| import OAI.NumberTheory.TwoPoint.Bounds.ResidueRestriction |
| import OAI.NumberTheory.TwoPoint.Bounds.ActualPaddingResidues |
|
|
| namespace OAI |
|
|
| /-! The actual padding estimates under the full tuple-and-padding residue |
| law. All dependence on the tuple coordinates is removed by the proved |
| product-law restriction identity. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset Filter |
| open scoped Classical |
|
|
| namespace ProhibitedPrimeFamily |
|
|
| variable {h J M B : ℕ} (data : ProhibitedPrimeFamily h J M) |
|
|
| lemma residue_average_padding_squares |
| (hB : ∀ p ∈ data.P ∪ data.Q, p ≤ B) (D : Finset ℕ) |
| (hD : D ⊆ retainedPrimeDivisors data.Q) (site : ℤ) |
| (η c : ℝ) (bins : Finset ℤ) : |
| (data.residueLaw B hB).average (fun x => |
| (actualPaddingVertex data.Q (data.residueOrigin x + site)) ^ 2 * |
| ∑ j ∈ bins, (paddingDensity D actualPaddingCoefficient (actualPaddingBin η c j) |
| (actualPaddingVertex data.Q) (data.residueOrigin x + site)) ^ 2) = |
| (data.paddingResidueLaw B hB).average (fun x => |
| (actualPaddingVertex data.Q (data.paddingResidueOrigin x + site)) ^ 2 * |
| ∑ j ∈ bins, (paddingDensity D actualPaddingCoefficient (actualPaddingBin η c j) |
| (actualPaddingVertex data.Q) (data.paddingResidueOrigin x + site)) ^ 2) := by |
| apply data.residue_average_padding_shift hB site (fun n => |
| (actualPaddingVertex data.Q n) ^ 2 * ∑ j ∈ bins, |
| (paddingDensity D actualPaddingCoefficient (actualPaddingBin η c j) |
| (actualPaddingVertex data.Q) n) ^ 2) |
| intro n m hnm |
| rw [actualPaddingVertex_residue_congr data.Q n m hnm] |
| congr 1 |
| apply sum_congr rfl |
| intro j _ |
| rw [actualPaddingDensity_residue_congr data.Q data.primeQ D hD _ n m hnm] |
|
|
| lemma residue_average_padding_degree_tail |
| (hB : ∀ p ∈ data.P ∪ data.Q, p ≤ B) (site : ℤ) (L : ℝ) : |
| (data.residueLaw B hB).average (fun x => |
| (actualPaddingVertex data.Q (data.residueOrigin x + site)) ^ 2 * |
| if ¬actualPaddingDegreeCut data.Q L (data.residueOrigin x + site) then 1 else 0) = |
| (data.paddingResidueLaw B hB).average (fun x => |
| (actualPaddingVertex data.Q (data.paddingResidueOrigin x + site)) ^ 2 * |
| if ¬actualPaddingDegreeCut data.Q L (data.paddingResidueOrigin x + site) then 1 else 0) := by |
| apply data.residue_average_padding_shift hB site (fun n => |
| (actualPaddingVertex data.Q n) ^ 2 * |
| if ¬actualPaddingDegreeCut data.Q L n then 1 else 0) |
| intro n m hnm |
| rw [actualPaddingVertex_residue_congr data.Q n m hnm, |
| actualPaddingDegreeCut_residue_congr data.Q L n m hnm] |
|
|
| end ProhibitedPrimeFamily |
|
|
| theorem ModFiveThetaInput.actual_family_padding_bin_square (hP : ModFiveThetaInput) |
| (E : Finset ℕ) : |
| ∃ C : ℝ, 0 < C ∧ ∀ (L η c : ℝ) (h J M B : ℕ) |
| (data : ProhibitedPrimeFamily h J M), data.Q = paddingPrimeSupply E L → |
| ∀ (site : ℤ) (bins : Finset ℤ), 1 ≤ L → 0 < η → η ≤ 1 → |
| ∀ (hB : ∀ p ∈ data.P ∪ data.Q, p ≤ B) |
| (D : Finset ℕ), D ⊆ retainedPrimeDivisors data.Q → |
| (data.residueLaw B hB).average (fun x => |
| (actualPaddingVertex data.Q (data.residueOrigin x + site)) ^ 2 * |
| ∑ j ∈ bins, (paddingDensity D actualPaddingCoefficient (actualPaddingBin η c j) |
| (actualPaddingVertex data.Q) (data.residueOrigin x + site)) ^ 2) / |
| paddingTiltNormalizer data.Q ≤ C / L := by |
| obtain ⟨C, hC, hb⟩ := hP.actual_padding_residue_bin_square E |
| refine ⟨C, hC, ?_⟩ |
| intro L η c h J M B data hQ site bins hL hη hηone hB D hD |
| rw [data.residue_average_padding_squares hB D hD site η c bins] |
| have hbQ : ∀ (Q : Finset ℕ), Q = paddingPrimeSupply E L → |
| ∀ (hprime : ∀ p ∈ Q, p.Prime) (hQB : ∀ p ∈ Q, p ≤ B) |
| (N : (Q → Fin B) → ℤ), |
| (∀ z (p : Q), (N z : ZMod p.val) = |
| ((z p).val : ZMod p.val) + (site : ZMod p.val)) → |
| D ⊆ retainedPrimeDivisors Q → |
| (FiniteLaw.independent (fun p : Q => uniformResidueLaw B p.val |
| (hprime p p.property).pos (hQB p p.property))).average (fun z => |
| (actualPaddingVertex Q (N z)) ^ 2 * ∑ j ∈ bins, |
| (paddingDensity D actualPaddingCoefficient (actualPaddingBin η c j) |
| (actualPaddingVertex Q) (N z)) ^ 2) / paddingTiltNormalizer Q ≤ C / L := by |
| intro Q hQ hprime hQB N hN hD |
| subst Q |
| exact hb L η c B site bins hL hη hηone hQB D hD N hN |
| apply hbQ data.Q hQ data.primeQ (fun p hp => hB p (mem_union_right _ hp)) |
| (fun z => data.paddingResidueOrigin z + site) _ hD |
| intro z p |
| rw [Int.cast_add, data.paddingResidueOrigin_spec z p] |
|
|
| theorem ModFiveThetaInput.eventually_actual_family_padding_degree_tail |
| (hP : ModFiveThetaInput) (E : Finset ℕ) : |
| ∀ᶠ L : ℝ in atTop, ∀ (h J M B : ℕ) (data : ProhibitedPrimeFamily h J M), |
| data.Q = paddingPrimeSupply E L → |
| ∀ (hB : ∀ p ∈ data.P ∪ data.Q, p ≤ B) (site : ℤ), |
| (data.residueLaw B hB).average (fun x => |
| (actualPaddingVertex data.Q (data.residueOrigin x + site)) ^ 2 * |
| if ¬actualPaddingDegreeCut data.Q L (data.residueOrigin x + site) then 1 else 0) / |
| paddingTiltNormalizer data.Q ≤ L ^ (-100 : ℝ) := by |
| filter_upwards [hP.eventually_actual_padding_degree_tail E] with L hb |
| intro h J M B data hQ hB site |
| rw [data.residue_average_padding_degree_tail hB site L] |
| have hbQ : ∀ (Q : Finset ℕ), Q = paddingPrimeSupply E L → |
| ∀ (hprime : ∀ p ∈ Q, p.Prime) (hQB : ∀ p ∈ Q, p ≤ B) |
| (N : (Q → Fin B) → ℤ), |
| (∀ z (p : Q), (N z : ZMod p.val) = |
| ((z p).val : ZMod p.val) + (site : ZMod p.val)) → |
| (FiniteLaw.independent (fun p : Q => uniformResidueLaw B p.val |
| (hprime p p.property).pos (hQB p p.property))).average (fun z => |
| (actualPaddingVertex Q (N z)) ^ 2 * |
| if ¬actualPaddingDegreeCut Q L (N z) then 1 else 0) / |
| paddingTiltNormalizer Q ≤ L ^ (-100 : ℝ) := by |
| intro Q hQ hprime hQB N hN |
| subst Q |
| exact hb B site hQB N hN |
| apply hbQ data.Q hQ data.primeQ (fun p hp => hB p (mem_union_right _ hp)) |
| (fun z => data.paddingResidueOrigin z + site) |
| intro z p |
| rw [Int.cast_add, data.paddingResidueOrigin_spec z p] |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
