| import OAI.NumberTheory.TwoPoint.ShortIntervals.MRTMixedPolynomial |
| import OAI.NumberTheory.TwoPoint.ShortIntervals.MRTRestrictedEnergy |
|
|
| namespace OAI |
|
|
| /-! Amplification on the multiscale classes where a preceding short-prime |
| polynomial is large. The inserted power is estimated by the actual mixed |
| prime/cofactor moment, not by an assumed large-value proposition. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset MeasureTheory |
| open scoped Classical |
|
|
| theorem mrt_amplified_energy (Q R : ℝ → ℂ) (hQ : Continuous Q) (hR : Continuous R) |
| (r : ℕ) {T V : ℝ} (hT : 0 ≤ T) (hV : 0 < V) |
| {E : Set ℝ} (hE : E ⊆ Set.Ioc (-T) T) |
| (hlarge : ∀ t ∈ E, V ≤ ‖Q t‖) : |
| (∫ t in E, ‖R t‖ ^ 2) ≤ |
| (∫ t in -T..T, ‖Q t ^ r * R t‖ ^ 2) / V ^ (2 * r) := by |
| have hprod : Continuous (fun t => Q t ^ r * R t) := (hQ.pow r).mul hR |
| have hi := mrt_continuous_square_integrable hprod hT hE |
| apply (le_div_iff₀ (pow_pos hV _)).mpr |
| calc |
| _ = ∫ t in E, V ^ (2 * r) * ‖R t‖ ^ 2 := by |
| rw [integral_const_mul] |
| ring |
| _ ≤ ∫ t in E, ‖Q t ^ r * R t‖ ^ 2 := by |
| apply setIntegral_mono_of_nonneg (fun _ _ => by positivity) _ hi |
| intro t ht |
| rw [norm_mul, norm_pow, mul_pow, ← pow_mul, Nat.mul_comm r 2] |
| exact mul_le_mul_of_nonneg_right |
| (pow_le_pow_left₀ hV.le (hlarge t ht) _) (sq_nonneg _) |
| _ ≤ _ := mrt_restricted_square_le_interval hprod hT hE |
|
|
| theorem mrt_large_prime_class_cofactor_energy (P M : Finset ℕ) |
| (hP : ∀ p ∈ P, p.Prime) (hM : ∀ m ∈ M, 0 < m) |
| (a b : ℕ → ℂ) (ha : ∀ p ∈ P, ‖a p‖ ≤ 1) (hb : ∀ m ∈ M, ‖b m‖ ≤ 1) |
| (r : ℕ) {L U : ℕ} (hL : 0 < L) (hLU : L ≤ U) |
| (hsupport : ∀ v ∈ Fintype.piFinset (fun _ : Fin r => P), ∀ m ∈ M, |
| L < (∏ i, v i) * m ∧ (∏ i, v i) * m ≤ U) |
| {T V : ℝ} (hT : 0 < T) (hV : 0 < V) |
| {E : Set ℝ} (hE : E ⊆ Set.Ioc (-T) T) |
| (hlarge : ∀ t ∈ E, V ≤ |
| ‖mrtExponentialPolynomial P (fun p => a p / (p : ℂ)) |
| (fun p => -Real.log (p : ℝ)) t‖) : |
| (∫ t in E, ‖mrtExponentialPolynomial M (fun m => b m / (m : ℂ)) |
| (fun m => -Real.log (m : ℝ)) t‖ ^ 2) ≤ |
| (16 * Real.exp 1 * ((T + U) / (L : ℝ)) * (r.factorial : ℝ) ^ 2 * |
| (∏ p ∈ P, (1 - (1 : ℝ) / p)⁻¹) ^ 3) / V ^ (2 * r) := by |
| apply (mrt_amplified_energy _ _ |
| (mrtExponentialPolynomial_continuous _ _ _) |
| (mrtExponentialPolynomial_continuous _ _ _) r hT.le hV hE hlarge).trans |
| exact div_le_div_of_nonneg_right |
| (mrt_mixed_prime_cofactor_mean_square P M hP hM a b ha hb r hL hLU hsupport hT) |
| (pow_nonneg hV.le _) |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
