| import OAI.Geometry.HeilbronnTriangle.ZeroGlobalCount |
| import OAI.Geometry.HeilbronnTriangle.AuxiliaryZeroWeights |
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| namespace OAI |
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| noncomputable section |
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| namespace Problem355.AuxiliaryZeroCount |
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| open Matrix PrimitiveNormal ZeroDeterminantCount LiftingProbability |
| open scoped BigOperators |
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| def countConstant : ℝ := |
| ZeroGlobalCount.countConstant 8 (2 * 2000 ^ 6) (4 * 2000 ^ 3) |
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| lemma countConstant_nonneg : 0 ≤ countConstant := |
| ZeroGlobalCount.countConstant_nonneg (by norm_num) (by positivity) (by positivity) |
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| theorem weighted_count_le_log |
| (D : RowData) (q N : ℕ) [Fact q.Prime] |
| (hN : 1 ≤ N) (hEq : IsUnit (D.E : ZMod q)) |
| (hparam : D.h ^ 26 ≤ (q : ℝ)) |
| (L : AuxiliarySampling.Law q ((D.B ^ D.k) ^ 2)) |
| (S : Finset (Matrix (Fin 3) (Fin 3) ℤ)) |
| (hdet : ∀ A ∈ S, A.det = 0) |
| (hrows : ∀ A ∈ S, ∀ i, A i ∈ D.L) |
| (hheight : ∀ A ∈ S, ∀ i, A 2 i ≠ 0) |
| (hproj : ∀ A ∈ S, ∀ i j : Fin 3, i ≠ j → |
| ((A 0 i : ℝ) / A 2 i, (A 1 i : ℝ) / A 2 i) ≠ |
| ((A 0 j : ℝ) / A 2 j, (A 1 j : ℝ) / A 2 j)) |
| (hbound : ∀ A ∈ S, ∀ i j, |A i j| ≤ (2 * N : ℕ)) : |
| (∑ A ∈ S, auxiliaryWeight q L.size L.weight L.sets |
| (A.map (Int.castRingHom (ZMod q))).col) ≤ |
| countConstant * (N : ℝ) ^ 6 / (D.I : ℝ) ^ 2 * |
| (Real.log (4 * (N : ℝ)) / Real.log 2) := by |
| apply ZeroGlobalCount.weighted_count_le_log D q N hN hEq hparam S |
| (fun A => auxiliaryWeight q L.size L.weight L.sets |
| (A.map (Int.castRingHom (ZMod q))).col) |
| 8 (2 * 2000 ^ 6) (4 * 2000 ^ 3) |
| (by norm_num) (by positivity) (by positivity) |
| (fun A _ => auxiliaryWeight_nonneg q L.size L.weight L.sets L.weight_nonneg _) |
| hdet hrows hheight hproj hbound |
| · intro A hA hpositive |
| obtain ⟨ω, hω⟩ := L.exists_outcome_of_weight_pos |
| (A.map (Int.castRingHom (ZMod q))).col hpositive |
| exact ⟨L.sets ω, L.isCap ω, hω⟩ |
| · intro A hA hAI |
| exact L.weight_le_eight _ hAI |
| · intro A hA |
| simpa only [RowData.h, Nat.cast_pow] using |
| L.weight_le_two_main_square (A.map (Int.castRingHom (ZMod q))).col |
| · intro A hA hne |
| simpa only [RowData.h, Nat.cast_pow] using |
| L.weight_le_four_main_square (A.map (Int.castRingHom (ZMod q))).col hne |
| · intro A hA x hx hkernel hAI hpositive |
| simpa only [RowData.h, Nat.cast_pow] using |
| L.kernel_norm_gt_main_square A hpositive hAI hx.ne_zero hkernel |
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| end Problem355.AuxiliaryZeroCount |
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| end |
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| end OAI |
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