| import OAI.MeasureTheory.Falconer.Angular.AngularFarDepth |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set Metric |
|
|
| theorem comparable_cells_strong_separation (P Q : DyadicSquare) |
| (hP : 0 < P.side) (hQ : 0 < Q.side) {l : ℝ} (hl : 0 < l) |
| (hsP : P.side ≤ l) (hsQ : Q.side ≤ l) {x y : Plane} |
| (hx : x ∈ P.carrier) (hy : y ∈ Q.carrier) |
| (hlo : 256*l < dist x y) (hhi : dist x y ≤ 1024*l) : |
| 100*l ≤ dist P.center Q.center ∧ dist P.center Q.center ≤ 2000*l := by |
| have hpx := (P.dist_le hP.le (P.center_mem hP) hx).trans (mul_le_mul_of_nonneg_left hsP (by norm_num)) |
| have hqy := (Q.dist_le hQ.le hy (Q.center_mem hQ)).trans (mul_le_mul_of_nonneg_left hsQ (by norm_num)) |
| have ht₁ := (dist_triangle P.center x Q.center).trans |
| (add_le_add (le_refl _) (dist_triangle x y Q.center)) |
| have ht₂ := (dist_triangle x P.center y).trans |
| (add_le_add (le_refl _) (dist_triangle P.center Q.center y)) |
| rw [dist_comm x P.center,dist_comm Q.center y] at ht₂ |
| constructor < |
|
|
| theorem comparable_dyadic_cells_strong_separation (Q₁ Q₂ : DyadicSquare) |
| (hQ₁ : Q₁.side = 1) (hQ₂ : Q₂.side = 1) (b : ℕ) (P Q : CellPath) |
| (hP : P.length = b) (hQ : Q.length = b) {x y : Plane} |
| (hx : x ∈ (Q₁.descendant P).carrier) (hy : y ∈ (Q₂.descendant Q).carrier) |
| (hlo : 256*(2:ℝ)^(-(b:ℝ)) < dist x y) (hhi : dist x y ≤ 1024*(2:ℝ)^(-(b:ℝ))) : |
| 100*(2:ℝ)^(-(b:ℝ)) ≤ dist (Q₁.descendant P).center (Q₂.descendant Q).center ∧ |
| dist (Q₁.descendant P).center (Q₂.descendant Q).center ≤ 2000*(2:ℝ)^(-(b:ℝ)) := by |
| have hp : 0 < Q₁.side := by rw [hQ₁] |
| have hq : 0 < Q₂.side := by rw [hQ₂] |
| apply comparable_cells_strong_separation (Q₁.descendant P) (Q₂.descendant Q) |
| (Q₁.descendant_side_pos hp P) (Q₂.descendant_side_pos hq Q) (by positivity) _ _ hx hy hlo hhi |
| · rw [Q₁.descendant_side,hP,hQ₁,Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),Real.rpow_natCast] |
| simp |
| · rw [Q₂.descendant_side,hQ,hQ₂,Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),Real.rpow_natCast] |
| simp |
|
|
| end PlanarFalconer |
|
|
| end OAI |
|
|
