| import OAI.Geometry.HeilbronnTriangle.AuxiliaryAffineWeights |
| import OAI.Geometry.HeilbronnTriangle.AuxiliarySet |
| import OAI.Geometry.HeilbronnTriangle.LiftingProbability |
|
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| namespace OAI |
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| noncomputable section |
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| namespace Problem355.AuxiliarySampling |
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| open scoped BigOperators |
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| structure Law (q H : ℕ) [Fact q.Prime] where |
| Outcome : Type |
| outcomeFintype : Fintype Outcome |
| weight : Outcome → ℝ |
| sets : Outcome → Finset (Fin 3 → ZMod q) |
| size : ℕ |
| size_pos : 0 < size |
| weight_nonneg : ∀ ω, 0 ≤ weight ω |
| weight_sum : @Finset.sum Outcome ℝ _ (@Finset.univ Outcome outcomeFintype) weight = 1 |
| sets_card : ∀ ω, (sets ω).card = size |
| size_lower : q ^ 2 ≤ 2000 ^ 3 * H ^ 6 * size |
| zero_not_mem : ∀ ω, (0 : Fin 3 → ZMod q) ∉ sets ω |
| isCap : ∀ ω, AuxiliaryCap.IsCap (sets ω : Set (Fin 3 → ZMod q)) |
| short_relation : ∀ ω (p : Fin 3 → (Fin 3 → ZMod q)), |
| (∀ i, p i ∈ sets ω) → Function.Injective p → |
| ∀ x : Fin 3 → ℤ, (∀ i, |x i| ≤ (H : ℤ)) → x ≠ 0 → |
| (∑ i, (x i : ZMod q) • p i) ≠ 0 |
| weight_le_eight : ∀ p : Fin 3 → (Fin 3 → ZMod q), |
| AffineIndependent (ZMod q) p → |
| @LiftingProbability.auxiliaryWeight Outcome (Fin 3 → ZMod q) outcomeFintype |
| q size weight sets p ≤ 8 |
| weight_le_four : ∀ (p : Fin 3 → (Fin 3 → ZMod q)) (i j : Fin 3), p i ≠ p j → |
| @LiftingProbability.auxiliaryWeight Outcome (Fin 3 → ZMod q) outcomeFintype |
| q size weight sets p ≤ 4 * ((q : ℝ) ^ 3 / size) |
| weight_le_two : ∀ p : Fin 3 → (Fin 3 → ZMod q), |
| @LiftingProbability.auxiliaryWeight Outcome (Fin 3 → ZMod q) outcomeFintype |
| q size weight sets p ≤ 2 * ((q : ℝ) ^ 3 / size) ^ 2 |
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| attribute [instance] Law.outcomeFintype |
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|
| private theorem affine_apply_inputShift {q : ℕ} [Fact q.Prime] |
| (e : (Fin 3 → ZMod q) ≃ᵃ[ZMod q] (Fin 3 → ZMod q)) |
| (x : Fin 3 → ZMod q) : |
| e x = (AffineTripleOrbit.inputShiftEquiv e).1 |
| ((AffineTripleOrbit.inputShiftEquiv e).2 + x) := by |
| have h := congrArg |
| (fun f : (Fin 3 → ZMod q) ≃ᵃ[ZMod q] (Fin 3 → ZMod q) => f x) |
| (AffineTripleOrbit.inputShiftEquiv.symm_apply_apply e) |
| exact h.symm.trans (AffineTripleOrbit.inputShiftEquiv_symm_apply |
| (AffineTripleOrbit.inputShiftEquiv e).1 (AffineTripleOrbit.inputShiftEquiv e).2 x) |
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| theorem nonempty_law_of_cap {q H w : ℕ} [Fact q.Prime] |
| (hH : 1 ≤ H) (hHq : H < q) |
| (S : Finset (Fin 3 → ZMod q)) (hS : S.Nonempty) |
| (hcap : AuxiliaryCap.IsCap (S : Set (Fin 3 → ZMod q))) |
| (hbox : ∀ v ∈ S, (v 0).val < w) |
| (hsize : q ^ 2 ≤ 2000 ^ 3 * H ^ 6 * S.card) |
| (A : Finset (Fin 3 → ZMod q)) (hA : q ^ 3 ≤ 2 * A.card) |
| (hallowed : ∀ a ∈ A, ∀ n : ℕ, 1 ≤ n → n ≤ 3 * H → |
| (n : ZMod q) * a 0 ∉ Auxiliary.integerWindow q (3 * H * w)) : |
| Nonempty (Law q H) := by |
| classical |
| let Aff := (Fin 3 → ZMod q) ≃ᵃ[ZMod q] (Fin 3 → ZMod q) |
| let : Finite Aff := Finite.of_injective |
| (fun e : Aff => (e : (Fin 3 → ZMod q) → (Fin 3 → ZMod q))) |
| DFunLike.coe_injective |
| let : Fintype Aff := Fintype.ofFinite Aff |
| let T : Finset Aff := AuxiliaryWeights.affineMapsWithInputShift A |
| have hA' : Fintype.card (Fin 3 → ZMod q) ≤ 2 * A.card := by |
| simpa using hA |
| have hT : T.Nonempty := AuxiliaryWeights.affineMapsWithInputShift_nonempty A hA' |
| have hTsize : Fintype.card Aff ≤ 2 * T.card := |
| AuxiliaryWeights.affineMapsWithInputShift_halfDensity A hA' |
| let μ : T → ℝ := fun _ => (T.card : ℝ)⁻¹ |
| let V : T → Finset (Fin 3 → ZMod q) := fun e => S.image e.1 |
| have hV (e : T) : V e = AuxiliarySet.transformed S |
| (AffineTripleOrbit.inputShiftEquiv e.1).2 |
| (AffineTripleOrbit.inputShiftEquiv e.1).1 := by |
| ext y |
| simp only [V, Finset.mem_image, AuxiliarySet.mem_transformed] |
| constructor |
| · rintro ⟨x, hx, rfl⟩ |
| exact ⟨x, hx, (affine_apply_inputShift e.1 x).symm⟩ |
| · rintro ⟨x, hx, rfl⟩ |
| exact ⟨x, hx, affine_apply_inputShift e.1 x⟩ |
| have hall (e : T) : ∀ n : ℕ, 1 ≤ n → n ≤ 3 * H → |
| (n : ZMod q) * (AffineTripleOrbit.inputShiftEquiv e.1).2 0 ∉ |
| Auxiliary.integerWindow q (3 * H * w) := by |
| apply hallowed |
| exact (Finset.mem_filter.mp e.2).2 |
| have hprops (e : T) := AuxiliarySet.transformed_properties hH hHq hcap hbox |
| (AffineTripleOrbit.inputShiftEquiv e.1).2 (hall e) |
| (AffineTripleOrbit.inputShiftEquiv e.1).1 |
| have hW (p : Fin 3 → (Fin 3 → ZMod q)) : |
| LiftingProbability.auxiliaryWeight q S.card μ V p = |
| AuxiliaryWeights.affineInclusionWeight S T p := by |
| rw [AuxiliaryWeights.affineInclusionWeight_eq_uniform_sum] |
| simp only [LiftingProbability.auxiliaryWeight, LiftingProbability.inclusionMass, |
| ZMod.card] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro e he |
| simp only [LiftingProbability.TripleIn, μ, V, Finset.mem_image] |
| split_ifs <;> simp_all |
| refine ⟨{ |
| Outcome := T |
| outcomeFintype := inferInstance |
| weight := μ |
| sets := V |
| size := S.card |
| size_pos := hS.card_pos |
| weight_nonneg := ?_ |
| weight_sum := ?_ |
| sets_card := ?_ |
| size_lower := hsize |
| zero_not_mem := ?_ |
| isCap := ?_ |
| short_relation := ?_ |
| weight_le_eight := ?_ |
| weight_le_four := ?_ |
| weight_le_two := ?_ }⟩ |
| · intro e |
| dsimp [μ] |
| positivity |
| · simp only [μ, Finset.sum_const, Finset.card_univ, Fintype.card_coe, nsmul_eq_mul] |
| exact mul_inv_cancel₀ (by exact_mod_cast hT.card_pos.ne') |
| · intro e |
| rw [hV] |
| exact (hprops e).1 |
| · intro e |
| rw [hV] |
| exact (hprops e).2.1 |
| · intro e |
| rw [hV] |
| exact (hprops e).2.2.1 |
| · intro e p hp hinj x hx hne |
| exact (hprops e).2.2.2 p (fun i => hV e ▸ hp i) hinj x hx hne |
| · intro p hp |
| rw [hW] |
| exact AuxiliaryWeights.affineInclusionWeight_le_eight S hS T hT hTsize p hp |
| · intro p i j hp |
| rw [hW] |
| simpa only [ZMod.card] using |
| AuxiliaryWeights.affineInclusionWeight_le_four_of_ne S hS T hT hTsize p i j hp |
| · intro p |
| rw [hW] |
| simpa only [ZMod.card] using |
| AuxiliaryWeights.affineInclusionWeight_le_two S hS T hT hTsize p |
|
|
| theorem nonempty_law {q H : ℕ} [Fact q.Prime] |
| (hH : 1 ≤ H) (hq : 2000 * H ^ 2 ≤ q) : Nonempty (Law q H) := by |
| classical |
| let w := q / (1000 * H ^ 2) |
| have hd : 0 < 1000 * H ^ 2 := by positivity |
| have hdq : 1000 * H ^ 2 ≤ q := by omega |
| have hwq : w ≤ q := Nat.div_le_self _ _ |
| have hq2 : q ≠ 2 := by |
| have hHsq : 1 ≤ H ^ 2 := by nlinarith |
| omega |
| obtain ⟨S, hbox, hsize, hcap⟩ := AuxiliaryCap.exists_cap_in_residueBox hq2 hwq |
| have hfloor : q ≤ 2000 * H ^ 2 * w := by |
| calc |
| q ≤ 2 * (1000 * H ^ 2) * w := AuxiliarySet.le_twice_mul_div hd hdq |
| _ = 2000 * H ^ 2 * w := by ring |
| have hc : q ^ 2 ≤ 2000 ^ 3 * H ^ 6 * S.card := by |
| have hcube : q ^ 3 ≤ (2000 * H ^ 2 * w) ^ 3 := by gcongr |
| have hm : 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 hm (Fact.out : q.Prime).pos |
| have hS : S.Nonempty := by |
| apply Finset.card_pos.mp |
| by_contra hn |
| have hz : S.card = 0 := by omega |
| have hp : 0 < q ^ 2 := pow_pos (Fact.out : q.Prime).pos 2 |
| simp only [hz, mul_zero] at hc |
| omega |
| let A := Auxiliary.vectorShifts (Auxiliary.allowedShifts |
| (Auxiliary.positiveMultipliers q H) (Auxiliary.integerWindow q (3 * H * w))) |
| have hw : 1000 * H ^ 2 * w ≤ q := Nat.mul_div_le _ _ |
| have hA : q ^ 3 ≤ 2 * A.card := |
| (Auxiliary.auxiliary_allowed_vector_card_gt_half q H w hH hq hw).le |
| have hmult : 3 * H < q := Auxiliary.auxiliary_multiplier_range q H hH hq |
| apply nonempty_law_of_cap hH (by omega) S hS hcap (fun v hv => hbox v hv 0) hc A hA |
| intro a ha n hn hnH |
| have ha' := (Auxiliary.mem_vectorShifts_iff _ a).mp ha |
| have ha'' := (Auxiliary.mem_allowedShifts_iff _ _ |
| (Auxiliary.positiveMultipliers_ne_zero q H hmult) (a 0)).mp ha' |
| apply ha'' |
| exact Finset.mem_image.mpr ⟨n, Finset.mem_Icc.mpr ⟨hn, hnH⟩, rfl⟩ |
|
|
| end Problem355.AuxiliarySampling |
|
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| end |
|
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| end OAI |
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|
