| import OAI.NumberTheory.Jacobsthal.Estimates.ConditionalPatternCounts |
| import OAI.NumberTheory.Jacobsthal.Primes.StableSourceChebyshev |
| |
| namespace OAI |
|
|
| namespace Erdos970 |
| open scoped _root_.Erdos970 |
|
|
|
|
| namespace ErdosLargePatternLaw |
| open NumberTheoryLean ErdosVarianceSmallModel ErdosVarianceMoments |
|
|
| attribute [local instance] Classical.propDecidable |
| attribute [local instance] Classical.decEq |
|
|
| theorem four_coordinate_count {A B C D : Type*} [Fintype A] [Fintype B] [Fintype C] [Fintype D] |
| (P : A → B → Prop) (Q : C → D → Prop) (n m : ℕ) |
| (hP : ∀ a,(Finset.univ.filter (P a)).card=n) |
| (hQ : ∀ c,(Finset.univ.filter (Q c)).card=m) : |
| (Finset.univ.filter (fun v : A × B × C × D => P v.1 v.2.1 ∧ Q v.2.2.1 v.2.2.2)).card = |
| Fintype.card A*n*Fintype.card C*m := by |
| have hi (a : A) (b : B) (c : C) (d : D) : |
| (if P a b ∧ Q c d then 1 else 0 : ℕ)=(if P a b then 1 else 0)*(if Q c d then 1 else 0) := by |
| by_cases hp : P a b < |
| have hp (a : A) : (∑ b : B,if P a b then 1 else 0)=n := by |
| simpa only [Finset.card_filter] using hP a |
| have hq (c : C) : (∑ d : D,if Q c d then 1 else 0)=m := by |
| simpa only [Finset.card_filter] using hQ c |
| have hd (a : A) (b : B) (c : C) : |
| (∑ d : D,(if P a b then 1 else 0)*(if Q c d then 1 else 0)) = |
| (if P a b then 1 else 0)*m := by |
| calc |
| _ = (if P a b then 1 else 0)*(∑ d : D,if Q c d then 1 else 0) := |
| (Finset.mul_sum _ _ _).symm |
| _ = _ := by rw [hq] |
| simp only [Finset.card_filter,Fintype.sum_prod_type,hi] |
| simp_rw [hd] |
| simp only [Finset.sum_const,Finset.card_univ,smul_eq_mul] |
| have hb (a : A) : (∑ b : B,Fintype.card C*((if P a b then 1 else 0)*m)) = |
| Fintype.card C*(n*m) := by |
| calc |
| _ = Fintype.card C*(∑ b : B,(if P a b then 1 else 0)*m) := (Finset.mul_sum _ _ _).symm |
| _ = Fintype.card C*((∑ b : B,if P a b then 1 else 0)*m) := |
| congrArg (fun x => Fintype.card C*x) (Finset.sum_mul _ _ _).symm |
| _ = _ := by rw [hp] |
| simp_rw [hb] |
| simp only [Finset.sum_const,Finset.card_univ,smul_eq_mul] |
| ring |
|
|
| theorem joint_condition_card (w : ℝ) (H qA : ℕ) (beta delta : ∀ t : ℕ,ZMod t) (S : Finset ℕ) |
| (hqA : ∀ t ∈ coprimePrimes w H,qA.Coprime t) : |
| (Finset.univ.filter (jointCondition w H qA beta delta S)).card = |
| (divisorModulus w H).totient*(∏ t ∈ divisorPrimes w H,(t-(positionResidues S t).card))* |
| (coprimeModulus w H).totient*(∏ t ∈ coprimePrimes w H,(t-(positionResidues S t).card)) := by |
| have he : Finset.univ.filter (jointCondition w H qA beta delta S) = |
| Finset.univ.filter (fun v : FullPattern w H => |
| divisorCondition w H beta S v.1 v.2.1 ∧ coprimeCondition w H qA delta S v.2.2.1 v.2.2.2) := by |
| apply Finset.filter_congr |
| rintro ⟨p0,k0,p1,m1⟩ _ |
| exact joint_condition_split w H qA beta delta S p0 k0 p1 m1 |
| rw [he,four_coordinate_count _ _ _ _ |
| (divisor_condition_card w H beta S) (fun p1 => coprime_condition_card w H qA delta S p1 hqA)] |
| simp only [ZMod.card_units_eq_totient] |
|
|
| theorem joint_density_split (w : ℝ) (H : ℕ) (S : Finset ℕ) : |
| jointDensity (divisorPrimes w H) S*jointDensity (coprimePrimes w H) S = |
| jointDensity (LargePrimeDeletion.cutoffPrimes ⌊w⌋₊) S := by |
| unfold jointDensity divisorPrimes coprimePrimes |
| exact Finset.prod_filter_mul_prod_filter_not _ _ _ |
|
|
| theorem joint_expectation (w : ℝ) (H qA : ℕ) (beta delta : ∀ t : ℕ,ZMod t) (S : Finset ℕ) |
| (hqA : ∀ t ∈ coprimePrimes w H,qA.Coprime t) : |
| expectation w H (fun v => if jointCondition w H qA beta delta S v then 1 else 0) = |
| jointDensity (LargePrimeDeletion.cutoffPrimes ⌊w⌋₊) S := by |
| have h0 : ∀ t ∈ divisorPrimes w H,t.Prime := fun t ht => |
| (LargePrimeDeletion.mem_cutoffPrimes.mp (Finset.mem_filter.mp ht).1).1 |
| have h1 : ∀ t ∈ coprimePrimes w H,t.Prime := fun t ht => |
| (LargePrimeDeletion.mem_cutoffPrimes.mp (Finset.mem_filter.mp ht).1).1 |
| have hc : (∑ v : FullPattern w H,if jointCondition w H qA beta delta S v then (1 : ℝ) else 0) = |
| ((Finset.univ.filter (jointCondition w H qA beta delta S)).card : ℝ) := by |
| simp only [Finset.card_filter,Nat.cast_sum,Nat.cast_ite,Nat.cast_one,Nat.cast_zero] |
| have hT0 : (divisorModulus w H : ℝ) ≠ 0 := by exact_mod_cast (divisorModulus_pos w H).ne' |
| have hT1 : (coprimeModulus w H : ℝ) ≠ 0 := by exact_mod_cast (coprimeModulus_pos w H).ne' |
| have hP0 : ((divisorModulus w H).totient : ℝ) ≠ 0 := by |
| exact_mod_cast (Nat.totient_pos.mpr (divisorModulus_pos w H)).ne' |
| have hP1 : ((coprimeModulus w H).totient : ℝ) ≠ 0 := by |
| exact_mod_cast (Nat.totient_pos.mpr (coprimeModulus_pos w H)).ne' |
| rw [expectation,← Finset.mul_sum,hc,joint_condition_card w H qA beta delta S hqA] |
| simp only [Nat.cast_mul] |
| rw [product_allowed_eq_density _ S h0,product_allowed_eq_density _ S h1] |
| change weight w H*(((divisorModulus w H).totient : ℝ)* |
| ((divisorModulus w H : ℝ)*jointDensity (divisorPrimes w H) S)* |
| ((coprimeModulus w H).totient : ℝ)* |
| ((coprimeModulus w H : ℝ)*jointDensity (coprimePrimes w H) S)) = _ |
| rw [← joint_density_split w H S] |
| unfold weight |
| field_simp |
|
|
| end ErdosLargePatternLaw |
|
|
|
|
|
|
| open _root_.Filter |
| namespace ErdosLargePatternLaw |
| open NumberTheoryLean ErdosVarianceSmallModel ErdosVarianceMoments ErdosRandomVariance |
|
|
| attribute [local instance] Classical.propDecidable |
| attribute [local instance] Classical.decEq |
|
|
| theorem single_expectation (w : ℝ) (H qA : ℕ) (beta delta : ∀ t : ℕ,ZMod t) (j : ℕ) |
| (hqA : ∀ t ∈ coprimePrimes w H,qA.Coprime t) : |
| expectation w H (indicator w H qA beta delta j) = |
| sieveProduct (LargePrimeDeletion.cutoffPrimes ⌊w⌋₊) := by |
| have h := joint_expectation w H qA beta delta {j} hqA |
| have he : (fun v => if jointCondition w H qA beta delta {j} v then (1 : ℝ) else 0) = |
| indicator w H qA beta delta j := by |
| funext v |
| simp only [jointCondition,Finset.mem_singleton,forall_eq,indicator] |
| rw [he,jointDensity_singleton] at h |
| exact h |
|
|
| theorem pair_expectation (w : ℝ) (H qA : ℕ) (beta delta : ∀ t : ℕ,ZMod t) (i j : ℕ) |
| (hqA : ∀ t ∈ coprimePrimes w H,qA.Coprime t) : |
| expectation w H (fun v => indicator w H qA beta delta i v*indicator w H qA beta delta j v) = |
| pairKernel (LargePrimeDeletion.cutoffPrimes ⌊w⌋₊) i j := by |
| have h := joint_expectation w H qA beta delta {i,j} hqA |
| have he : (fun v => indicator w H qA beta delta i v*indicator w H qA beta delta j v) = |
| (fun v => if jointCondition w H qA beta delta {i,j} v then 1 else 0) := by |
| funext v |
| simp only [jointCondition,Finset.forall_mem_insert,Finset.mem_singleton,forall_eq,indicator] |
| by_cases hi : survives w H qA beta delta i v < |
| by_cases hj : survives w H qA beta delta j v < |
| rw [he,h,jointDensity_pair] |
| |
| theorem count_variance (w : ℝ) (H qA J : ℕ) (beta delta : ∀ t : ℕ,ZMod t) |
| (hqA : ∀ t ∈ coprimePrimes w H,qA.Coprime t) : |
| expectation w H (fun v => ((survivorCount w H qA J beta delta v : ℝ)- |
| (J : ℝ)*sieveProduct (LargePrimeDeletion.cutoffPrimes ⌊w⌋₊))^2) ≤ |
| (J : ℝ)*sieveProduct (LargePrimeDeletion.cutoffPrimes ⌊w⌋₊) := by |
| have hs (j : ℕ) := single_expectation w H qA beta delta j hqA |
| have hp (i j : ℕ) := pair_expectation w H qA beta delta i j hqA |
| have hh := stable_prime_variance (Finset.univ : Finset (FullPattern w H)) (fun _ => weight w H) |
| J (indicator w H qA beta delta) (LargePrimeDeletion.cutoffPrimes ⌊w⌋₊) |
| (fun p hp => (LargePrimeDeletion.mem_cutoffPrimes.mp hp).1) 0 (le_refl _) |
| (fun j _ v _ => indicator_binary w H qA beta delta j v) |
| (by simpa only [add_zero] using (weight_sum w H).le) |
| (fun j _ => by simpa only [sub_zero,one_mul,expectation,weightedMoment] using (hs j).ge) |
| (fun j _ => by simpa only [add_zero,one_mul,expectation,weightedMoment] using (hs j).le) |
| (fun i _ j _ _ => by simpa only [add_zero,one_mul,expectation,weightedMoment] using (hp i j).le) |
| simpa only [mul_zero,zero_mul,one_mul,add_zero,indicatorSum,← count_eq_indicator_sum,expectation,weightedMoment] using hh |
|
|
| theorem uniform_pattern_chebyshev (a eta tau : ℝ) (ha : 0 < a) (heta : 0 < eta) (htau : 0 < tau) : |
| ∀ᶠ w : ℝ in atTop,1 < w ∧ ∀ (H qA : ℕ) (beta delta : ∀ t : ℕ,ZMod t), |
| (∀ t ∈ coprimePrimes w H,qA.Coprime t) → ∀ J : ℕ,w^a ≤ (J : ℝ) → |
| (∑ v : FullPattern w H with eta*((J : ℝ)*SmallSieveFinite.smallEuler ⌊w⌋₊) ≤ |
| |(survivorCount w H qA J beta delta v : ℝ)-(J : ℝ)*SmallSieveFinite.smallEuler ⌊w⌋₊|, |
| weight w H) ≤ tau := by |
| have heps : 0 < tau*eta^2 := mul_pos htau (sq_pos_of_pos heta) |
| filter_upwards [stable_source_chebyshev a (tau*eta^2) ha heps] with w hw |
| refine ⟨hw.1,?_⟩ |
| intro H qA beta delta hqA J hJ |
| have hcut : variancePrimes w=LargePrimeDeletion.cutoffPrimes ⌊w⌋₊ := by |
| ext p |
| simp only [variancePrimes,LargePrimeDeletion.mem_cutoffPrimes,Finset.mem_filter,Finset.mem_Ioc] |
| exact ⟨fun h => ⟨h.2,h.1.2⟩,fun h => ⟨⟨h.1.pos,h.2⟩,h.1⟩⟩ |
| have hs (j : ℕ) := single_expectation w H qA beta delta j hqA |
| have hp (i j : ℕ) := pair_expectation w H qA beta delta i j hqA |
| have hh := hw.2 J hJ (FullPattern w H) Finset.univ (fun _ => weight w H) |
| (indicator w H qA beta delta) 0 eta (fun _ _ => (weight_positive w H).le) |
| (le_refl _) zero_le_one heta (fun j _ v _ => indicator_binary w H qA beta delta j v) |
| (by simpa only [add_zero] using (weight_sum w H).le) |
| (fun j _ => by simpa only [sub_zero,one_mul,hcut,expectation,weightedMoment] using (hs j).ge) |
| (fun j _ => by simpa only [add_zero,one_mul,hcut,expectation,weightedMoment] using (hs j).le) |
| (fun i _ j _ _ => by simpa only [add_zero,one_mul,hcut,expectation,weightedMoment] using (hp i j).le) |
| have hV : sieveProduct (variancePrimes w)=SmallSieveFinite.smallEuler ⌊w⌋₊ := by |
| rw [hcut] |
| simp only [sieveProduct,SmallSieveFinite.smallEuler,one_div] |
| have hb : (tau*eta^2+4*0)/eta^2=tau := by |
| field_simp [heta.ne'] |
| ring |
| simpa only [hV,hb,indicatorSum,← count_eq_indicator_sum] using hh |
| |
| end ErdosLargePatternLaw |
| |
| |
| end Erdos970 |
| |
| end OAI |
| |
