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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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 | import OAI.Geometry.HeilbronnTriangle.AuxiliarySamplingLaw
import OAI.Geometry.HeilbronnTriangle.AuxiliaryIntegralSupport
namespace OAI
noncomputable section
namespace Problem355.AuxiliarySampling
open scoped BigOperators
open LiftingProbability
namespace Law
variable {q H : ℕ} [Fact q.Prime]
theorem exists_outcome_of_weight_pos (L : Law q H) (p : Fin 3 → Fin 3 → ZMod q)
(hpositive : 0 < auxiliaryWeight q L.size L.weight L.sets p) :
∃ ω : L.Outcome, ∀ i, p i ∈ L.sets ω := by
classical
have hinc : 0 < inclusionMass L.weight L.sets p := by
by_contra hn
have hn' : inclusionMass L.weight L.sets p ≤ 0 := le_of_not_gt hn
have hbad : auxiliaryWeight q L.size L.weight L.sets p ≤ 0 := by
unfold auxiliaryWeight
exact mul_nonpos_of_nonneg_of_nonpos (by positivity) hn'
exact (not_le_of_gt hpositive) hbad
by_contra hn
have hnone : ∀ ω : L.Outcome, ¬ (∀ i, p i ∈ L.sets ω) := by
intro ω hω
exact hn ⟨ω, hω⟩
have hz : inclusionMass L.weight L.sets p = 0 := by
unfold inclusionMass
apply Finset.sum_eq_zero
intro ω hω
exact ite_eq_right (hnone ω)
exact (ne_of_gt hinc) hz
theorem support_of_weight_pos (L : Law q H) (p : Fin 3 → Fin 3 → ZMod q)
(hpositive : 0 < auxiliaryWeight q L.size L.weight L.sets p) :
(∀ i, p i ≠ 0) ∧
(AffineIndependent (ZMod q) p ∨ ∃ i j, i ≠ j ∧ p i = p j) ∧
(AffineIndependent (ZMod q) p → ∀ x : Fin 3 → ℤ,
(∀ i, |x i| ≤ (H : ℤ)) → x ≠ 0 →
(∑ i, (x i : ZMod q) • p i) ≠ 0) := by
classical
obtain ⟨ω, hω⟩ := L.exists_outcome_of_weight_pos p hpositive
refine ⟨?_, ?_, ?_⟩
· intro i hi
exact L.zero_not_mem ω (hi ▸ hω i)
· by_cases hinj : Function.Injective p
· exact Or.inl (AuxiliarySet.affineIndependent_of_mem_cap (L.isCap ω) hω hinj)
· right
simp only [Function.Injective] at hinj
push Not at hinj
obtain ⟨i, j, heq, hne⟩ := hinj
exact ⟨i, j, hne, heq⟩
· intro hAI x hx hne
exact L.short_relation ω p hω hAI.injective x hx hne
theorem matrix_short_kernel_excluded (L : Law q H)
(A : Matrix (Fin 3) (Fin 3) (ZMod q))
(hpositive : 0 < auxiliaryWeight q L.size L.weight L.sets A.col)
(hAI : AffineIndependent (ZMod q) A.col)
(x : Fin 3 → ℤ) (hx : ∀ i, |x i| ≤ (H : ℤ)) (hne : x ≠ 0) :
A.mulVec (fun i => (x i : ZMod q)) ≠ 0 := by
intro hrel
have hno := (L.support_of_weight_pos A.col hpositive).2.2 hAI x hx hne
apply hno
funext i
have hi := congrFun hrel i
simpa [Matrix.mulVec, Matrix.col, dotProduct, Finset.sum_apply, mul_comm] using hi
theorem kernel_norm_gt_of_weight_pos (L : Law q H)
(A : Matrix (Fin 3) (Fin 3) ℤ)
(hpositive : 0 < auxiliaryWeight q L.size L.weight L.sets
(A.map (Int.castRingHom (ZMod q))).col)
(hAI : AffineIndependent (ZMod q) (A.map (Int.castRingHom (ZMod q))).col)
{x : PrimitiveNormal.Vector} (hne : x ≠ 0) (hkernel : A.mulVec x = 0) :
(H : ℝ) < ‖PrimitiveNormal.toEuclidean x‖ := by
by_contra hn
have hx := AuxiliarySet.coordinate_abs_le_of_euclidean_norm_le (le_of_not_gt hn)
have hno := L.matrix_short_kernel_excluded (A.map (Int.castRingHom (ZMod q)))
hpositive hAI x hx hne
exact hno (AuxiliarySet.cast_mulVec_eq_zero hkernel)
theorem kernel_norm_gt_main_square {h : ℕ} (L : Law q (h ^ 2))
(A : Matrix (Fin 3) (Fin 3) ℤ)
(hpositive : 0 < auxiliaryWeight q L.size L.weight L.sets
(A.map (Int.castRingHom (ZMod q))).col)
(hAI : AffineIndependent (ZMod q) (A.map (Int.castRingHom (ZMod q))).col)
{x : PrimitiveNormal.Vector} (hne : x ≠ 0) (hkernel : A.mulVec x = 0) :
(h : ℝ) ^ 2 < ‖PrimitiveNormal.toEuclidean x‖ := by
simpa only [Nat.cast_pow] using L.kernel_norm_gt_of_weight_pos A hpositive hAI hne hkernel
end Law
end Problem355.AuxiliarySampling
end
end OAI
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