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caec230 | 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 | import OAI.MeasureTheory.Falconer.Angular.AngularNearNeighbors
namespace OAI
namespace PlanarFalconer
open Set Metric MeasureTheory
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
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')
end PlanarFalconer
end OAI
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