| import OAI.MeasureTheory.Falconer.Angular.AngularLengthPartition |
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| namespace OAI |
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| namespace PlanarFalconer |
| open Set Metric |
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| 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 |
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| 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] |
| have h2 := (Q₂.descendant B).dist_le |
| (Q₂.descendant_side_pos (by rw [hQ₂] |
| 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 |
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| 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 |
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| end PlanarFalconer |
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| end OAI |
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