File size: 5,481 Bytes
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 | import OAI.NumberTheory.Ostmann.Arithmetic.BulkResidueIntegral
import OAI.NumberTheory.Ostmann.ZeroDensity.BulkHaarDensity
import OAI.NumberTheory.Ostmann.Arithmetic.BulkSeparatedKernelBound
namespace OAI
/-! # Integrating the nonnegative arithmetic norm under the original Page density -/
namespace Ostmann
open MeasureTheory
open scoped Classical BigOperators
/-- The absolute residue average is integrated with the true harmonic box
volume. The Page factors remain inside the arithmetic norm estimate. -/
theorem bulk_absolute_integral_bound {J : Type*} [Fintype J]
(P : PublishedProgressionInput) (Q M : ℕ) [NeZero M]
(u v : J → ℝ) (hu : ∀ j, 0 < u j)
(K : BulkIntegrand J) (a : (J → (ZMod M)ˣ) → ℂ) (W A : ℝ)
(hW : 0 ≤ W) (hK : ∀ y, ‖K y‖ ≤ W)
(hA : ∀ y : J → ℝ, (∀ j, y j ∈ Set.Ioc (u j) (v j)) →
(Fintype.card (J → (ZMod M)ˣ) : ℝ)⁻¹ *
∑ z, ‖a z * ∏ j, pageGiantWeight P Q M (z j).val.val (y j)‖ ≤ A) :
‖∑ z : J → (ZMod M)ˣ, ∫ y, (K.constMul (a z)) y ∂Measure.pi (fun j =>
primeGiantMeasure P Q M (z j).val.val (u j) (v j))‖ ≤
(W * A) * ∏ j, ∫ x in Set.Ioc (u j) (v j), (x : ℝ)⁻¹ := by
let μ : J → Measure ℝ := fun j => volume.restrict (Set.Ioc (u j) (v j))
have hi (j : J) : Integrable (fun x : ℝ => x⁻¹) (μ j) := by
have hc : ContinuousOn (fun x : ℝ => x⁻¹) (Set.Icc (u j) (v j)) :=
continuousOn_inv₀.mono (fun x hx => ne_of_gt (lt_of_lt_of_le (hu j) hx.1))
exact hc.integrableOn_Icc.mono_set Set.Ioc_subset_Icc_self
have hp : Integrable (fun y : J → ℝ => ∏ j, (y j)⁻¹) (Measure.pi μ) :=
Integrable.fintype_prod hi
rw [bulk_page_residue_integral P Q (fun _ => M) (fun z j => (z j).val.val) u v hu (fun z => K.constMul (a z))]
have hb : ∀ᵐ y ∂Measure.pi μ,
‖∑ z, (K.constMul (a z)) y *
∏ j, (selectedPrimeLogDensity P Q M (z j).val.val (y j) : ℂ)‖ ≤
(W * A) * ∏ j, (y j)⁻¹ := by
have hm : ∀ᵐ y ∂Measure.pi μ, ∀ j, y j ∈ Set.Ioc (u j) (v j) :=
Filter.eventually_all.mpr (fun j =>
(Measure.tendsto_eval_ae_ae (μ := μ) (i := j)).eventually (ae_restrict_mem measurableSet_Ioc))
filter_upwards [hm] with y hy
have hny (j : J) : 0 < y j := (hu j).trans (hy j).1
have he : (∑ z, (K.constMul (a z)) y *
∏ j, (selectedPrimeLogDensity P Q M (z j).val.val (y j) : ℂ)) =
K y * ∑ z, a z * ∏ j, (selectedPrimeLogDensity P Q M (z j).val.val (y j) : ℂ) := by
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro z _
change (a z * K y) * _ = _
ring
rw [he, bulk_prime_density_haar P Q M y a, norm_mul, norm_mul]
have hmean : ‖(Fintype.card (J → (ZMod M)ˣ) : ℂ)⁻¹ *
∑ z, a z * ∏ j, pageGiantWeight P Q M (z j).val.val (y j)‖ ≤ A := by
rw [norm_mul, norm_inv, Complex.norm_natCast]
exact (mul_le_mul_of_nonneg_left (norm_sum_le _ _) (by positivity)).trans (hA y hy)
have hprod : ‖∏ j, (y j : ℂ)⁻¹‖ = ∏ j, (y j)⁻¹ := by
rw [norm_prod]
apply Finset.prod_congr rfl
intro j _
rw [norm_inv, Complex.norm_real, Real.norm_of_nonneg (hny j).le]
rw [hprod, ← mul_assoc]
exact mul_le_mul_of_nonneg_right (mul_le_mul (hK y) hmean (norm_nonneg _) hW)
(Finset.prod_nonneg fun j _ => inv_nonneg.mpr (hny j).le)
have hn := norm_integral_le_of_norm_le (hp.const_mul (W * A)) hb
simpa only [integral_const_mul, integral_fintype_prod_eq_prod] using hn
/-- Sum all original boxes with their unchanged reciprocal-prime normalizers. -/
theorem bulk_absolute_mixture_bound {J C : Type*} [Fintype J] [Fintype C]
(P : PublishedProgressionInput) (Q M : ℕ) [NeZero M]
(u v : J → C → ℝ) (hu : ∀ j c, 0 < u j c)
(Z : J → ℝ) (hZ : ∀ j, 0 ≤ Z j)
(hmass : ∀ j, Z j * ∑ c, ∫ x in Set.Ioc (u j c) (v j c), (x : ℝ)⁻¹ ≤ 2)
(K : BulkIntegrand J) (a : (J → (ZMod M)ˣ) → ℂ) (W A : ℝ)
(hW : 0 ≤ W) (hA0 : 0 ≤ A) (hK : ∀ y, ‖K y‖ ≤ W)
(hA : ∀ y : J → ℝ, (∀ j, 0 ≤ y j) →
(Fintype.card (J → (ZMod M)ˣ) : ℝ)⁻¹ *
∑ z, ‖a z * ∏ j, pageGiantWeight P Q M (z j).val.val (y j)‖ ≤ A) :
‖∑ z : J → (ZMod M)ˣ, ∫ y, (K.constMul (a z)) y ∂Measure.pi (fun j =>
bulkCellMixture (Z j) (fun c => primeGiantMeasure P Q M (z j).val.val (u j c) (v j c)))‖ ≤
(W * A) * 2 ^ Fintype.card J := by
let μ := fun (z : J → (ZMod M)ˣ) j c =>
primeGiantMeasure P Q M (z j).val.val (u j c) (v j c)
let _ : ∀ z j c, IsFiniteMeasure (μ z j c) := fun z j c =>
finite_primeGiantMeasure P Q _ _ _ _ (hu j c)
change ‖∑ z, ∫ y, (K.constMul (a z)) y ∂Measure.pi (fun j => bulkCellMixture (Z j) (μ z j))‖ ≤ _
rw [bulk_residue_mixture_integral Z hZ μ (fun z => K.constMul (a z))]
let H := fun j c => ∫ x in Set.Ioc (u j c) (v j c), (x : ℝ)⁻¹
have hH (j : J) (c : C) : 0 ≤ H j c := by
apply integral_nonneg_of_ae
filter_upwards [ae_restrict_mem measurableSet_Ioc] with x hx
exact inv_nonneg.mpr ((hu j c).trans hx.1).le
apply bulk_box_sum_bound Z hZ H hH hmass (W * A) (mul_nonneg hW hA0)
intro c
exact bulk_absolute_integral_bound P Q M (fun j => u j (c j)) (fun j => v j (c j))
(fun j => hu j (c j)) K a W A hW hK (fun y hy => hA y (fun j => ((hu j (c j)).trans (hy j).1).le))
end Ostmann
end OAI
|