
by_cases hab : a = b
· subst b
exact ih _ (Q.child_side_pos hQ a.1 a.2) (by simpa using hlen)
(by intro he; exact hne (he ▸ rfl))
· exact (Q.children_disjoint hab).mono
((Q.child a.1 a.2).descendant_subset (Q.child_side_pos hQ a.1 a.2) p)
((Q.child b.1 b.2).descendant_subset (Q.child_side_pos hQ b.1 b.2) q)
theorem DyadicSquare.mem_descendant_at_depth (Q : DyadicSquare) (n : ℕ)
{z : Plane} (hz : z ∈ Q.carrier) :
∃ p ∈ dyadicPaths n, z ∈ (Q.descendant p).carrier := by
induction n generalizing Q with
| zero => exact ⟨[],by simp, hz⟩
| succ n ih =>
have step (i j : Bool) (hz : z ∈ (Q.child i j).carrier) :
∃ p ∈ dyadicPaths (n+1), z ∈ (Q.descendant p).carrier := by
obtain ⟨p,hp,hz⟩ := ih _ hz
exact ⟨(i,j) :: p,by simpa using hp,hz⟩
rw [Q.cover_children] at hz
rcases hz with (hz | hz) | (hz | hz)
· exact step false false hz
· exact step false true hz
· exact step true false hz
· exact step true true hz
def DyadicSquare.box (Q : DyadicSquare) : Set (ℝ × ℝ) :=
Ico Q.x (Q.x+Q.side) ×ˢ Ico Q.y (Q.y+Q.side)
def pairToPlane (z : ℝ × ℝ) : Plane := WithLp.toLp 2 ![z.1,z.2]
@[simp] theorem pairToPlane_mem (Q : DyadicSquare) (z : ℝ × ℝ) :
pairToPlane z ∈ Q.carrier ↔ z ∈ Q.box := by
simp only [pairToPlane, DyadicSquare.carrier, DyadicSquare.box,
mem_ofPred_eq, mem_prod, mem_Ico,
Matrix.cons_val_zero, Matrix.cons_val_one]
tauto
theorem DyadicSquare.box_measurable (Q : DyadicSquare) : MeasurableSet Q.box :=
measurableSet_Ico.prod measurableSet_Ico
theorem DyadicSquare.box_area (Q : DyadicSquare) :
(volume.prod volume) Q.box = ENNReal.ofReal Q.side * ENNReal.ofReal Q.side := by
simp [box, Measure.prod_prod, Real.volume_Ico]
theorem DyadicSquare.boxes_disjoint (Q : DyadicSquare) (hQ : 0 < Q.side)
{p q : List (Bool × Bool)} (hlen : p.length = q.length) (hne : p ≠ q) :
Disjoint (Q.descendant p).box (Q.descendant q).box := by
apply Set.disjoint_left.mpr
intro z hp hq
exact Set.disjoint_left.mp (Q.descendants_disjoint hQ hlen hne)
((pairToPlane_mem _ _).mpr hp) ((pairToPlane_mem _ _).mpr hq)
theorem plane_coordinate_dist_le (z x : Plane) (i : Fin 2) :
|z i - x i| ≤ dist z x := by
simpa only [Real.dist_eq] using PiLp.dist_apply_le z x i
theorem box_subset_of_meets_ball (Q : DyadicSquare) (x : Plane) {r w : ℝ}
(hside : Q.side = w) (hr : r ≤ w)
(hmeet : (Q.carrier ∩ ball x r).Nonempty) :
Q.box ⊆ Icc (x 0-2*w) (x 0+2*w) ×ˢ Icc (x 1-2*w) (x 1+2*w) := by
obtain ⟨z,hz,hball⟩ := hmeet
have h0 := (plane_coordinate_dist_le z x 0).trans_lt hball
have h1 := (plane_coordinate_dist_le z x 1).trans_lt hball
have hx0 := abs_lt.mp h0
have hx1 := abs_lt.mp h1
intro u hu
simp only [DyadicSquare.box, mem_prod, mem_Ico] at hu
simp only [DyadicSquare.carrier, mem_ofPred_eq] at hz
simp only [mem_prod, mem_Icc]
rw [hside] at hz hu
constructor <;> constructor <;> linarith [hu.1.1,hu.1.2,hu.2.1,hu.2.2,
hz.1,hz.2.1,hz.2.2.1,hz.2.2.2,hx0.1,hx0.2,hx1.1,hx1.2]
noncomputable def cellsMeeting (Q : DyadicSquare) (n : ℕ) (x : Plane) (r : ℝ) :
Finset (List (Bool × Bool)) := by
classical
exact (dyadicPaths n).filter fun p =>
((Q.descendant p).carrier ∩ ball x r).Nonempty
theorem dyadic_ball_card_le (Q : DyadicSquare) (hQ : 0 < Q.side)
(n : ℕ) (x : Plane) {r : ℝ} (hr : r ≤ Q.side/(2 : ℝ)^n) :
(cellsMeeting Q n x r).card ≤ 16 := by
classical
let S := (dyadicPaths n).filter fun p =>
((Q.descendant p).carrier ∩ ball x r).Nonempty
let w := Q.side/(2 : ℝ)^n
have hw : 0 < w := by dsimp [w]; positivity
have hlen {path : List (Bool × Bool)} (hp : path ∈ S) : path.length = n :=
(mem_dyadicPaths n path).mp (Finset.mem_filter.mp hp).1
have hside {path : List (Bool × Bool)} (hp : path ∈ S) :
(Q.descendant path).side = w := by
rw [Q.descendant_side, hlen hp]
have hdis : PairwiseDisjoint (S : Set (List (Bool × Bool)))
(fun p => (Q.descendant p).box) := by
intro p hp q hq hpq
exact Q.boxes_disjoint hQ ((hlen hp).trans (hlen hq).symm) hpq
have harea : (∑ p ∈ S, ENNReal.ofReal w * ENNReal.ofReal w) ≤
ENNReal.ofReal (4*w) * ENNReal.ofReal (4*w) := by
calc
_ = ∑ p ∈ S, (volume.prod volume) (Q.descendant p).box := by
apply Finset.sum_congr rfl
intro p hp
rw [DyadicSquare.box_area, hside hp]
_ = (volume.prod volume) (⋃ p ∈ S, (Q.descendant p).box) :=
(measure_biUnion_finset hdis (fun p _ => (Q.descendant p).box_measurable)).symm
_ ≤ (volume.prod volume)
(Icc (x 0-2*w) (x 0+2*w) ×ˢ Icc (x 1-2*w) (x 1+2*w)) := by
apply measure_mono
intro u hu
obtain ⟨p,hp,hu⟩ := mem_iUnion₂.mp hu
exact box_subset_of_meets_ball _ x (hside hp) hr
(Finset.mem_filter.mp hp).2 hu
_ = _ := by
rw [Measure.prod_prod, Real.volume_Icc, Real.volume_Icc]
congr 2 <;> ring
have hreal := ENNReal.toReal_mono (ENNReal.mul_ne_top ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top) harea
simp only [Finset.sum_const, nsmul_eq_mul, ENNReal.toReal_mul,
ENNReal.toReal_natCast, ENNReal.toReal_ofReal hw.le,
ENNReal.toReal_ofReal (by positivity : 0 ≤ 4*w)] at hreal
have hcard : (S.card : ℝ) ≤ 16 := by
by_contra h
have hpos := mul_pos (show 0 < (S.card : ℝ)-16 by linarith) (mul_pos hw hw)
nlinarith [hreal]
exact_mod_cast hcard
theorem dyadic_ball_measure_le (Q : DyadicSquare) (hQ : 0 < Q.side)
(μ : Measure Plane) (hcarry : μ Q.carrierᶜ = 0)
(s C : ℝ) (hC : 0 ≤ C) (n : ℕ)
(hb : ∀ p : List (Bool × Bool), p.length = n →
μ (Q.descendant p).carrier ≤ ENNReal.ofReal (C*(Q.descendant p).side^s))
(x : Plane) {r : ℝ} (hr : r ≤ Q.side/(2 : ℝ)^n) :
μ (ball x r) ≤ ENNReal.ofReal (16*C*(Q.side/(2 : ℝ)^n)^s) := by
classical
let S := (dyadicPaths n).filter fun p =>
((Q.descendant p).carrier ∩ ball x r).Nonempty
have hcover : ball x r ⊆ (⋃ p ∈ S, (Q.descendant p).carrier) ∪ Q.carrierᶜ := by
intro z hz
by_cases hQz : z ∈ Q.carrier
· obtain ⟨p,hp,hzp⟩ := Q.mem_descendant_at_depth n hQz
exact Or.inl (mem_iUnion₂.mpr ⟨p,Finset.mem_filter.mpr ⟨hp,⟨z,hzp,hz⟩⟩,hzp⟩)
· exact Or.inr hQz
calc
μ (ball x r) ≤ μ ((⋃ p ∈ S, (Q.descendant p).carrier) ∪ Q.carrierᶜ) := measure_mono hcover
_ ≤ μ (⋃ p ∈ S, (Q.descendant p).carrier) + μ Q.carrierᶜ := measure_union_le _ _
_ = μ (⋃ p ∈ S, (Q.descendant p).carrier) := by rw [hcarry, add_zero]
_ ≤ ∑ p ∈ S, μ (Q.descendant p).carrier := measure_biUnion_finset_le _ _
_ ≤ ∑ p ∈ S, ENNReal.ofReal (C*(Q.side/(2 : ℝ)^n)^s) := by
apply Finset.sum_le_sum
intro p hp
have hl := (mem_dyadicPaths n p).mp (Finset.mem_filter.mp hp).1
simpa only [Q.descendant_side, hl] using hb p hl
_ ≤ (16 : ℝ≥0∞) * ENNReal.ofReal (C*(Q.side/(2 : ℝ)^n)^s) := by
simp only [Finset.sum_const, nsmul_eq_mul]
exact mul_le_mul_left (by exact_mod_cast dyadic_ball_card_le Q hQ n x hr) _
_ = _ := by
rw [show (16 : ℝ≥0∞) = ENNReal.ofReal (16 : ℝ) by norm_num,
← ENNReal.ofReal_mul' (mul_nonneg hC (Real.rpow_nonneg (by positivity) _))]
congr 1
ring
end PlanarFalconer
end OAI
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