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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 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 | import OAI.NumberTheory.TwoPointCorrelations.ModFiveContourKernel
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
namespace OAI
/-! A short multiplicative smoothing of Perron's kernel. The difference
has both the first-order bound and an integrable inverse-square tail. -/
namespace TwoPointCorrelations
open Complex
noncomputable def halaszPerronWindowKernel (x δ : ℝ) (s : ℂ) : ℂ :=
(((1 + δ : ℝ) : ℂ) * modFivePerronKernel ((1 + δ) * x) s -
modFivePerronKernel x s) / (δ : ℂ)
lemma halasz_exp_window_difference {δ : ℝ} (hδ : 0 < δ) (hδ1 : δ ≤ 1) (t : ℝ) :
‖Complex.exp (((2 : ℂ) + (t : ℂ) * Complex.I) * (Real.log (1 + δ) : ℂ)) - 1‖ ≤
δ * (3 + |t|) := by
have hy : 0 < 1 + δ := by linarith
have hlog0 : 0 ≤ Real.log (1 + δ) := Real.log_nonneg (by linarith)
have hlog : Real.log (1 + δ) ≤ δ := by
simpa only [add_sub_cancel_left] using Real.log_le_sub_one_of_pos hy
let z := Complex.exp (((t * Real.log (1 + δ) : ℝ) : ℂ) * Complex.I)
have hz : ‖z‖ = 1 := Complex.norm_exp_ofReal_mul_I _
have hz1 : ‖z - 1‖ ≤ |t| * Real.log (1 + δ) := by
have h := Real.norm_exp_I_mul_ofReal_sub_one_le (x := t * Real.log (1 + δ))
simpa only [z, mul_comm Complex.I, Real.norm_eq_abs, abs_mul, abs_of_nonneg hlog0] using h
have he : Complex.exp (((2 : ℂ) + (t : ℂ) * Complex.I) *
(Real.log (1 + δ) : ℂ)) = (((1 + δ) ^ 2 : ℝ) : ℂ) * z := by
rw [show ((2 : ℂ) + (t : ℂ) * Complex.I) * (Real.log (1 + δ) : ℂ) =
((2 * Real.log (1 + δ) : ℝ) : ℂ) +
(((t * Real.log (1 + δ) : ℝ) : ℂ) * Complex.I) by push_cast; ring,
Complex.exp_add, ← Complex.ofReal_exp]
congr 1
exact_mod_cast (show Real.exp (2 * Real.log (1 + δ)) = (1 + δ) ^ 2 by
simpa only [Nat.cast_ofNat, Real.exp_log hy] using Real.exp_nat_mul (Real.log (1 + δ)) 2)
rw [he, show (((1 + δ) ^ 2 : ℝ) : ℂ) * z - 1 =
(((1 + δ) ^ 2 - 1 : ℝ) : ℂ) * z + (z - 1) by push_cast; ring]
calc
_ ≤ ‖(((1 + δ) ^ 2 - 1 : ℝ) : ℂ) * z‖ + ‖z - 1‖ := norm_add_le _ _
_ ≤ ((1 + δ) ^ 2 - 1) + |t| * Real.log (1 + δ) := by
rw [norm_mul, Complex.norm_real, Real.norm_eq_abs, hz, mul_one,
abs_of_nonneg (by nlinarith : 0 ≤ (1 + δ) ^ 2 - 1)]
exact add_le_add le_rfl hz1
_ ≤ δ * (3 + |t|) := by
have hm := mul_le_mul_of_nonneg_left hlog (abs_nonneg t)
nlinarith
lemma halasz_perron_window_factor {x δ : ℝ} (hx : 0 < x) (hδ : 0 < δ) (s : ℂ) :
halaszPerronWindowKernel x δ s =
(x : ℂ) ^ s * (Complex.exp ((s + 1) * (Real.log (1 + δ) : ℂ)) - 1) /
((δ : ℂ) * s * (s + 1)) := by
have hy : 0 < 1 + δ := by linarith
have hyexp : ((1 + δ : ℝ) : ℂ) * ((1 + δ : ℝ) : ℂ) ^ s =
Complex.exp ((s + 1) * (Real.log (1 + δ) : ℂ)) := by
calc
_ = ((1 + δ : ℝ) : ℂ) ^ (s + 1) := by
rw [Complex.cpow_add _ _ (Complex.ofReal_ne_zero.mpr hy.ne'), Complex.cpow_one]
ring
_ = _ := by
rw [Complex.cpow_def_of_ne_zero (Complex.ofReal_ne_zero.mpr hy.ne'),
← Complex.ofReal_log hy.le]
congr 1
ring
unfold halaszPerronWindowKernel modFivePerronKernel
rw [Complex.ofReal_mul, Complex.mul_cpow_ofReal_nonneg hy.le hx.le]
calc
_ = (x : ℂ) ^ s *
(((1 + δ : ℝ) : ℂ) * ((1 + δ : ℝ) : ℂ) ^ s - 1) /
((δ : ℂ) * s * (s + 1)) := by
simp only [div_eq_mul_inv, mul_inv_rev]
ring
_ = _ := by rw [hyexp]
theorem halasz_perron_window_bound {x δ : ℝ} (hx : 0 < x) (hδ : 0 < δ)
(hδ1 : δ ≤ 1) (t : ℝ) :
‖halaszPerronWindowKernel x δ (1 + (t : ℂ) * Complex.I)‖ ≤
4 * x / ‖1 + (t : ℂ) * Complex.I‖ := by
let s : ℂ := 1 + (t : ℂ) * Complex.I
have hs : 1 ≤ ‖s‖ := by simpa [s] using Complex.abs_re_le_norm s
have hs1 : 2 ≤ ‖s + 1‖ := by
have h := Complex.abs_re_le_norm (s + 1)
norm_num [s] at h
exact h
have ht : |t| ≤ ‖s + 1‖ := by simpa [s] using Complex.abs_im_le_norm (s + 1)
have hs0 : 0 < ‖s‖ := by linarith
have hs10 : 0 < ‖s + 1‖ := by linarith
have he : ‖Complex.exp ((s + 1) * (Real.log (1 + δ) : ℂ)) - 1‖ ≤
δ * (4 * ‖s + 1‖) := by
have he := halasz_exp_window_difference hδ hδ1 t
have hshape : s + 1 = (2 : ℂ) + (t : ℂ) * Complex.I := by dsimp [s]; ring
rw [hshape]
apply he.trans
have hnorm : 3 + |t| ≤ 4 * ‖s + 1‖ := by linarith
rw [← hshape]
exact mul_le_mul_of_nonneg_left hnorm hδ.le
rw [halasz_perron_window_factor hx hδ, norm_div, norm_mul,
Complex.norm_cpow_eq_rpow_re_of_pos hx]
change x ^ s.re * ‖Complex.exp ((s + 1) * (Real.log (1 + δ) : ℂ)) - 1‖ /
‖(δ : ℂ) * s * (s + 1)‖ ≤ 4 * x / ‖s‖
have hre : s.re = 1 := by simp [s]
rw [hre, Real.rpow_one, norm_mul, norm_mul, Complex.norm_real,
Real.norm_eq_abs, abs_of_pos hδ]
calc
_ ≤ x * (δ * (4 * ‖s + 1‖)) / (δ * ‖s‖ * ‖s + 1‖) := by
exact div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left he hx.le) (by positivity)
_ = _ := by field_simp [ne_of_gt hδ, ne_of_gt hs0, ne_of_gt hs10]
theorem halasz_perron_window_tail {x δ : ℝ} (hx : 0 < x) (hδ : 0 < δ)
(hδ1 : δ ≤ 1) (t : ℝ) :
‖halaszPerronWindowKernel x δ (1 + (t : ℂ) * Complex.I)‖ ≤
20 * x / (δ * (1 + t ^ 2)) := by
have hy : 0 < 1 + δ := by linarith
have hk1 := modFive_perron_kernel_vertical (mul_pos hy hx) (by norm_num : (1 / 2 : ℝ) ≤ 1) t
have hk0 := modFive_perron_kernel_vertical hx (by norm_num : (1 / 2 : ℝ) ≤ 1) t
simp only [Real.rpow_one] at hk1 hk0
unfold halaszPerronWindowKernel
rw [norm_div, Complex.norm_real, Real.norm_eq_abs, abs_of_pos hδ]
apply (div_le_div_of_nonneg_right (norm_sub_le _ _) hδ.le).trans
rw [norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_pos hy]
have htop : (1 + δ) * ‖modFivePerronKernel ((1 + δ) * x)
(1 + (t : ℂ) * Complex.I)‖ + ‖modFivePerronKernel x (1 + (t : ℂ) * Complex.I)‖ ≤
20 * x / (1 + t ^ 2) := by
calc
_ ≤ (1 + δ) * (4 * ((1 + δ) * x) / (1 + t ^ 2)) + 4 * x / (1 + t ^ 2) :=
add_le_add (mul_le_mul_of_nonneg_left hk1 hy.le) hk0
_ ≤ _ := by
have hδ2 : (1 + δ) ^ 2 ≤ 4 := by nlinarith
have hnum : (1 + δ) * (4 * ((1 + δ) * x)) + 4 * x ≤ 20 * x := by
nlinarith [mul_nonneg hx.le (sub_nonneg.mpr hδ2)]
calc
_ = ((1 + δ) * (4 * ((1 + δ) * x)) + 4 * x) / (1 + t ^ 2) := by ring
_ ≤ _ := div_le_div_of_nonneg_right hnum (by positivity)
exact (div_le_div_of_nonneg_right htop hδ.le).trans_eq (by
field_simp [ne_of_gt hδ, show (1 + t ^ 2 : ℝ) ≠ 0 by positivity])
end TwoPointCorrelations
end OAI
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