File size: 2,390 Bytes
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 | import OAI.Geometry.HeilbronnTriangle.AuxiliarySupport
import OAI.Geometry.HeilbronnTriangle.PrimitiveNormal
namespace OAI
namespace Problem355.AuxiliarySet
open scoped BigOperators
theorem coordinate_abs_le_of_euclidean_norm_le
{x : PrimitiveNormal.Vector} {H : ℕ}
(hx : ‖PrimitiveNormal.toEuclidean x‖ ≤ (H : ℝ)) :
∀ i, |x i| ≤ (H : ℤ) := by
intro i
have hi := (PiLp.norm_apply_le (PrimitiveNormal.toEuclidean x) i).trans hx
have hi' : |(x i : ℝ)| ≤ (H : ℝ) := by
simpa [PrimitiveNormal.toEuclidean, Real.norm_eq_abs] using hi
exact_mod_cast hi'
theorem cast_mulVec_eq_zero {q : ℕ}
{A : Matrix (Fin 3) (Fin 3) ℤ} {x : Fin 3 → ℤ}
(hx : A.mulVec x = 0) :
(A.map (Int.castRingHom (ZMod q))).mulVec (fun i => (x i : ZMod q)) = 0 := by
funext i
have hi := congrArg (fun z : ℤ => (z : ZMod q)) (congrFun hx i)
simpa [Matrix.mulVec, dotProduct, Matrix.map_apply] using hi
theorem kernel_norm_gt_of_positive_auxiliary_inclusion
{q H w : ℕ} [Fact q.Prime] (hH : 1 ≤ H) (hHq : H < q)
{S : Finset (Fin 3 → ZMod q)}
(hS : AuxiliaryCap.IsCap (S : Set (Fin 3 → ZMod q)))
(hbox : ∀ v ∈ S, (v 0).val < w)
{Θ : Type*} (T : Finset Θ) (μ : Θ → ℝ)
(a : Θ → Fin 3 → ZMod q)
(G : Θ → ((Fin 3 → ZMod q) ≃ₗ[ZMod q] (Fin 3 → ZMod q)))
(ha : ∀ θ ∈ T, ∀ n : ℕ, 1 ≤ n → n ≤ 3 * H →
(n : ZMod q) * a θ 0 ∉ Auxiliary.integerWindow q (3 * H * w))
(A : Matrix (Fin 3) (Fin 3) ℤ)
[DecidablePred (fun θ =>
∀ j, (fun i => (A i j : ZMod q)) ∈ transformed S (a θ) (G θ))]
(hpositive : 0 < ∑ θ ∈ T,
if ∀ j, (fun i => (A i j : ZMod q)) ∈ transformed S (a θ) (G θ)
then μ θ else 0)
(hAI : AffineIndependent (ZMod q) (fun j => fun i => (A i j : ZMod q)))
{x : PrimitiveNormal.Vector} (hne : x ≠ 0) (hkernel : A.mulVec x = 0) :
(H : ℝ) < ‖PrimitiveNormal.toEuclidean x‖ := by
classical
by_contra hnot
have hnorm : ‖PrimitiveNormal.toEuclidean x‖ ≤ (H : ℝ) := le_of_not_gt hnot
have hsupport := matrix_support_of_inclusion_sum_pos hH hHq hS hbox T μ a G ha
(fun i j => (A i j : ZMod q)) hpositive
have hno := hsupport.2.2 hAI x (coordinate_abs_le_of_euclidean_norm_le hnorm) hne
exact hno (cast_mulVec_eq_zero hkernel)
end Problem355.AuxiliarySet
end OAI
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