| import OAI.NumberTheory.TwoPoint.ShortIntervals.MRTMultiscaleEnergy |
|
|
| namespace OAI |
|
|
| /-! The explicit amplification order used between two successive MRT |
| prime scales. The integer ceiling satisfies the exact support inequalities |
| needed by the mixed cofactor moment. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset MeasureTheory |
| open scoped Classical |
|
|
| noncomputable def mrtAmplificationOrder (Y u : ℝ) : ℕ := |
| ⌈Real.log u / Real.log Y⌉₊ |
|
|
| lemma mrt_nat_pow_eq_exp {Y : ℝ} (hY : 0 < Y) (r : ℕ) : |
| Y ^ r = Real.exp ((r : ℝ) * Real.log Y) := by |
| rw [← Real.rpow_natCast, Real.rpow_def_of_pos hY] |
| congr 1 |
| ring |
|
|
| theorem mrt_amplification_order_bounds {Y u : ℝ} (hY : 1 < Y) (hu : 1 ≤ u) : |
| u ≤ Y ^ mrtAmplificationOrder Y u ∧ |
| Y ^ mrtAmplificationOrder Y u ≤ u * Y ∧ |
| (mrtAmplificationOrder Y u : ℝ) < Real.log u / Real.log Y + 1 := by |
| have hY0 : 0 < Y := by linarith |
| have hu0 : 0 < u := by linarith |
| have hlogY : 0 < Real.log Y := Real.log_pos hY |
| have hquot : 0 ≤ Real.log u / Real.log Y := |
| div_nonneg (Real.log_nonneg hu) hlogY.le |
| have hlo : Real.log u / Real.log Y ≤ (mrtAmplificationOrder Y u : ℝ) := |
| Nat.le_ceil _ |
| have hhi : (mrtAmplificationOrder Y u : ℝ) < Real.log u / Real.log Y + 1 := |
| Nat.ceil_lt_add_one hquot |
| refine ⟨?_, ?_, hhi⟩ |
| · calc |
| u = Real.exp (Real.log u) := (Real.exp_log hu0).symm |
| _ ≤ Real.exp ((mrtAmplificationOrder Y u : ℝ) * Real.log Y) := |
| Real.exp_le_exp.mpr ((div_le_iff₀ hlogY).mp hlo) |
| _ = _ := (mrt_nat_pow_eq_exp hY0 _).symm |
| · have hh : (mrtAmplificationOrder Y u : ℝ) * Real.log Y < |
| Real.log u + Real.log Y := by |
| have hh := (mul_lt_mul_of_pos_right hhi hlogY) |
| rw [add_mul, div_mul_cancel₀ _ hlogY.ne', one_mul] at hh |
| exact hh |
| calc |
| _ = Real.exp ((mrtAmplificationOrder Y u : ℝ) * Real.log Y) := |
| mrt_nat_pow_eq_exp hY0 _ |
| _ ≤ Real.exp (Real.log u + Real.log Y) := Real.exp_le_exp.mpr hh.le |
| _ = u * Y := by rw [Real.exp_add, Real.exp_log hu0, Real.exp_log hY0] |
| |
| theorem mrt_short_class_cofactor_energy_chosen (P Q : Finset ℕ) |
| (hP : ∀ p ∈ P, p.Prime) {Y N : ℕ} (hY : 1 < Y) (hN : 0 < N) |
| (hbin : P ⊆ Icc Y (2 * Y)) (a : ℕ → ℂ) (ha : ∀ p ∈ P, ‖a p‖ ≤ 1) |
| (F : ℕ → ℂ) (hF : OneBounded F) {u : ℝ} (hu : 1 ≤ 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, ‖mrtCofactorPolynomial Q F N u t‖ ^ 2) ≤ |
| (16 * Real.exp 10 * |
| (T / (N : ℝ) + (2 : ℝ) ^ (mrtAmplificationOrder Y u + 1) * Y) * |
| ((mrtAmplificationOrder Y u).factorial : ℝ) ^ 2) / |
| V ^ (2 * mrtAmplificationOrder Y u) := by |
| obtain ⟨hlo, hhi, _⟩ := mrt_amplification_order_bounds |
| (show (1 : ℝ) < Y by exact_mod_cast hY) hu |
| exact mrt_short_class_cofactor_energy P Q hP (by omega) hN hbin a ha F hF hu _ |
| hlo hhi hT hV hE hlarge |
| |
| end TwoPointCorrelations |
| |
| end OAI |
| |
