| import OAI.MeasureTheory.Falconer.Angular.AngularNearNeighbors |
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| namespace OAI |
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| namespace PlanarFalconer |
| open Set Metric MeasureTheory |
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| theorem dyadic_far_points (Q : DyadicSquare) (hQ : Q.side = 1) |
| (p : ℕ) (P P' : CellPath) (hP : P.length = p) (hP' : P'.length = p) |
| (hfar : 20/(2:ℝ)^p < dist (Q.descendant P).center (Q.descendant P').center) |
| (x x' : Plane) (hx : x ∈ (Q.descendant P).carrier) (hx' : x' ∈ (Q.descendant P').carrier) : |
| 16/(2:ℝ)^p < dist x x' := by |
| have hQpos : 0 < Q.side := by rw [hQ]; norm_num |
| have h1 := (Q.descendant P).dist_le (Q.descendant_side_pos hQpos P).le |
| ((Q.descendant P).center_mem (Q.descendant_side_pos hQpos P)) hx |
| have h2 := (Q.descendant P').dist_le (Q.descendant_side_pos hQpos P').le hx' |
| ((Q.descendant P').center_mem (Q.descendant_side_pos hQpos P')) |
| rw [Q.descendant_side,hP,hQ,mul_one_div] at h1 |
| rw [Q.descendant_side,hP',hQ,mul_one_div] at h2 |
| have ht := (dist_triangle (Q.descendant P).center x (Q.descendant P').center).trans |
| (add_le_add (le_refl _) (dist_triangle x x' (Q.descendant P').center)) |
| have he : 20/(2:ℝ)^p = 16/(2:ℝ)^p+2/(2:ℝ)^p+2/(2:ℝ)^p := by ring |
| rw [he] at hfar |
| linarith |
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| theorem angular_far_displacement (Q₁ Q₂ : DyadicSquare) |
| (hQ₁ : Q₁.side = 1) (hQ₂ : Q₂.side = 1) (p : ℕ) (row col : CellPath × CellPath) |
| (hr : row ∈ (dyadicPaths p) ×ˢ (dyadicPaths p)) |
| (hc : col ∈ (dyadicPaths p) ×ˢ (dyadicPaths p)) |
| (hfar : ¬ angularNear Q₁ Q₂ p row col) |
| (x x' y y' : Plane) (hx : x ∈ (Q₁.descendant row.1).carrier) |
| (hx' : x' ∈ (Q₁.descendant col.1).carrier) (hy : y ∈ (Q₂.descendant row.2).carrier) |
| (hy' : y' ∈ (Q₂.descendant col.2).carrier) : |
| 16/(2:ℝ)^p < dist x x' ∨ 16/(2:ℝ)^p < dist y y' := by |
| simp only [Finset.mem_product,mem_dyadicPaths] at hr hc |
| unfold angularNear at hfar |
| by_cases h1 : dist (Q₁.descendant row.1).center (Q₁.descendant col.1).center ≤ 20/(2:ℝ)^p |
| · exact Or.inr (dyadic_far_points Q₂ hQ₂ p row.2 col.2 hr.2 hc.2 |
| (lt_of_not_ge (fun h2 => hfar ⟨h1,h2⟩)) y y' hy hy') |
| · exact Or.inl (dyadic_far_points Q₁ hQ₁ p row.1 col.1 hr.1 hc.1 (lt_of_not_ge h1) x x' hx hx') |
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| end PlanarFalconer |
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| end OAI |
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