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f4b3ad1 | 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 | import OAI.NumberTheory.Ostmann.Arithmetic.LogCellPartitionReplacement
import OAI.NumberTheory.Ostmann.Arithmetic.LogCellPartitionTuples
import OAI.NumberTheory.Ostmann.Arithmetic.PrimeCellJointReplacementAlgebra
namespace OAI
open _root_.Erdos970 _root_.OAI.Erdos970
open Erdos970.Erdos970Dependency.SiegelWalfisz
noncomputable section
namespace Ostmann.Arithmetic.LogCellPartition
open PrimeProgression PrimeCellReplacement
open scoped BigOperators
def assignedJointComplexTestSum {ι : Type*} [Fintype ι] [DecidableEq ι]
(N : ι → ℕ) (M : ℕ) [NeZero M] (lo hi η Z : ι → ℝ)
(j : GridBoxIndex lo hi η) (F : (ι → (ZMod M)ˣ) → ℂ) : ℂ :=
∑ p : AssignedPrimeTuple N lo hi η j,
(∏ i, (((Z i * ((p i).val : ℝ))⁻¹ : ℝ) : ℂ)) *
jointUnitTest F (fun i => ((p i).val : ZMod M))
theorem assignedJointComplexTestSum_eq {ι : Type*} [Fintype ι] [DecidableEq ι]
(N : ι → ℕ) (M : ℕ) [NeZero M] (lo hi η Z : ι → ℝ)
(j : GridBoxIndex lo hi η) (F : (ι → (ZMod M)ˣ) → ℂ) :
assignedJointComplexTestSum N M lo hi η Z j F =
∑ u : ι → (ZMod M)ˣ,
(∏ i, (assignedResidueMass (N i) M (u i) (lo i) (hi i) (η i) (Z i) (j i) : ℂ)) * F u := by
classical
have hpart := finite_product_prior_partition
(fun i (p : {p : ℕ // p ∈ assignedPrimeSupport (N i) (lo i) (hi i) (η i) (j i)}) =>
(p.val : ZMod M))
(fun i (p : {p : ℕ // p ∈ assignedPrimeSupport (N i) (lo i) (hi i) (η i) (j i)}) =>
(((Z i * (p.val : ℝ))⁻¹ : ℝ) : ℂ)) F
change assignedJointComplexTestSum N M lo hi η Z j F = _ at hpart
rw [hpart]
apply Finset.sum_congr rfl
intro u hu
congr 1
apply Finset.prod_congr rfl
intro i hi'
rw [Finset.sum_coe_sort (assignedPrimeSupport (N i) (lo i) (hi i) (η i) (j i))
(fun p : ℕ => if (u i : ZMod M) = (p : ZMod M) then
(((Z i * (p : ℝ))⁻¹ : ℝ) : ℂ) else 0)]
simp only [assignedResidueMass, Complex.ofReal_sum, Finset.sum_filter]
apply Finset.sum_congr rfl
intro p hp
by_cases he : (p : ZMod M) = (u i : ZMod M)
· simp [he]
· simp [he, Ne.symm he]
theorem assignedJointComplexTestSum_error {ι : Type*} [Fintype ι] [DecidableEq ι]
(N : ι → ℕ) (M : ℕ) [NeZero M] (lo hi η Z : ι → ℝ)
(j : GridBoxIndex lo hi η) (main E : ι → ℝ)
(F : (ι → (ZMod M)ˣ) → ℂ) {ε B : ℝ} (hε : 0 ≤ ε) (hB : 1 ≤ B)
(hE : ∀ i, E i ≤ ε) (hbound : ∀ i, |main i| + E i ≤ B)
(hmass : ∀ i (u : (ZMod M)ˣ),
|assignedResidueMass (N i) M u (lo i) (hi i) (η i) (Z i) (j i) - main i| ≤ E i) :
‖assignedJointComplexTestSum N M lo hi η Z j F -
(∏ i, (main i : ℂ)) * ∑ u : ι → (ZMod M)ˣ, F u‖ ≤
Fintype.card ι * ε * B ^ Fintype.card ι * ∑ u : ι → (ZMod M)ˣ, ‖F u‖ := by
classical
have hμ : ∀ i (u : (ZMod M)ˣ),
‖(assignedResidueMass (N i) M u (lo i) (hi i) (η i) (Z i) (j i) : ℂ)‖ ≤ B := by
intro i u
have ht := abs_add_le
(assignedResidueMass (N i) M u (lo i) (hi i) (η i) (Z i) (j i) - main i) (main i)
rw [sub_add_cancel] at ht
simp only [Complex.norm_real, Real.norm_eq_abs]
linarith [hmass i u, hbound i]
have hν : ∀ i (u : (ZMod M)ˣ), ‖(main i : ℂ)‖ ≤ B := by
intro i u
have he0 : 0 ≤ E i := (abs_nonneg _).trans (hmass i 1)
simp only [Complex.norm_real, Real.norm_eq_abs]
linarith [hbound i]
have he : ∀ i (u : (ZMod M)ˣ),
‖(assignedResidueMass (N i) M u (lo i) (hi i) (η i) (Z i) (j i) : ℂ) - (main i : ℂ)‖ ≤ ε := by
intro i u
simpa only [← Complex.ofReal_sub, Complex.norm_real, Real.norm_eq_abs] using
(hmass i u).trans (hE i)
have h := joint_product_test_error
(fun i (u : (ZMod M)ˣ) =>
(assignedResidueMass (N i) M u (lo i) (hi i) (η i) (Z i) (j i) : ℂ))
(fun i (_ : (ZMod M)ˣ) => (main i : ℂ)) F hε hB hμ hν he
rw [assignedJointComplexTestSum_eq]
simpa only [Finset.mul_sum] using h
theorem exists_assigned_joint_primeCell_replacement_constants :
∃ d K L₀ : ℝ, 0 < d ∧ 0 < K ∧ 1 ≤ L₀ ∧
∀ (ι : Type*) [Fintype ι] [DecidableEq ι] (N : ι → ℕ) (M : ℕ) [NeZero M]
(lo hi η Z : ι → ℝ),
(∀ i, L₀ ≤ lo i ∧ lo i ≤ hi i ∧ 0 < η i ∧ η i ≤ 1 ∧
⌊Real.exp (hi i)⌋₊ ≤ N i ∧ 0 < Z i ∧
(M : ℝ) ≤ Real.exp (d * (lo i) ^ (1 / 3 : ℝ))) →
∀ (j : GridBoxIndex lo hi η) (ε B : ℝ), 0 ≤ ε → 1 ≤ B →
(∀ i, (K / Z i) * Real.exp (-d * (gridPoint (lo i) (hi i) (η i) (j i).val) ^ (1 / 3 : ℝ)) +
(Z i*Real.exp (gridPoint (lo i) (hi i) (η i) (j i).val))⁻¹ ≤ ε) →
(∀ i, |harmonicIntegral M (gridPoint (lo i) (hi i) (η i) (j i).val)
(gridPoint (lo i) (hi i) (η i) ((j i).val+1)) / Z i| +
((K / Z i) * Real.exp (-d * (gridPoint (lo i) (hi i) (η i) (j i).val) ^ (1 / 3 : ℝ)) +
(Z i*Real.exp (gridPoint (lo i) (hi i) (η i) (j i).val))⁻¹) ≤ B) →
∀ F : (ι → (ZMod M)ˣ) → ℂ,
‖assignedJointComplexTestSum N M lo hi η Z j F -
(∏ i, ((harmonicIntegral M (gridPoint (lo i) (hi i) (η i) (j i).val)
(gridPoint (lo i) (hi i) (η i) ((j i).val+1)) / Z i : ℝ) : ℂ)) *
∑ u : ι → (ZMod M)ˣ, F u‖ ≤
Fintype.card ι * ε * B ^ Fintype.card ι *
∑ u : ι → (ZMod M)ˣ, ‖F u‖ := by
obtain ⟨d,K,L₀,hd,hK,hL₀,h⟩ := exists_assigned_residueMass_error_constants
refine ⟨d,K,L₀,hd,hK,hL₀,?_⟩
intro ι _ _ N M _ lo hi η Z hcell j ε B hε hB hE hbound F
apply assignedJointComplexTestSum_error N M lo hi η Z j
(fun i => harmonicIntegral M (gridPoint (lo i) (hi i) (η i) (j i).val)
(gridPoint (lo i) (hi i) (η i) ((j i).val+1)) / Z i)
(fun i => (K / Z i) * Real.exp (-d * (gridPoint (lo i) (hi i) (η i) (j i).val) ^ (1 / 3 : ℝ)) +
(Z i*Real.exp (gridPoint (lo i) (hi i) (η i) (j i).val))⁻¹) F hε hB hE hbound
intro i u
obtain ⟨hlo,hlohi,hη,hη1,hN,hZ,hmod⟩ := hcell i
exact h (lo i) (hi i) (η i) hlo hlohi hη hη1 (N i) M (Z i) hN hZ hmod (j i) u u.isUnit
end Ostmann.Arithmetic.LogCellPartition
end
end OAI
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