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8769036 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 | 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
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