| import OAI.MeasureTheory.Falconer.Estimates.ComparablePositiveRebin |
| import OAI.MeasureTheory.Falconer.Geometry.ScaledDyadicPacking |
|
|
| namespace OAI |
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|
| namespace PlanarFalconer |
| open Set Metric MeasureTheory |
| attribute [local instance] Classical.propDecidable |
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|
| theorem dyadic_two_thousand_neighbors (Q : DyadicSquare) (hQ : Q.side = 1) |
| (p : ℕ) (x : Plane) : |
| (((dyadicPaths p).filter (fun c => dist x (Q.descendant c).center ≤ 2000/(2:ℝ)^p)).card : ℝ) ≤ 16032016 := by |
| classical |
| have hpos : 0 < Q.side := by rw [hQ]; norm_num |
| have hsub : (dyadicPaths p).filter (fun c => dist x (Q.descendant c).center ≤ 2000/(2:ℝ)^p) ⊆ |
| cellsMeeting Q p x (2001/(2:ℝ)^p) := by |
| intro c hc |
| obtain ⟨hc,hb⟩ := Finset.mem_filter.mp hc |
| refine Finset.mem_filter.mpr ⟨hc,(Q.descendant c).center, |
| (Q.descendant c).center_mem (Q.descendant_side_pos hpos c),?_⟩ |
| rw [mem_ball,dist_comm] |
| exact hb.trans_lt (div_lt_div_of_pos_right (by norm_num) (by positivity)) |
| have hh := dyadic_scaled_ball_card_le Q hpos p x (C₀ := 2001) (by norm_num) |
| (r := 2001/(2:ℝ)^p) (by rw [hQ]; ring_nf; exact le_rfl) |
| norm_num at hh |
| exact (Nat.cast_le.mpr (Finset.card_le_card hsub)).trans (by exact_mod_cast hh) |
|
|
| theorem angularSeparated_degree_real (Q₁ Q₂ : DyadicSquare) (hQ₁ : Q₁.side = 1) (hQ₂ : Q₂.side = 1) |
| (p : ℕ) (C : Finset (CellPath × CellPath)) |
| (hC : C ⊆ (dyadicPaths p) ×ˢ (dyadicPaths p)) (row : CellPath × CellPath) : |
| ((C.filter (angularSeparated Q₁ Q₂ p row)).card : ℝ) ≤ 257025537024256 := by |
| classical |
| let S₁ := (dyadicPaths p).filter (fun c => |
| dist (Q₁.descendant row.1).center (Q₁.descendant c).center ≤ 2000/(2:ℝ)^p) |
| let S₂ := (dyadicPaths p).filter (fun c => |
| dist (Q₂.descendant row.2).center (Q₂.descendant c).center ≤ 2000/(2:ℝ)^p) |
| have hsub : C.filter (angularSeparated Q₁ Q₂ p row) ⊆ S₁ ×ˢ S₂ := by |
| intro col hc |
| obtain ⟨hc,hnear⟩ := Finset.mem_filter.mp hc |
| simp only [angularSeparated,Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),Real.rpow_natCast,← div_eq_mul_inv] at hnear |
| obtain ⟨h1,h2⟩ := Finset.mem_product.mp (hC hc) |
| exact Finset.mem_product.mpr ⟨Finset.mem_filter.mpr ⟨h1,hnear.1.2⟩, |
| Finset.mem_filter.mpr ⟨h2,hnear.2.2⟩⟩ |
| have h1 := dyadic_two_thousand_neighbors Q₁ hQ₁ p (Q₁.descendant row.1).center |
| have h2 := dyadic_two_thousand_neighbors Q₂ hQ₂ p (Q₂.descendant row.2).center |
| have hh := Nat.cast_le (α := ℝ).mpr (Finset.card_le_card hsub) |
| rw [Finset.card_product,Nat.cast_mul] at hh |
| exact hh.trans (by nlinarith [mul_le_mul h1 h2 (Nat.cast_nonneg S₂.card) (by norm_num : (0:ℝ) ≤ 16032016)]) |
|
|
| theorem angularSeparated_degree (Q₁ Q₂ : DyadicSquare) (hQ₁ : Q₁.side = 1) (hQ₂ : Q₂.side = 1) |
| (p : ℕ) (C : Finset (CellPath × CellPath)) |
| (hC : C ⊆ (dyadicPaths p) ×ˢ (dyadicPaths p)) (row : CellPath × CellPath) : |
| (C.filter (angularSeparated Q₁ Q₂ p row)).card ≤ 257025537024256 := by |
| exact_mod_cast angularSeparated_degree_real Q₁ Q₂ hQ₁ hQ₂ p C hC row |
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|
| end PlanarFalconer |
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| end OAI |
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