| 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 |
|
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| end OAI |
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