| import OAI.Geometry.HeilbronnTriangle.OrbitSampling |
| import OAI.Geometry.HeilbronnTriangle.PrimePowerData |
| import OAI.Geometry.HeilbronnTriangle.OrbitRowLattice |
| import OAI.Geometry.HeilbronnTriangle.LiftedBadEvent |
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| namespace OAI |
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| noncomputable section |
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|
| namespace Problem355.ConditionalBadTripleBound |
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|
| open OrbitSampling IntegerSampling LiftingProbability |
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|
| def orbitTransposeEquiv (h : ℕ) (C : Fin 3 → Fin 3 → ZMod h) : |
| MulAction.orbit (MainGroup h) C ≃ Section04Orbit.slOrbit (Matrix.transpose C) where |
| toFun A := ⟨Matrix.transpose A.val, by |
| obtain ⟨G, hG⟩ := A.property |
| exact ⟨G, by rw [← hG, sl_action_transpose]⟩⟩ |
| invFun A := ⟨Matrix.transpose A.val, by |
| obtain ⟨G, hG⟩ := A.property |
| refine ⟨G, ?_⟩ |
| have ht := (sl_action_transpose h G C).trans hG |
| exact funext fun i => funext fun j => congrFun (congrFun ht j) i⟩ |
| left_inv A := by apply Subtype.ext; exact Matrix.transpose_transpose _ |
| right_inv A := by apply Subtype.ext; exact Matrix.transpose_transpose _ |
|
|
| theorem card_sampling_orbit (h : ℕ) [NeZero h] |
| (C : Fin 3 → Fin 3 → ZMod h) : |
| (orbitFinset (MainGroup h) C).card = |
| Nat.card (Section04Orbit.slOrbit (Matrix.transpose C)) := by |
| rw [card_orbitFinset] |
| exact Nat.card_congr (orbitTransposeEquiv h C) |
|
|
| theorem sampling_orbit_lower_bound {p k : ℕ} [NeZero (p ^ k)] |
| (hp : p.Prime) (hk : 0 < k) |
| (C : Fin 3 → Fin 3 → ZMod (p ^ k)) |
| (d : PrimePowerData p k (Matrix.transpose C)) : |
| (21 / 64 : ℝ) * ((p ^ k : ℕ) : ℝ) ^ 8 / |
| (((p ^ d.b : ℕ) : ℝ) ^ 3 * ((p ^ d.e : ℕ) : ℝ) ^ 2) ≤ |
| ((orbitFinset (MainGroup (p ^ k)) C).card : ℝ) := by |
| let : NeZero (p ^ k) := ⟨pow_ne_zero _ hp.ne_zero⟩ |
| rw [card_sampling_orbit] |
| exact d.orbit_lower_bound hp hk |
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|
| theorem integral_det_residue (h q L : ℕ) [NeZero h] |
| (shift : Fin 3 → ℤ) (C : Fin 3 → Fin 3 → ZMod h) |
| (x : Fin 3 → Box (L * (h * q)) 3 shift) |
| (hx : (fun i => residue h (x i)) ∈ orbitFinset (MainGroup h) C) : |
| ((integralMatrix x).det : ZMod h) = (Matrix.transpose C).det := by |
| obtain ⟨G, hG⟩ := integralMatrix_reduction_of_mem_orbit h q L shift C x hx |
| apply OrbitRowLattice.det_cast_eq_of_reduction_mem_slOrbit h (Matrix.transpose C) |
| (integralMatrix x) |
| exact ⟨G, hG.symm⟩ |
|
|
| theorem conditional_small_mass_le {Ω : Type*} [Fintype Ω] |
| {p k : ℕ} [NeZero (p ^ k)] (hp : p.Prime) (hk : 0 < k) |
| (q L s : ℕ) (hq : 0 < q) (hL : 0 < L) (hs : 0 < s) |
| (shift : Fin 3 → ℤ) (C : Fin 3 → Fin 3 → ZMod (p ^ k)) |
| (d : PrimePowerData p k (Matrix.transpose C)) |
| (w : Ω → ℝ) (V : Ω → Finset (Fin 3 → ZMod q)) |
| (hw : ∀ ω, 0 ≤ w ω) |
| (S : Finset (Fin 3 → Box (L * (p ^ k * q)) 3 shift)) |
| {τ : ℤ} (hwidth : 2 * τ < (p ^ k : ℕ)) |
| {K R : ℝ} (hK : 0 ≤ K) |
| (hcount : ∀ t : ℤ, |t| ≤ τ → |
| (∑ x ∈ (S.filter (fun x => |
| (fun i => residue (p ^ k) (x i)) ∈ orbitFinset (MainGroup (p ^ k)) C)).filter |
| (fun x => (integralMatrix x).det = t), |
| auxiliaryWeight q s w V (fun i => residue q (x i))) ≤ |
| K * R ^ 2 * ((L * (p ^ k * q) : ℕ) : ℝ) ^ 6 / |
| (((p ^ d.b : ℕ) : ℝ) ^ 2 * ((p ^ d.e : ℕ) : ℝ) ^ 2)) : |
| (∑ x ∈ S.filter (fun x => |(integralMatrix x).det| ≤ τ), |
| liftedMass (fun x i => residue (p ^ k) (x i)) |
| (fun x i => residue q (x i)) (orbitFinset (MainGroup (p ^ k)) C) |
| w V s (L ^ 9) x) ≤ |
| (K / (21 / 64)) * R ^ 2 * ((p ^ k : ℕ) : ℝ) * ((p ^ d.b : ℕ) : ℝ) / |
| ((L * (p ^ k * q) : ℕ) : ℝ) ^ 3 := by |
| classical |
| let : NeZero (p ^ k) := ⟨pow_ne_zero _ hp.ne_zero⟩ |
| exact lifted_small_event_le (D := ((p ^ d.b : ℕ) : ℝ)) |
| (E := ((p ^ d.e : ℕ) : ℝ)) |
| (fun x i => residue (p ^ k) (x i)) (fun x i => residue q (x i)) |
| (fun x => (integralMatrix x).det) |
| (orbitFinset (MainGroup (p ^ k)) C) (orbitFinset_nonempty C) w V |
| s (p ^ k) q L hs (pow_pos hp.pos _) hq hL hw S hwidth |
| (Matrix.transpose C).det |
| (fun x _ hxo => integral_det_residue (p ^ k) q L shift C x hxo) |
| (by exact_mod_cast pow_pos hp.pos d.b) |
| (by exact_mod_cast pow_pos hp.pos d.e) |
| (by norm_num) hK (sampling_orbit_lower_bound hp hk C d) hcount |
|
|
| end Problem355.ConditionalBadTripleBound |
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|
| end |
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| end OAI |
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|
