| import OAI.Geometry.HeilbronnTriangle.AuxiliaryCap |
| import OAI.Geometry.HeilbronnTriangle.TranslateAverage |
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| namespace OAI |
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| namespace Problem355.AuxiliaryCap |
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| variable {F : Type*} [Field F] [Fintype F] |
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| 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⟩ |
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| 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) |
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| noncomputable def residueInterval (q w : ℕ) : Finset (ZMod q) := by |
| classical |
| exact (Finset.range w).image (fun x : ℕ => (x : ZMod q)) |
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| 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 |
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| @[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⟩ |
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| noncomputable def residueBox (q w : ℕ) : Finset (Fin 3 → ZMod q) := |
| Fintype.piFinset (fun _ => residueInterval q w) |
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| @[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] |
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| @[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] |
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| 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 |
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| 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) |
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| end Problem355.AuxiliaryCap |
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| end OAI |
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