File size: 3,120 Bytes
180bc03 | 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 | 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
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