File size: 3,150 Bytes
d4317d7 | 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 46 47 48 49 50 51 52 53 54 55 56 57 58 59 | import OAI.MeasureTheory.Falconer.Estimates.ComparablePositiveRebin
import OAI.MeasureTheory.Falconer.Geometry.ScaledDyadicPacking
namespace OAI
namespace PlanarFalconer
open Set Metric MeasureTheory
attribute [local instance] Classical.propDecidable
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
end PlanarFalconer
end OAI
|