| import OAI.Geometry.HeilbronnTriangle.CrossNormal |
| import OAI.Geometry.HeilbronnTriangle.NormalGroupCount |
| import OAI.Geometry.HeilbronnTriangle.LatticeReciprocal |
|
|
| namespace OAI |
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|
| 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 |
|
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| end OAI |
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|
