| import OAI.MeasureTheory.Falconer.DistanceSet |
| import Mathlib.Tactic.Linarith |
| import Mathlib.Tactic.Positivity |
| import Mathlib.Tactic.FinCases |
| import Mathlib.Analysis.Normed.Lp.MeasurableSpace |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set Metric |
|
|
| structure DyadicSquare where |
| x : ℝ |
| y : ℝ |
| side : ℝ |
|
|
| def DyadicSquare.carrier (Q : DyadicSquare) : Set Plane := |
| {z | Q.x ≤ z 0 ∧ z 0 < Q.x + Q.side ∧ |
| Q.y ≤ z 1 ∧ z 1 < Q.y + Q.side} |
|
|
| noncomputable def DyadicSquare.child (Q : DyadicSquare) (i j : Bool) : DyadicSquare := |
| ⟨Q.x + (if i then Q.side / 2 else 0), |
| Q.y + (if j then Q.side / 2 else 0), Q.side / 2⟩ |
|
|
| theorem DyadicSquare.child_side_pos (Q : DyadicSquare) (hQ : 0 < Q.side) |
| (i j : Bool) : 0 < (Q.child i j).side := by |
| dsimp [child] |
| positivity |
|
|
| theorem DyadicSquare.measurable (Q : DyadicSquare) : MeasurableSet Q.carrier := by |
| have h0 : Continuous (fun z : Plane => z 0) := (EuclideanSpace.proj 0).continuous |
| have h1 : Continuous (fun z : Plane => z 1) := (EuclideanSpace.proj 1).continuous |
| exact (isClosed_le continuous_const h0).measurableSet.inter |
| ((isOpen_lt h0 continuous_const).measurableSet.inter |
| ((isClosed_le continuous_const h1).measurableSet.inter |
| (isOpen_lt h1 continuous_const).measurableSet)) |
|
|
| theorem DyadicSquare.child_subset (Q : DyadicSquare) (hQ : 0 ≤ Q.side) |
| (i j : Bool) : (Q.child i j).carrier ⊆ Q.carrier := by |
| have halfSide_nonneg : 0 ≤ Q.side / 2 := div_nonneg hQ (by norm_num) |
| intro z hz |
| cases i <;> cases j <;> |
| simp only [carrier, child, Bool.false_eq_true, ↓reduceIte, |
| add_zero, mem_ofPred_eq] at * <;> |
| refine ⟨by linarith [halfSide_nonneg], by linarith [halfSide_nonneg], |
| by linarith [halfSide_nonneg], by linarith [halfSide_nonneg]⟩ |
|
|
| theorem DyadicSquare.cover_children (Q : DyadicSquare) : |
| Q.carrier = ((Q.child false false).carrier ∪ (Q.child false true).carrier) ∪ |
| ((Q.child true false).carrier ∪ (Q.child true true).carrier) := by |
| ext z |
| simp only [carrier, child, Bool.false_eq_true, ↓reduceIte, |
| add_zero, mem_ofPred_eq, mem_union] |
| constructor |
| · intro hz |
| by_cases hx : z 0 < Q.x + Q.side / 2 <;> |
| by_cases hy : z 1 < Q.y + Q.side / 2 |
| · exact Or.inl (Or.inl ⟨hz.1,hx,hz.2.2.1,hy⟩) |
| · exact Or.inl (Or.inr ⟨hz.1,hx,by linarith,by linarith [hz.2.2.2]⟩) |
| · exact Or.inr (Or.inl ⟨by linarith,by linarith [hz.2.1],hz.2.2.1,hy⟩) |
| · exact Or.inr (Or.inr ⟨by linarith,by linarith [hz.2.1], |
| by linarith,by linarith [hz.2.2.2]⟩) |
| · intro hz |
| rcases hz with (hz | hz) | (hz | hz) <;> |
| refine ⟨by linarith [hz.1,hz.2.1], by linarith [hz.1,hz.2.1], |
| by linarith [hz.2.2.1,hz.2.2.2], by linarith [hz.2.2.1,hz.2.2.2]⟩ |
|
|
| theorem DyadicSquare.children_disjoint (Q : DyadicSquare) |
| {i j i' j' : Bool} (h : (i,j) ≠ (i',j')) : |
| Disjoint (Q.child i j).carrier (Q.child i' j').carrier := by |
| apply Set.disjoint_left.mpr |
| intro z hz hz' |
| cases i <;> cases j <;> cases i' <;> cases j' <;> |
| simp_all only [carrier, child, Bool.false_eq_true, ↓reduceIte, |
| add_zero, mem_ofPred_eq, ne_eq, not_true_eq_false] <;> linarith |
|
|
| theorem DyadicSquare.dist_le (Q : DyadicSquare) (hQ : 0 ≤ Q.side) |
| {z w : Plane} (hz : z ∈ Q.carrier) (hw : w ∈ Q.carrier) : |
| dist z w ≤ 2 * Q.side := by |
| have h0 : |z 0 - w 0| ≤ Q.side := by |
| apply abs_le.mpr; constructor <;> linarith [hz.1,hz.2.1,hw.1,hw.2.1] |
| have h1 : |z 1 - w 1| ≤ Q.side := by |
| apply abs_le.mpr; constructor <;> |
| linarith [hz.2.2.1,hz.2.2.2,hw.2.2.1,hw.2.2.2] |
| have hs := EuclideanSpace.dist_sq_eq z w |
| simp only [Fin.sum_univ_two, Real.dist_eq, sq_abs] at hs |
| have hs0 := sq_le_sq₀ (abs_nonneg _) hQ |>.mpr h0 |
| have hs1 := sq_le_sq₀ (abs_nonneg _) hQ |>.mpr h1 |
| rw [sq_abs] at hs0 hs1 |
| nlinarith [dist_nonneg (x := z) (y := w)] |
|
|
| theorem DyadicSquare.ediam_le (Q : DyadicSquare) (hQ : 0 ≤ Q.side) : |
| ediam Q.carrier ≤ ENNReal.ofReal (2 * Q.side) := by |
| apply Metric.ediam_le |
| intro z hz w hw |
| rw [edist_dist] |
| exact ENNReal.ofReal_le_ofReal (Q.dist_le hQ hz hw) |
|
|
| end PlanarFalconer |
|
|
| end OAI |
|
|
