File size: 2,893 Bytes
7d9acbd | 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 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 | import OAI.Geometry.HeilbronnTriangle.CrossNormal
import OAI.Geometry.HeilbronnTriangle.NormalGroupCount
import OAI.Geometry.HeilbronnTriangle.LatticeReciprocal
namespace OAI
noncomputable section
namespace Problem355.Anisotropic
open Matrix
open scoped Matrix
open PrimitiveNormal
abbrev IntVector := Fin 3 → ℤ
abbrev length (v : IntVector) : ℝ := ‖LatticeBox.realVector v‖
structure PlaneData (y : IntVector) where
first : ℝ
second : ℝ
one_le_first : 1 ≤ first
first_le_second : first ≤ second
second_le_length : second ≤ length y
first_le : ∀ v : IntVector, v ⬝ᵥ y = 0 → v ≠ 0 → first ≤ length v
second_le : ∀ v w : IntVector, v ⬝ᵥ y = 0 → w ⬝ᵥ y = 0 →
v ⨯₃ w ≠ 0 → second ≤ max (length v) (length w)
shortest : ∃ z : IntVector, IsPrimitive z ∧ z ⬝ᵥ y = 0 ∧ length z = first
large_count : ∀ (S : Finset IntVector) (R : ℝ), second ≤ R →
(∀ v ∈ S, v ⬝ᵥ y = 0 ∧ length v ≤ R) →
(S.card : ℝ) ≤ 9 * Real.pi * R ^ 2 / length y
small_count : ∀ (S : Finset IntVector) (R : ℝ), first ≤ R → R < second →
(∀ v ∈ S, v ⬝ᵥ y = 0 ∧ length v ≤ R) →
(S.card : ℝ) ≤ 3 * R / first
theorem affine_plane_card_le (y : IntVector) (d : PlaneData y)
(S : Finset IntVector) (center : EuclideanSpace ℝ (Fin 3))
(R : ℝ) (hR : d.second ≤ R)
(hsize : ∀ v ∈ S, ‖LatticeBox.realVector v - center‖ ≤ R)
(hplane : ∀ v ∈ S, ∀ w ∈ S, (v - w) ⬝ᵥ y = 0) :
(S.card : ℝ) ≤ 36 * Real.pi * R ^ 2 / length y := by
classical
have hR0 : 0 ≤ R := (show (0 : ℝ) ≤ 1 by norm_num).trans
(d.one_le_first.trans (d.first_le_second.trans hR))
by_cases hS : S.Nonempty
· obtain ⟨w, hw⟩ := hS
let T := S.image (fun v => v - w)
have hcard : T.card = S.card := Finset.card_image_of_injective _
(fun _ _ h => sub_left_inj.mp h)
have hT (v : IntVector) (hv : v ∈ T) :
v ⬝ᵥ y = 0 ∧ length v ≤ 2 * R := by
obtain ⟨u, hu, rfl⟩ := Finset.mem_image.mp hv
refine ⟨hplane u hu w hw, ?_⟩
have hcast : LatticeBox.realVector (u - w) =
LatticeBox.realVector u - LatticeBox.realVector w := by
ext i
simp [LatticeBox.realVector]
change ‖LatticeBox.realVector (u - w)‖ ≤ 2 * R
rw [hcast]
rw [show LatticeBox.realVector u - LatticeBox.realVector w =
(LatticeBox.realVector u - center) - (LatticeBox.realVector w - center) by abel]
exact (norm_sub_le _ _).trans (by linarith [hsize u hu, hsize w hw])
have h := d.large_count T (2 * R) (by linarith) hT
rw [hcard] at h
convert h using 1; ring
· have hzero : S = ∅ := Finset.not_nonempty_iff_eq_empty.mp hS
rw [hzero]
simp only [Finset.card_empty, Nat.cast_zero]
positivity
end Problem355.Anisotropic
end
end OAI
|