| import OAI.Geometry.HeilbronnTriangle.AuxiliaryCap |
| import OAI.Geometry.HeilbronnTriangle.ShortRelations |
| import OAI.Geometry.HeilbronnTriangle.AllowedShifts |
| import OAI.Geometry.HeilbronnTriangle.CapInBox |
|
|
| namespace OAI |
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|
|
| namespace Problem355.AuxiliarySet |
|
|
| open scoped BigOperators |
|
|
| variable {F : Type*} [Field F] |
|
|
| theorem affineIndependent_of_mem_cap |
| {S : Set (Fin 3 → F)} (hS : AuxiliaryCap.IsCap S) |
| {p : Fin 3 → (Fin 3 → F)} (hp : ∀ i, p i ∈ S) |
| (hinj : Function.Injective p) : AffineIndependent F p := by |
| have hcol := hS (p 0) (hp 0) (p 1) (hp 1) (p 2) (hp 2) |
| (hinj.ne (by decide)) (hinj.ne (by decide)) (hinj.ne (by decide)) |
| have h := affineIndependent_iff_not_collinear_set.mpr hcol |
| convert h using 1 |
| funext i |
| fin_cases i <;> rfl |
|
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| noncomputable def transformed (S : Finset (Fin 3 → F)) (a : Fin 3 → F) |
| (G : (Fin 3 → F) ≃ₗ[F] (Fin 3 → F)) : Finset (Fin 3 → F) := by |
| classical |
| exact S.image (fun v => G (a + v)) |
|
|
| @[simp] theorem mem_transformed (S : Finset (Fin 3 → F)) (a : Fin 3 → F) |
| (G : (Fin 3 → F) ≃ₗ[F] (Fin 3 → F)) (u : Fin 3 → F) : |
| u ∈ transformed S a G ↔ ∃ v ∈ S, G (a + v) = u := by |
| classical |
| simp [transformed] |
|
|
| @[simp] theorem card_transformed (S : Finset (Fin 3 → F)) (a : Fin 3 → F) |
| (G : (Fin 3 → F) ≃ₗ[F] (Fin 3 → F)) : |
| (transformed S a G).card = S.card := by |
| classical |
| apply Finset.card_image_iff.mpr |
| intro v hv w hw heq |
| exact add_left_cancel (G.injective heq) |
|
|
| theorem isCap_transformed {S : Finset (Fin 3 → F)} |
| (hS : AuxiliaryCap.IsCap (S : Set (Fin 3 → F))) |
| (a : Fin 3 → F) (G : (Fin 3 → F) ≃ₗ[F] (Fin 3 → F)) : |
| AuxiliaryCap.IsCap (transformed S a G : Set (Fin 3 → F)) := by |
| have h := (hS.translate a).image G.toAffineEquiv.toAffineMap G.injective |
| convert h using 1 |
| ext v |
| simp only [Finset.mem_coe, mem_transformed, Set.mem_image] |
| constructor |
| · rintro ⟨w, hw, rfl⟩ |
| exact ⟨a + w, ⟨w, hw, rfl⟩, rfl⟩ |
| · rintro ⟨_, ⟨w, hw, rfl⟩, rfl⟩ |
| exact ⟨w, hw, rfl⟩ |
|
|
| theorem zero_not_mem_transformed {S : Finset (Fin 3 → F)} {a : Fin 3 → F} |
| (ha : ∀ v ∈ S, a + v ≠ 0) |
| (G : (Fin 3 → F) ≃ₗ[F] (Fin 3 → F)) : |
| (0 : Fin 3 → F) ∉ transformed S a G := by |
| rintro h |
| obtain ⟨v, hv, hzero⟩ := (mem_transformed S a G 0).mp h |
| exact ha v hv (G.map_eq_zero_iff.mp hzero) |
|
|
| theorem short_sum_zero_relation_excluded |
| {q H : ℕ} [Fact q.Prime] |
| {S : Finset (Fin 3 → ZMod q)} |
| (hS : AuxiliaryCap.IsCap (S : Set (Fin 3 → ZMod q))) |
| (a : Fin 3 → ZMod q) |
| (G : (Fin 3 → ZMod q) ≃ₗ[ZMod q] (Fin 3 → ZMod q)) |
| {p : Fin 3 → (Fin 3 → ZMod q)} |
| (hp : ∀ i, p i ∈ transformed S a G) (hinj : Function.Injective p) |
| {x : Fin 3 → ℤ} (hsum : ∑ i, x i = 0) |
| (hshort : ∀ i, |x i| ≤ (H : ℤ)) (hHq : H < q) (hne : x ≠ 0) : |
| (∑ i, (x i : ZMod q) • p i) ≠ 0 := by |
| exact ShortRelations.sum_smul_ne_zero_of_bounded_coefficients |
| (affineIndependent_of_mem_cap (isCap_transformed hS a G) hp hinj) |
| hsum hshort hHq hne |
|
|
| theorem short_nonzero_sum_relation_excluded |
| {q H w : ℕ} [Fact q.Prime] |
| {S : Finset (Fin 3 → ZMod q)} |
| (hbox : ∀ v ∈ S, (v 0).val < w) |
| (a : Fin 3 → ZMod q) |
| (ha : ∀ n : ℕ, 1 ≤ n → n ≤ 3 * H → |
| (n : ZMod q) * a 0 ∉ Auxiliary.integerWindow q (3 * H * w)) |
| (G : (Fin 3 → ZMod q) ≃ₗ[ZMod q] (Fin 3 → ZMod q)) |
| {p : Fin 3 → (Fin 3 → ZMod q)} |
| (hp : ∀ i, p i ∈ transformed S a G) |
| {x : Fin 3 → ℤ} (hsum : ∑ i, x i ≠ 0) |
| (hshort : ∀ i, |x i| ≤ (H : ℤ)) : |
| (∑ i, (x i : ZMod q) • p i) ≠ 0 := by |
| classical |
| choose v hv heq using fun i => (mem_transformed S a G (p i)).mp (hp i) |
| let z : Fin 3 → Fin 3 → ℤ := fun i j => (v i j).val |
| have hz : ∀ i, (fun j => (z i j : ZMod q)) = v i := by |
| intro i |
| funext j |
| simp [z] |
| have hb : ∀ i, 0 ≤ z i 0 ∧ z i 0 ≤ (w : ℤ) := by |
| intro i |
| constructor |
| · exact Int.natCast_nonneg _ |
| · dsimp [z] |
| exact_mod_cast (hbox (v i) (hv i)).le |
| have h := Auxiliary.shifted_vector_short_relation_ne_zero q H w a ha x z |
| hshort hb hsum G |
| simpa only [hz, heq] using h |
|
|
| theorem zero_not_mem_of_allowed |
| {q H w : ℕ} [Fact q.Prime] (hH : 1 ≤ H) |
| {S : Finset (Fin 3 → ZMod q)} |
| (hbox : ∀ v ∈ S, (v 0).val < w) |
| (a : Fin 3 → ZMod q) |
| (ha : ∀ n : ℕ, 1 ≤ n → n ≤ 3 * H → |
| (n : ZMod q) * a 0 ∉ Auxiliary.integerWindow q (3 * H * w)) |
| (G : (Fin 3 → ZMod q) ≃ₗ[ZMod q] (Fin 3 → ZMod q)) : |
| (0 : Fin 3 → ZMod q) ∉ transformed S a G := by |
| intro hzero |
| have h := short_nonzero_sum_relation_excluded hbox a ha G |
| (p := fun _ => 0) (fun _ => hzero) (x := ![1, 0, 0]) |
| (by norm_num [Fin.sum_univ_succ]) |
| (by |
| intro i |
| fin_cases i |
| · simpa using (show (1 : ℤ) ≤ H by exact_mod_cast hH) |
| · simp |
| · simp) |
| exact h (by simp) |
|
|
| theorem short_relation_excluded |
| {q H w : ℕ} [Fact q.Prime] |
| {S : Finset (Fin 3 → ZMod q)} |
| (hS : AuxiliaryCap.IsCap (S : Set (Fin 3 → ZMod q))) |
| (hbox : ∀ v ∈ S, (v 0).val < w) |
| (a : Fin 3 → ZMod q) |
| (ha : ∀ n : ℕ, 1 ≤ n → n ≤ 3 * H → |
| (n : ZMod q) * a 0 ∉ Auxiliary.integerWindow q (3 * H * w)) |
| (G : (Fin 3 → ZMod q) ≃ₗ[ZMod q] (Fin 3 → ZMod q)) |
| {p : Fin 3 → (Fin 3 → ZMod q)} |
| (hp : ∀ i, p i ∈ transformed S a G) (hinj : Function.Injective p) |
| {x : Fin 3 → ℤ} (hshort : ∀ i, |x i| ≤ (H : ℤ)) |
| (hHq : H < q) (hne : x ≠ 0) : |
| (∑ i, (x i : ZMod q) • p i) ≠ 0 := by |
| by_cases hsum : ∑ i, x i = 0 |
| · exact short_sum_zero_relation_excluded hS a G hp hinj hsum hshort hHq hne |
| · exact short_nonzero_sum_relation_excluded hbox a ha G hp hsum hshort |
|
|
| theorem transformed_properties |
| {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) |
| (a : Fin 3 → ZMod q) |
| (ha : ∀ n : ℕ, 1 ≤ n → n ≤ 3 * H → |
| (n : ZMod q) * a 0 ∉ Auxiliary.integerWindow q (3 * H * w)) |
| (G : (Fin 3 → ZMod q) ≃ₗ[ZMod q] (Fin 3 → ZMod q)) : |
| (transformed S a G).card = S.card ∧ |
| (0 : Fin 3 → ZMod q) ∉ transformed S a G ∧ |
| AuxiliaryCap.IsCap (transformed S a G : Set (Fin 3 → ZMod q)) ∧ |
| ∀ p : Fin 3 → (Fin 3 → ZMod q), |
| (∀ i, p i ∈ transformed S a G) → Function.Injective p → |
| ∀ x : Fin 3 → ℤ, (∀ i, |x i| ≤ (H : ℤ)) → x ≠ 0 → |
| (∑ i, (x i : ZMod q) • p i) ≠ 0 := by |
| refine ⟨card_transformed S a G, zero_not_mem_of_allowed hH hbox a ha G, |
| isCap_transformed hS a G, ?_⟩ |
| intro p hp hinj x hx hne |
| exact short_relation_excluded hS hbox a ha G hp hinj hx hHq hne |
|
|
| theorem exists_set_excluding_short_relations |
| {q H w : ℕ} [Fact q.Prime] (hH : 1 ≤ H) |
| (hq : 2000 * H ^ 2 ≤ q) (hw : 1000 * H ^ 2 * w ≤ q) |
| {S : Finset (Fin 3 → ZMod q)} |
| (hS : AuxiliaryCap.IsCap (S : Set (Fin 3 → ZMod q))) |
| (hbox : ∀ v ∈ S, (v 0).val < w) : |
| ∃ V : Finset (Fin 3 → ZMod q), V.card = S.card ∧ |
| (0 : Fin 3 → ZMod q) ∉ V ∧ |
| AuxiliaryCap.IsCap (V : Set (Fin 3 → ZMod q)) ∧ |
| ∀ p : Fin 3 → (Fin 3 → ZMod q), |
| (∀ i, p i ∈ V) → Function.Injective p → |
| ∀ x : Fin 3 → ℤ, (∀ i, |x i| ≤ (H : ℤ)) → x ≠ 0 → |
| (∑ i, (x i : ZMod q) • p i) ≠ 0 := by |
| obtain ⟨a₀, ha₀⟩ := Auxiliary.exists_auxiliary_allowed_shift q H w hH hq hw |
| let a : Fin 3 → ZMod q := ![a₀, 0, 0] |
| let G := LinearEquiv.refl (ZMod q) (Fin 3 → ZMod q) |
| refine ⟨transformed S a G, transformed_properties hH ?_ hS hbox a ?_ G⟩ |
| · have h := Auxiliary.auxiliary_multiplier_range q H hH hq |
| omega |
| · simpa only [a, Matrix.cons_val_zero] using ha₀ |
|
|
| theorem le_twice_mul_div {q d : ℕ} (hd : 0 < d) (hdq : d ≤ q) : |
| q ≤ 2 * d * (q / d) := by |
| have hquot : 1 ≤ q / d := (Nat.le_div_iff_mul_le hd).mpr (by simpa using hdq) |
| have hrem := Nat.mod_lt q hd |
| have hsplit := Nat.mod_add_div q d |
| have hprod : d ≤ d * (q / d) := by |
| simpa using Nat.mul_le_mul_left d hquot |
| nlinarith |
|
|
| theorem exists_large_auxiliary_set |
| {q H : ℕ} [Fact q.Prime] (hH : 1 ≤ H) (hq : 2000 * H ^ 2 ≤ q) : |
| ∃ V : Finset (Fin 3 → ZMod q), |
| q ^ 2 ≤ 2000 ^ 3 * H ^ 6 * V.card ∧ |
| (0 : Fin 3 → ZMod q) ∉ V ∧ |
| AuxiliaryCap.IsCap (V : Set (Fin 3 → ZMod q)) ∧ |
| ∀ p : Fin 3 → (Fin 3 → ZMod q), |
| (∀ i, p i ∈ V) → Function.Injective p → |
| ∀ x : Fin 3 → ℤ, (∀ i, |x i| ≤ (H : ℤ)) → x ≠ 0 → |
| (∑ i, (x i : ZMod q) • p i) ≠ 0 := by |
| let w := q / (1000 * H ^ 2) |
| have hHsq : 1 ≤ H ^ 2 := by nlinarith |
| have hd : 0 < 1000 * H ^ 2 := by positivity |
| have hdq : 1000 * H ^ 2 ≤ q := by omega |
| have hq2 : q ≠ 2 := by omega |
| have hwq : w ≤ q := Nat.div_le_self _ _ |
| obtain ⟨S, hbox, hsize, hcap⟩ := AuxiliaryCap.exists_cap_in_residueBox hq2 hwq |
| have hw : 1000 * H ^ 2 * w ≤ q := Nat.mul_div_le _ _ |
| obtain ⟨V, hcard, hzero, hVcap, hrel⟩ := |
| exists_set_excluding_short_relations hH hq hw hcap (fun v hv => hbox v hv 0) |
| refine ⟨V, ?_, hzero, hVcap, hrel⟩ |
| rw [hcard] |
| have hfloor : q ≤ 2000 * H ^ 2 * w := by |
| calc |
| q ≤ 2 * (1000 * H ^ 2) * w := le_twice_mul_div hd hdq |
| _ = 2000 * H ^ 2 * w := by ring |
| have hcube : q ^ 3 ≤ (2000 * H ^ 2 * w) ^ 3 := by gcongr |
| have hmul : q * q ^ 2 ≤ q * (2000 ^ 3 * H ^ 6 * S.card) := by |
| calc |
| q * q ^ 2 = q ^ 3 := by ring |
| _ ≤ (2000 * H ^ 2 * w) ^ 3 := hcube |
| _ = (2000 ^ 3 * H ^ 6) * w ^ 3 := by ring |
| _ ≤ (2000 ^ 3 * H ^ 6) * (q * S.card) := Nat.mul_le_mul_left _ hsize |
| _ = q * (2000 ^ 3 * H ^ 6 * S.card) := by ring |
| exact Nat.le_of_mul_le_mul_left hmul (Fact.out : q.Prime).pos |
|
|
| theorem exists_large_auxiliary_set_real |
| {q H : ℕ} [Fact q.Prime] (hH : 1 ≤ H) (hq : 2000 * H ^ 2 ≤ q) : |
| ∃ V : Finset (Fin 3 → ZMod q), |
| (q : ℝ) ^ 2 / (2000 ^ 3 * (H : ℝ) ^ 6) ≤ V.card ∧ V.Nonempty ∧ |
| (0 : Fin 3 → ZMod q) ∉ V ∧ |
| AuxiliaryCap.IsCap (V : Set (Fin 3 → ZMod q)) ∧ |
| ∀ p : Fin 3 → (Fin 3 → ZMod q), |
| (∀ i, p i ∈ V) → Function.Injective p → |
| ∀ x : Fin 3 → ℤ, (∀ i, |x i| ≤ (H : ℤ)) → x ≠ 0 → |
| (∑ i, (x i : ZMod q) • p i) ≠ 0 := by |
| obtain ⟨V, hsize, hzero, hcap, hrel⟩ := exists_large_auxiliary_set hH hq |
| have hHp : (0 : ℝ) < H := by exact_mod_cast (show 0 < H by omega) |
| have hsizeR : (q : ℝ) ^ 2 ≤ (2000 ^ 3 * (H : ℝ) ^ 6) * V.card := by |
| exact_mod_cast hsize |
| have hVpos : 0 < V.card := by |
| by_contra hnot |
| have hv : V.card = 0 := by omega |
| have hqp : 0 < q ^ 2 := pow_pos (Fact.out : q.Prime).pos 2 |
| simp only [hv, mul_zero] at hsize |
| omega |
| refine ⟨V, ?_, Finset.card_pos.mp hVpos, hzero, hcap, hrel⟩ |
| exact (div_le_iff₀ (by positivity)).mpr (by simpa [mul_comm] using hsizeR) |
|
|
| end Problem355.AuxiliarySet |
|
|
| end OAI |
|
|
