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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 99 100 101 | import OAI.Geometry.HeilbronnTriangle.AuxiliaryCap
import OAI.Geometry.HeilbronnTriangle.TranslateAverage
namespace OAI
namespace Problem355.AuxiliaryCap
variable {F : Type*} [Field F] [Fintype F]
theorem exists_cap_subset (hF : ringChar F ≠ 2) (B : Finset (Fin 3 → F)) :
∃ S : Finset (Fin 3 → F), S ⊆ B ∧
B.card ≤ Fintype.card F * S.card ∧ IsCap (S : Set (Fin 3 → F)) := by
classical
obtain ⟨G, hG, hcap⟩ := exists_finset_cap hF
obtain ⟨a, ha⟩ := TranslateAverage.exists_large_image_inter G B
refine ⟨G.image (a + ·) ∩ B, Finset.inter_subset_right, ?_, ?_⟩
· rw [hG, Fintype.card_fun, Fintype.card_fin] at ha
have hmul : Fintype.card F ^ 2 * B.card ≤
Fintype.card F ^ 2 *
(Fintype.card F * (G.image (a + ·) ∩ B).card) := by
simpa [pow_succ, mul_assoc] using ha
exact Nat.le_of_mul_le_mul_left hmul (pow_pos Fintype.card_pos 2)
· apply (hcap.translate a).mono
intro p hp
obtain ⟨q, hq, heq⟩ := Finset.mem_image.mp (Finset.mem_inter.mp hp).1
exact ⟨q, hq, heq⟩
theorem exists_cap_subset_real (hF : ringChar F ≠ 2) (B : Finset (Fin 3 → F)) :
∃ S : Finset (Fin 3 → F), S ⊆ B ∧
(B.card : ℝ) / Fintype.card F ≤ S.card ∧ IsCap (S : Set (Fin 3 → F)) := by
obtain ⟨S, hS, hcard, hcap⟩ := exists_cap_subset hF B
refine ⟨S, hS, ?_, hcap⟩
apply (div_le_iff₀ (by exact_mod_cast Fintype.card_pos (α := F))).mpr
exact_mod_cast (by simpa [mul_comm] using hcard : B.card ≤ S.card * Fintype.card F)
noncomputable def residueInterval (q w : ℕ) : Finset (ZMod q) := by
classical
exact (Finset.range w).image (fun x : ℕ => (x : ZMod q))
theorem card_residueInterval {q w : ℕ} (hw : w ≤ q) :
(residueInterval q w).card = w := by
classical
calc
(residueInterval q w).card = (Finset.range w).card := by
apply Finset.card_image_iff.mpr
intro a ha b hb hab
have hav : a < q := lt_of_lt_of_le (Finset.mem_range.mp ha) hw
have hbv : b < q := lt_of_lt_of_le (Finset.mem_range.mp hb) hw
have hv := congrArg ZMod.val hab
simpa [ZMod.val_natCast_of_lt hav, ZMod.val_natCast_of_lt hbv] using hv
_ = w := Finset.card_range w
@[simp] theorem mem_residueInterval {q w : ℕ} [NeZero q] (hw : w ≤ q) (x : ZMod q) :
x ∈ residueInterval q w ↔ x.val < w := by
classical
constructor
· intro hx
obtain ⟨a, ha, rfl⟩ := Finset.mem_image.mp hx
have haw := Finset.mem_range.mp ha
simpa [ZMod.val_natCast_of_lt (lt_of_lt_of_le haw hw)] using haw
· intro hx
exact Finset.mem_image.mpr ⟨x.val, Finset.mem_range.mpr hx, ZMod.natCast_zmod_val x⟩
noncomputable def residueBox (q w : ℕ) : Finset (Fin 3 → ZMod q) :=
Fintype.piFinset (fun _ => residueInterval q w)
@[simp] theorem card_residueBox {q w : ℕ} (hw : w ≤ q) :
(residueBox q w).card = w ^ 3 := by
rw [residueBox, Fintype.card_piFinset_const, card_residueInterval hw]
@[simp] theorem mem_residueBox {q w : ℕ} [NeZero q] (hw : w ≤ q)
(p : Fin 3 → ZMod q) : p ∈ residueBox q w ↔ ∀ i, (p i).val < w := by
simp [residueBox, Fintype.mem_piFinset, mem_residueInterval hw]
theorem exists_cap_in_residueBox {q w : ℕ} [Fact q.Prime]
(hq : q ≠ 2) (hw : w ≤ q) :
∃ S : Finset (Fin 3 → ZMod q),
(∀ p ∈ S, ∀ i, (p i).val < w) ∧ w ^ 3 ≤ q * S.card ∧
IsCap (S : Set (Fin 3 → ZMod q)) := by
have hchar : ringChar (ZMod q) ≠ 2 := by simpa only [ZMod.ringChar_zmod_n] using hq
obtain ⟨S, hSB, hcard, hcap⟩ := exists_cap_subset hchar (residueBox q w)
refine ⟨S, ?_, ?_, hcap⟩
· intro p hp
exact (mem_residueBox hw p).mp (hSB hp)
· simpa [card_residueBox hw, ZMod.card] using hcard
theorem exists_cap_in_residueBox_real {q w : ℕ} [Fact q.Prime]
(hq : q ≠ 2) (hw : w ≤ q) :
∃ S : Finset (Fin 3 → ZMod q),
(∀ p ∈ S, ∀ i, (p i).val < w) ∧ (w : ℝ) ^ 3 / q ≤ S.card ∧
IsCap (S : Set (Fin 3 → ZMod q)) := by
obtain ⟨S, hS, hcard, hcap⟩ := exists_cap_in_residueBox hq hw
refine ⟨S, hS, ?_, hcap⟩
apply (div_le_iff₀ (by exact_mod_cast (Fact.out : q.Prime).pos)).mpr
exact_mod_cast (by simpa [mul_comm] using hcard : w ^ 3 ≤ S.card * q)
end Problem355.AuxiliaryCap
end OAI
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