| 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 |
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|
