| import OAI.Geometry.HeilbronnTriangle.AnisotropicGeometry |
| import OAI.Geometry.HeilbronnTriangle.AnisotropicDyadic |
| import OAI.Geometry.HeilbronnTriangle.PlaneFiberAssembly |
| import OAI.Geometry.HeilbronnTriangle.IntegerNormalGroups |
| import OAI.Geometry.HeilbronnTriangle.AnisotropicPlaneData |
|
|
| namespace OAI |
|
|
|
|
| noncomputable section |
|
|
| namespace Problem355.Anisotropic |
|
|
| open Matrix |
| open scoped Matrix |
| open PrimitiveNormal |
|
|
| abbrev Triple := PlaneFiberAssembly.Triple |
|
|
| theorem thick_fiber_card_le |
| (S : Finset Triple) (y : IntVector) (d : PlaneData y) (t : ℤ) |
| (R₁ R₂ R₃ : ℝ) (h₂₁ : R₂ ≤ R₁) (h₃₂ : R₃ ≤ R₂) |
| (hlarge : d.second ≤ R₃) |
| (hdet : ∀ x ∈ S, x.1 ⬝ᵥ (x.2.1 ⨯₃ x.2.2) = t) |
| (hcross : ∀ x ∈ S, ∃ a : ℤ, a ≠ 0 ∧ x.2.1 ⨯₃ x.2.2 = a • y) |
| (hplane : ∀ x ∈ S, x.2.1 ⬝ᵥ y = 0 ∧ x.2.2 ⬝ᵥ y = 0) |
| (hnorm : ∀ x ∈ S, length x.1 ≤ R₁ ∧ |
| length x.2.1 ≤ R₂ ∧ length x.2.2 ≤ R₃) : |
| (S.card : ℝ) ≤ 2916 * Real.pi ^ 3 * (R₁ * R₂ * R₃) ^ 2 / length y ^ 3 := by |
| have h := PlaneFiberAssembly.triple_count_of_plane_counts S y t R₁ R₂ R₃ |
| (36 * Real.pi * R₁ ^ 2 / length y) |
| (9 * Real.pi * R₂ ^ 2 / length y) |
| (9 * Real.pi * R₃ ^ 2 / length y) |
| (by positivity) (by positivity) hdet hcross hplane hnorm |
| (fun T center hT => affine_plane_card_le y d T center R₁ |
| (hlarge.trans (h₃₂.trans h₂₁)) (fun v hv => (hT v hv).2) |
| (fun v hv w hw => by rw [sub_dotProduct, (hT v hv).1, (hT w hw).1, sub_self])) |
| (fun T hT => d.large_count T R₂ (hlarge.trans h₃₂) hT) |
| (fun T hT => d.large_count T R₃ hlarge hT) |
| convert h using 1; ring |
|
|
| theorem thin_fiber_card_le |
| (S : Finset Triple) (y : IntVector) (d : PlaneData y) (t : ℤ) |
| (R₁ R₂ R₃ : ℝ) (h₂₁ : R₂ ≤ R₁) |
| (hlarge : d.second ≤ R₂) (hfirst : d.first ≤ R₃) (hthin : R₃ < d.second) |
| (hdet : ∀ x ∈ S, x.1 ⬝ᵥ (x.2.1 ⨯₃ x.2.2) = t) |
| (hcross : ∀ x ∈ S, ∃ a : ℤ, a ≠ 0 ∧ x.2.1 ⨯₃ x.2.2 = a • y) |
| (hplane : ∀ x ∈ S, x.2.1 ⬝ᵥ y = 0 ∧ x.2.2 ⬝ᵥ y = 0) |
| (hnorm : ∀ x ∈ S, length x.1 ≤ R₁ ∧ |
| length x.2.1 ≤ R₂ ∧ length x.2.2 ≤ R₃) : |
| (S.card : ℝ) ≤ 972 * Real.pi ^ 2 * R₁ ^ 2 * R₂ ^ 2 * R₃ / |
| (length y ^ 2 * d.first) := by |
| have h := PlaneFiberAssembly.triple_count_of_plane_counts S y t R₁ R₂ R₃ |
| (36 * Real.pi * R₁ ^ 2 / length y) |
| (9 * Real.pi * R₂ ^ 2 / length y) (3 * R₃ / d.first) |
| (by positivity) (by positivity) hdet hcross hplane hnorm |
| (fun T center hT => affine_plane_card_le y d T center R₁ |
| (hlarge.trans h₂₁) (fun v hv => (hT v hv).2) |
| (fun v hv w hw => by rw [sub_dotProduct, (hT v hv).1, (hT w hw).1, sub_self])) |
| (fun T hT => d.large_count T R₂ hlarge hT) |
| (fun T hT => d.small_count T R₃ hfirst hthin hT) |
| convert h using 1; ring |
|
|
| theorem scales_le_of_nonempty |
| (S : Finset Triple) (y : IntVector) (d : PlaneData y) (t : ℤ) (ht : t ≠ 0) |
| (R₂ R₃ : ℝ) (h₃₂ : R₃ ≤ R₂) (hS : S.Nonempty) |
| (hdet : ∀ x ∈ S, x.1 ⬝ᵥ (x.2.1 ⨯₃ x.2.2) = t) |
| (hplane : ∀ x ∈ S, x.2.1 ⬝ᵥ y = 0 ∧ x.2.2 ⬝ᵥ y = 0) |
| (hnorm : ∀ x ∈ S, length x.2.1 ≤ R₂ ∧ length x.2.2 ≤ R₃) : |
| d.first ≤ R₃ ∧ d.second ≤ R₂ := by |
| obtain ⟨x, hx⟩ := hS |
| have hc : x.2.1 ⨯₃ x.2.2 ≠ 0 := by |
| intro hc |
| apply ht |
| have h := hdet x hx |
| simpa [hc] using h.symm |
| have hv : x.2.2 ≠ 0 := by |
| intro hv |
| apply hc |
| simp [hv] |
| refine ⟨(d.first_le _ (hplane x hx).2 hv).trans (hnorm x hx).2, ?_⟩ |
| exact (d.second_le _ _ (hplane x hx).1 (hplane x hx).2 hc).trans |
| (max_le (hnorm x hx).1 ((hnorm x hx).2.trans h₃₂)) |
|
|
| theorem thin_shell_sum_le |
| (S : Finset IntVector) (first weight : IntVector → ℝ) |
| (R₁ R₂ R₃ U Y C₀ C₁ : ℝ) |
| (hU : 0 < U) (hY : 0 < Y) (hR₃ : 0 ≤ R₃) (hUR : U ≤ R₃) |
| (hC₀ : 0 ≤ C₀) (hC₁ : 0 ≤ C₁) |
| (hcard : (S.card : ℝ) ≤ C₀ * U ^ 2 * Y ^ 2) |
| (hrange : ∀ y ∈ S, U ≤ first y ∧ Y ≤ length y) |
| (hweight : ∀ y ∈ S, weight y ≤ |
| C₁ * R₁ ^ 2 * R₂ ^ 2 * R₃ / (length y ^ 2 * first y)) : |
| ∑ y ∈ S, weight y ≤ C₀ * C₁ * (R₁ * R₂ * R₃) ^ 2 := by |
| have hB : 0 ≤ C₁ * R₁ ^ 2 * R₂ ^ 2 * R₃ / (Y ^ 2 * U) := by positivity |
| calc |
| ∑ y ∈ S, weight y ≤ |
| ∑ _y ∈ S, C₁ * R₁ ^ 2 * R₂ ^ 2 * R₃ / (Y ^ 2 * U) := by |
| apply Finset.sum_le_sum |
| intro y hy |
| apply (hweight y hy).trans |
| apply div_le_div_of_nonneg_left (by positivity) (by positivity) |
| exact mul_le_mul (pow_le_pow_left₀ hY.le (hrange y hy).2 2) |
| (hrange y hy).1 hU.le (sq_nonneg _) |
| _ = (S.card : ℝ) * (C₁ * R₁ ^ 2 * R₂ ^ 2 * R₃ / (Y ^ 2 * U)) := by simp |
| _ ≤ (C₀ * U ^ 2 * Y ^ 2) * |
| (C₁ * R₁ ^ 2 * R₂ ^ 2 * R₃ / (Y ^ 2 * U)) := |
| mul_le_mul_of_nonneg_right hcard hB |
| _ = (C₀ * C₁) * U ^ 2 * Y ^ 2 * (R₁ ^ 2 / Y) * |
| (R₂ ^ 2 / Y) * (R₃ / U) := by field_simp |
| _ ≤ (C₀ * C₁) * (R₁ * R₂ * R₃) ^ 2 := |
| thin_normal_group_numeric hU hY hR₃ hUR (mul_nonneg hC₀ hC₁) |
|
|
| theorem thin_normal_sum_le |
| (S : Finset IntVector) (first weight : IntVector → ℝ) |
| (R₁ R₂ R₃ C₀ C₁ : ℝ) |
| (h₂ : 1 ≤ R₂) (h₃ : 1 ≤ R₃) |
| (h₂₁ : R₂ ≤ R₁) (h₃₁ : R₃ ≤ R₁) |
| (hC₀ : 0 ≤ C₀) (hC₁ : 0 ≤ C₁) |
| (hweight0 : ∀ y ∈ S, 0 ≤ weight y) |
| (hrange : ∀ y ∈ S, 1 ≤ first y ∧ first y ≤ R₃ ∧ |
| 1 ≤ length y ∧ length y ≤ R₂ * R₃) |
| (hweight : ∀ y ∈ S, weight y ≤ |
| C₁ * R₁ ^ 2 * R₂ ^ 2 * R₃ / (length y ^ 2 * first y)) |
| (hgroup : ∀ (T : Finset IntVector), T ⊆ S → ∀ U Y : ℝ, |
| 1 ≤ U → 1 ≤ Y → |
| (∀ y ∈ T, U ≤ first y ∧ first y ≤ 2 * U ∧ |
| Y ≤ length y ∧ length y ≤ 2 * Y) → |
| (T.card : ℝ) ≤ C₀ * U ^ 2 * Y ^ 2) : |
| ∑ y ∈ S, weight y ≤ |
| 2 * C₀ * C₁ * (R₁ * R₂ * R₃) ^ 2 * |
| (Real.log (2 * R₁) / Real.log 2) ^ 2 := by |
| classical |
| have hprod : 1 ≤ R₂ * R₃ := by nlinarith |
| have hA : 0 ≤ C₀ * C₁ * (R₁ * R₂ * R₃) ^ 2 := by positivity |
| have h := sum_le_of_dyadic_shell_estimates S first length weight h₃ hprod hA |
| hweight0 hrange (by |
| intro i j |
| let T := S.filter fun y => (2 : ℝ) ^ i ≤ first y ∧ first y < 2 ^ (i + 1) ∧ |
| (2 : ℝ) ^ j ≤ length y ∧ length y < 2 ^ (j + 1) |
| have hU : 1 ≤ (2 : ℝ) ^ i := one_le_pow₀ (by norm_num) |
| have hY : 1 ≤ (2 : ℝ) ^ j := one_le_pow₀ (by norm_num) |
| have hTS : T ⊆ S := Finset.filter_subset _ _ |
| have hT (y : IntVector) (hy : y ∈ T) : |
| (2 : ℝ) ^ i ≤ first y ∧ first y ≤ 2 * 2 ^ i ∧ |
| (2 : ℝ) ^ j ≤ length y ∧ length y ≤ 2 * 2 ^ j := by |
| obtain ⟨_, h₁, h₂, h₃, h₄⟩ := Finset.mem_filter.mp hy |
| exact ⟨h₁, by simpa [pow_succ, mul_comm] using h₂.le, |
| h₃, by simpa [pow_succ, mul_comm] using h₄.le⟩ |
| change ∑ y ∈ T, weight y ≤ _ |
| by_cases hne : T.Nonempty |
| · obtain ⟨y, hy⟩ := hne |
| exact thin_shell_sum_le T first weight R₁ R₂ R₃ _ _ C₀ C₁ |
| (by positivity) (by positivity) (by linarith) |
| ((hT y hy).1.trans (hrange y (hTS hy)).2.1) hC₀ hC₁ |
| (hgroup T hTS _ _ hU hY hT) |
| (fun z hz => ⟨(hT z hz).1, (hT z hz).2.2.1⟩) |
| (fun z hz => hweight z (hTS hz)) |
| · rw [Finset.not_nonempty_iff_eq_empty.mp hne] |
| simpa using hA) |
| calc |
| ∑ y ∈ S, weight y ≤ |
| (C₀ * C₁ * (R₁ * R₂ * R₃) ^ 2) * |
| ((Real.log (2 * R₃) / Real.log 2) * |
| (Real.log (2 * (R₂ * R₃)) / Real.log 2)) := by |
| simpa only [mul_assoc] using h |
| _ ≤ (C₀ * C₁ * (R₁ * R₂ * R₃) ^ 2) * |
| (2 * (Real.log (2 * R₁) / Real.log 2) ^ 2) := |
| mul_le_mul_of_nonneg_left (dyadic_log_product_le h₂ h₃ h₂₁ h₃₁) hA |
| _ = _ := by ring |
|
|
| theorem triple_card_le_of_normal_assignment |
| (hplanes : ∀ y : IntVector, IsPrimitive y → Nonempty (PlaneData y)) |
| (S : Finset Triple) (normal : Triple → IntVector) (t : ℤ) (ht : t ≠ 0) |
| (R₁ R₂ R₃ : ℝ) (h₃ : 1 ≤ R₃) (h₃₂ : R₃ ≤ R₂) (h₂₁ : R₂ ≤ R₁) |
| (hdet : ∀ x ∈ S, x.1 ⬝ᵥ (x.2.1 ⨯₃ x.2.2) = t) |
| (hprim : ∀ x ∈ S, IsPrimitive (normal x)) |
| (hcross : ∀ x ∈ S, ∃ a : ℤ, a ≠ 0 ∧ x.2.1 ⨯₃ x.2.2 = a • normal x) |
| (hplane : ∀ x ∈ S, x.2.1 ⬝ᵥ normal x = 0 ∧ x.2.2 ⬝ᵥ normal x = 0) |
| (hnormal : ∀ x ∈ S, length (normal x) ≤ R₂ * R₃) |
| (hnorm : ∀ x ∈ S, length x.1 ≤ R₁ ∧ |
| length x.2.1 ≤ R₂ ∧ length x.2.2 ≤ R₃) : |
| (S.card : ℝ) ≤ 35721000 * Real.pi ^ 3 * (R₁ * R₂ * R₃) ^ 2 * |
| (Real.log (2 * R₁) / Real.log 2) ^ 2 := by |
| classical |
| let Y := S.image normal |
| let F (y : IntVector) := S.filter (fun x => normal x = y) |
| let weight (y : IntVector) : ℝ := (F y).card |
| let data (y : IntVector) (hy : IsPrimitive y) : PlaneData y := |
| Classical.choice (hplanes y hy) |
| let first (y : IntVector) := if hy : IsPrimitive y then (data y hy).first else 1 |
| let second (y : IntVector) := if hy : IsPrimitive y then (data y hy).second else 1 |
| have hY (y : IntVector) (hy : y ∈ Y) : IsPrimitive y ∧ |
| 1 ≤ length y ∧ length y ≤ R₂ * R₃ ∧ (F y).Nonempty := by |
| obtain ⟨x, hx, rfl⟩ := Finset.mem_image.mp hy |
| exact ⟨hprim x hx, LatticeBox.one_le_norm_realVector (hprim x hx).ne_zero, |
| hnormal x hx, ⟨x, Finset.mem_filter.mpr ⟨hx, rfl⟩⟩⟩ |
| have hFdet (y : IntVector) : ∀ x ∈ F y, |
| x.1 ⬝ᵥ (x.2.1 ⨯₃ x.2.2) = t := |
| fun x hx => hdet x (Finset.mem_filter.mp hx).1 |
| have hFcross (y : IntVector) : ∀ x ∈ F y, |
| ∃ a : ℤ, a ≠ 0 ∧ x.2.1 ⨯₃ x.2.2 = a • y := by |
| intro x hx |
| obtain ⟨hxS, hxy⟩ := Finset.mem_filter.mp hx |
| simpa only [hxy] using hcross x hxS |
| have hFplane (y : IntVector) : ∀ x ∈ F y, |
| x.2.1 ⬝ᵥ y = 0 ∧ x.2.2 ⬝ᵥ y = 0 := by |
| intro x hx |
| obtain ⟨hxS, hxy⟩ := Finset.mem_filter.mp hx |
| simpa only [hxy] using hplane x hxS |
| have hFnorm (y : IntVector) : ∀ x ∈ F y, length x.1 ≤ R₁ ∧ |
| length x.2.1 ≤ R₂ ∧ length x.2.2 ≤ R₃ := |
| fun x hx => hnorm x (Finset.mem_filter.mp hx).1 |
| have hscale (y : IntVector) (hy : y ∈ Y) : |
| 1 ≤ first y ∧ first y ≤ R₃ ∧ second y ≤ R₂ := by |
| have hp := (hY y hy).1 |
| have hs := scales_le_of_nonempty (F y) y (data y hp) t ht R₂ R₃ h₃₂ |
| (hY y hy).2.2.2 (hFdet y) (hFplane y) (fun x hx => (hFnorm y x hx).2) |
| simpa only [first, second, dite_eq_left hp] using |
| And.intro (data y hp).one_le_first hs |
| let E := Y.filter (fun y => second y ≤ R₃) |
| let T := Y.filter (fun y => ¬ second y ≤ R₃) |
| have h₂ : 1 ≤ R₂ := h₃.trans h₃₂ |
| have h₁ : 1 ≤ R₁ := h₂.trans h₂₁ |
| have hRprod : 1 ≤ R₂ * R₃ := by nlinarith |
| have hthin : ∑ y ∈ T, weight y ≤ |
| 2 * (18000 * Real.pi) * (972 * Real.pi ^ 2) * (R₁ * R₂ * R₃) ^ 2 * |
| (Real.log (2 * R₁) / Real.log 2) ^ 2 := by |
| apply thin_normal_sum_le T first weight R₁ R₂ R₃ _ _ h₂ h₃ h₂₁ (h₃₂.trans h₂₁) |
| (by positivity) (by positivity) |
| · intro y hy |
| exact Nat.cast_nonneg _ |
| · intro y hy |
| have hyY := (Finset.mem_filter.mp hy).1 |
| exact ⟨(hscale y hyY).1, (hscale y hyY).2.1, |
| (hY y hyY).2.1, (hY y hyY).2.2.1⟩ |
| · intro y hy |
| obtain ⟨hyY, hyt⟩ := Finset.mem_filter.mp hy |
| have hp := (hY y hyY).1 |
| have hlarge : (data y hp).second ≤ R₂ := by |
| simpa only [second, dite_eq_left hp] using (hscale y hyY).2.2 |
| have hfirst : (data y hp).first ≤ R₃ := by |
| simpa only [first, dite_eq_left hp] using (hscale y hyY).2.1 |
| have hlow : R₃ < (data y hp).second := by |
| simpa only [second, dite_eq_left hp] using lt_of_not_ge hyt |
| simpa only [first, dite_eq_left hp, weight] using |
| thin_fiber_card_le (F y) y (data y hp) t R₁ R₂ R₃ h₂₁ |
| hlarge hfirst hlow (hFdet y) (hFcross y) (hFplane y) (hFnorm y) |
| · intro A hAT U Z hU hZ hA |
| apply normal_shell_card_le hplanes hU (by linarith) A |
| intro y hy |
| have hyY := (Finset.mem_filter.mp (hAT hy)).1 |
| have hp := (hY y hyY).1 |
| refine ⟨hp, (hA y hy).2.2.2, data y hp, ?_, ?_⟩ |
| · simpa only [first, dite_eq_left hp] using (hA y hy).1 |
| · simpa only [first, dite_eq_left hp] using (hA y hy).2.1 |
| have heasy : ∑ y ∈ E, weight y ≤ |
| (2916 * Real.pi ^ 3 * (R₁ * R₂ * R₃) ^ 2) * |
| ((125 / Real.log 2) * Real.log (2 * (R₂ * R₃))) := by |
| calc |
| ∑ y ∈ E, weight y ≤ ∑ y ∈ E, |
| (2916 * Real.pi ^ 3 * (R₁ * R₂ * R₃) ^ 2) / length y ^ 3 := by |
| apply Finset.sum_le_sum |
| intro y hy |
| obtain ⟨hyY, hye⟩ := Finset.mem_filter.mp hy |
| have hp := (hY y hyY).1 |
| have hlarge : (data y hp).second ≤ R₃ := by |
| simpa only [second, dite_eq_left hp] using hye |
| exact thick_fiber_card_le (F y) y (data y hp) t R₁ R₂ R₃ h₂₁ h₃₂ |
| hlarge (hFdet y) (hFcross y) (hFplane y) (hFnorm y) |
| _ = (2916 * Real.pi ^ 3 * (R₁ * R₂ * R₃) ^ 2) * |
| ∑ y ∈ E, 1 / length y ^ 3 := by |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro y hy |
| ring |
| _ ≤ _ := mul_le_mul_of_nonneg_left |
| (LatticeBox.reciprocal_cube_sum_le_log hRprod E (by |
| intro y hy |
| have h := hY y (Finset.mem_filter.mp hy).1 |
| exact ⟨h.1.ne_zero, h.2.2.1⟩)) (by positivity) |
| have hlog2 : 0 < Real.log 2 := Real.log_pos (by norm_num) |
| have hlogprod : 0 ≤ Real.log (2 * (R₂ * R₃)) / Real.log 2 := by |
| apply div_nonneg _ hlog2.le |
| exact Real.log_nonneg (by nlinarith) |
| have hlog3 : 1 ≤ Real.log (2 * R₃) / Real.log 2 := by |
| apply (le_div_iff₀ hlog2).mpr |
| simpa using Real.log_le_log (by norm_num : (0 : ℝ) < 2) (by linarith : 2 ≤ 2 * R₃) |
| have hlogbound : Real.log (2 * (R₂ * R₃)) / Real.log 2 ≤ |
| 2 * (Real.log (2 * R₁) / Real.log 2) ^ 2 := by |
| exact (le_mul_of_one_le_left hlogprod hlog3).trans |
| (dyadic_log_product_le h₂ h₃ h₂₁ (h₃₂.trans h₂₁)) |
| have heasy' : ∑ y ∈ E, weight y ≤ |
| 250 * 2916 * Real.pi ^ 3 * (R₁ * R₂ * R₃) ^ 2 * |
| (Real.log (2 * R₁) / Real.log 2) ^ 2 := by |
| calc |
| ∑ y ∈ E, weight y ≤ _ := heasy |
| _ = (125 * 2916 * Real.pi ^ 3 * (R₁ * R₂ * R₃) ^ 2) * |
| (Real.log (2 * (R₂ * R₃)) / Real.log 2) := by ring |
| _ ≤ (125 * 2916 * Real.pi ^ 3 * (R₁ * R₂ * R₃) ^ 2) * |
| (2 * (Real.log (2 * R₁) / Real.log 2) ^ 2) := |
| mul_le_mul_of_nonneg_left hlogbound (by positivity) |
| _ = _ := by ring |
| have hcard : (S.card : ℝ) = ∑ y ∈ Y, weight y := by |
| dsimp only [weight, F] |
| exact_mod_cast (Finset.card_eq_sum_card_fiberwise |
| (s := S) (t := Y) (f := normal) (fun x hx => Finset.mem_image_of_mem normal hx)) |
| have hsplit : (∑ y ∈ E, weight y) + (∑ y ∈ T, weight y) = |
| ∑ y ∈ Y, weight y := Finset.sum_filter_add_sum_filter_not Y _ _ |
| rw [hcard, ← hsplit] |
| calc |
| _ ≤ _ := add_le_add heasy' hthin |
| _ = _ := by ring |
|
|
| theorem anisotropic_count_of_planeData |
| (hplanes : ∀ y : IntVector, IsPrimitive y → Nonempty (PlaneData y)) |
| (S : Finset Triple) (t : ℤ) (ht : t ≠ 0) |
| (R₁ R₂ R₃ : ℝ) (h₃ : 1 ≤ R₃) (h₃₂ : R₃ ≤ R₂) (h₂₁ : R₂ ≤ R₁) |
| (hdet : ∀ x ∈ S, x.1 ⬝ᵥ (x.2.1 ⨯₃ x.2.2) = t) |
| (hnorm : ∀ x ∈ S, length x.1 ≤ R₁ ∧ |
| length x.2.1 ≤ R₂ ∧ length x.2.2 ≤ R₃) : |
| (S.card : ℝ) ≤ (35721000 * Real.pi ^ 3 / (Real.log 2) ^ 2) * |
| (R₁ * R₂ * R₃) ^ 2 * Real.log (2 * R₁) ^ 2 := by |
| classical |
| have hex : ∀ x : Triple, ∃ y : IntVector, x ∈ S → |
| IsPrimitive y ∧ (∃ a : ℤ, a ≠ 0 ∧ x.2.1 ⨯₃ x.2.2 = a • y) ∧ |
| (x.2.1 ⬝ᵥ y = 0 ∧ x.2.2 ⬝ᵥ y = 0) ∧ length y ≤ R₂ * R₃ := by |
| intro x |
| by_cases hx : x ∈ S |
| · have hnz : x.2.1 ⨯₃ x.2.2 ≠ 0 := by |
| intro hc |
| apply ht |
| have hd := hdet x hx |
| simpa [hc] using hd.symm |
| obtain ⟨a, ha, y, hyp, hfactor, horth₁, horth₂, hyl, _⟩ := |
| CrossNormal.exists_bounded_primitive_cross_factor x.2.1 x.2.2 hnz |
| refine ⟨y, fun _ => ⟨hyp, ⟨a, ne_of_gt ha, hfactor⟩, ⟨horth₁, horth₂⟩, ?_⟩⟩ |
| exact hyl.trans (mul_le_mul (hnorm x hx).2.1 (hnorm x hx).2.2 |
| (norm_nonneg _) ((norm_nonneg _).trans (hnorm x hx).2.1)) |
| · exact ⟨0, fun h => (hx h).elim⟩ |
| choose normal hnormal using hex |
| have h := triple_card_le_of_normal_assignment hplanes S normal t ht |
| R₁ R₂ R₃ h₃ h₃₂ h₂₁ hdet (fun x hx => (hnormal x hx).1) |
| (fun x hx => (hnormal x hx).2.1) (fun x hx => (hnormal x hx).2.2.1) |
| (fun x hx => (hnormal x hx).2.2.2) hnorm |
| convert h using 1; simp only [div_pow]; ring |
|
|
| theorem anisotropic_count |
| (S : Finset Triple) (t : ℤ) (ht : t ≠ 0) |
| (R₁ R₂ R₃ : ℝ) (h₃ : 1 ≤ R₃) (h₃₂ : R₃ ≤ R₂) (h₂₁ : R₂ ≤ R₁) |
| (hdet : ∀ x ∈ S, x.1 ⬝ᵥ (x.2.1 ⨯₃ x.2.2) = t) |
| (hnorm : ∀ x ∈ S, length x.1 ≤ R₁ ∧ |
| length x.2.1 ≤ R₂ ∧ length x.2.2 ≤ R₃) : |
| (S.card : ℝ) ≤ (35721000 * Real.pi ^ 3 / (Real.log 2) ^ 2) * |
| (R₁ * R₂ * R₃) ^ 2 * Real.log (2 * R₁) ^ 2 := |
| anisotropic_count_of_planeData exists_planeData S t ht R₁ R₂ R₃ |
| h₃ h₃₂ h₂₁ hdet hnorm |
|
|
| theorem anisotropic_constant_pos : |
| 0 < 35721000 * Real.pi ^ 3 / (Real.log 2) ^ 2 := by |
| have h : 0 < Real.log 2 := Real.log_pos (by norm_num) |
| positivity |
|
|
| end Problem355.Anisotropic |
|
|
| end |
|
|
| end OAI |
|
|
