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7f37411 | 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 | import OAI.MeasureTheory.Falconer.Angular.AngularLengthPartition
namespace OAI
namespace PlanarFalconer
open Set Metric
theorem angularLengthShell_cover {x : Fin 4 → Plane} {a h : ℕ}
(hc : x ∈ angularLengthComparable)
(hupper : max (dist (x 0) (x 1)) (dist (x 2) (x 3)) ≤ 2*(2:ℝ)^(-(a:ℝ)))
(hfar : 16*(2:ℝ)^(-(h:ℝ)) < max (dist (x 0) (x 1)) (dist (x 2) (x 3))) :
∃ b ∈ Finset.Icc (a+9) (h+5), x ∈ angularLengthShell b := by
let d := max (dist (x 0) (x 1)) (dist (x 2) (x 3))
have hd : 0 < d := (by positivity : 0 < 16*(2:ℝ)^(-(h:ℝ))).trans hfar
obtain ⟨n,hnlo,hnhi⟩ := exists_displacement_depth hd
have hpow (k : ℝ) : (2:ℝ)^(k-(n:ℝ)-10) = (2:ℝ)^k*(2:ℝ)^(-((n:ℝ)+10)) := by
rw [show k-(n:ℝ)-10 = k+(-((n:ℝ)+10)) by ring,Real.rpow_add (by norm_num : (0:ℝ) < 2)]
have haexp : (2:ℝ)^(-(n:ℝ)-1) < (2:ℝ)^(1-(a:ℝ)) := by
have he : (2:ℝ)^(1-(a:ℝ)) = 2*(2:ℝ)^(-(a:ℝ)) := by
rw [sub_eq_add_neg,Real.rpow_add (by norm_num : (0:ℝ) < 2),Real.rpow_one]
rw [he]
exact hnlo.trans_le hupper
have hhexp : (2:ℝ)^(4-(h:ℝ)) < (2:ℝ)^(-(n:ℝ)) := by
have he : (2:ℝ)^(4-(h:ℝ)) = 16*(2:ℝ)^(-(h:ℝ)) := by
rw [sub_eq_add_neg,Real.rpow_add (by norm_num : (0:ℝ) < 2)]
norm_num
rw [he]
exact hfar.trans_le hnhi
have haR := (Real.rpow_lt_rpow_left_iff (by norm_num : (1:ℝ) < 2)).mp haexp
have hhR := (Real.rpow_lt_rpow_left_iff (by norm_num : (1:ℝ) < 2)).mp hhexp
have haZ : (a:ℤ)-2 < n := by exact_mod_cast (show (a:ℝ)-2 < (n:ℝ) by linarith)
have hhZ : n < (h:ℤ)-4 := by exact_mod_cast (show (n:ℝ) < (h:ℝ)-4 by linarith)
have hb0 : 0 ≤ n+10 := by omega
let b := (n+10).toNat
have hbZ : (b:ℤ) = n+10 := Int.toNat_of_nonneg hb0
have hbR : (b:ℝ) = (n:ℝ)+10 := by exact_mod_cast hbZ
refine ⟨b,Finset.mem_Icc.mpr ⟨by omega,by omega⟩,hc,?_,?_⟩
· have he : (2:ℝ)^(-(n:ℝ)-1) = 512*(2:ℝ)^(-(b:ℝ)) := by
rw [hbR,show -(n:ℝ)-1 = 9-(n:ℝ)-10 by ring,hpow]
norm_num
rw [← he]
exact hnlo
· have he : (2:ℝ)^(-(n:ℝ)) = 1024*(2:ℝ)^(-(b:ℝ)) := by
rw [hbR,show -(n:ℝ) = 10-(n:ℝ)-10 by ring,hpow]
norm_num
rw [← he]
exact hnhi
theorem parent_cell_displacement_bound (Q₁ Q₂ : DyadicSquare)
(hQ₁ : Q₁.side = 1) (hQ₂ : Q₂.side = 1) (a : ℕ) (A B : CellPath)
(hA : A.length = a) (hB : B.length = a) (x : Fin 4 → Plane)
(hx₀ : x 0 ∈ (Q₁.descendant A).carrier) (hx₁ : x 1 ∈ (Q₁.descendant A).carrier)
(hx₂ : x 2 ∈ (Q₂.descendant B).carrier) (hx₃ : x 3 ∈ (Q₂.descendant B).carrier) :
max (dist (x 0) (x 1)) (dist (x 2) (x 3)) ≤ 2*(2:ℝ)^(-(a:ℝ)) := by
have h1 := (Q₁.descendant A).dist_le
(Q₁.descendant_side_pos (by rw [hQ₁]; norm_num) A).le hx₀ hx₁
have h2 := (Q₂.descendant B).dist_le
(Q₂.descendant_side_pos (by rw [hQ₂]; norm_num) B).le hx₂ hx₃
rw [Q₁.descendant_side,hA,hQ₁] at h1
rw [Q₂.descendant_side,hB,hQ₂] at h2
simp only [Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),Real.rpow_natCast,
one_div] at *
exact max_le h1 h2
theorem angular_far_max_displacement (Q₁ Q₂ : DyadicSquare)
(hQ₁ : Q₁.side = 1) (hQ₂ : Q₂.side = 1) (h : ℕ) (row col : CellPath × CellPath)
(hr : row ∈ (dyadicPaths h) ×ˢ (dyadicPaths h))
(hc : col ∈ (dyadicPaths h) ×ˢ (dyadicPaths h))
(hfar : ¬ angularNear Q₁ Q₂ h row col) (x : Fin 4 → Plane)
(hx₀ : x 0 ∈ (Q₁.descendant row.1).carrier) (hx₁ : x 1 ∈ (Q₁.descendant col.1).carrier)
(hx₂ : x 2 ∈ (Q₂.descendant row.2).carrier) (hx₃ : x 3 ∈ (Q₂.descendant col.2).carrier) :
16*(2:ℝ)^(-(h:ℝ)) < max (dist (x 0) (x 1)) (dist (x 2) (x 3)) := by
have hh := angular_far_displacement Q₁ Q₂ hQ₁ hQ₂ h row col hr hc hfar
(x 0) (x 1) (x 2) (x 3) hx₀ hx₁ hx₂ hx₃
simpa only [Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),Real.rpow_natCast,
← div_eq_mul_inv,lt_max_iff] using hh
end PlanarFalconer
end OAI
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