| import OAI.Geometry.HeilbronnTriangle.AuxiliarySamplingLaw |
| import OAI.Geometry.HeilbronnTriangle.AuxiliaryIntegralSupport |
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| namespace OAI |
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| noncomputable section |
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| namespace Problem355.AuxiliarySampling |
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| open scoped BigOperators |
| open LiftingProbability |
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| namespace Law |
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| variable {q H : ℕ} [Fact q.Prime] |
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| 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 |
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| 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 |
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| 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 |
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| 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) |
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| 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 |
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| end Law |
| end Problem355.AuxiliarySampling |
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| end |
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| end OAI |
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