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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 | import OAI.Geometry.HeilbronnTriangle.OrbitSampling
import OAI.Geometry.HeilbronnTriangle.PrimePowerData
import OAI.Geometry.HeilbronnTriangle.OrbitRowLattice
import OAI.Geometry.HeilbronnTriangle.LiftedBadEvent
namespace OAI
noncomputable section
namespace Problem355.ConditionalBadTripleBound
open OrbitSampling IntegerSampling LiftingProbability
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
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
end
end OAI
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