| import OAI.NumberTheory.TwoPointCorrelations.HalaszPhaseIntegral | |
| namespace OAI | |
| /-! A uniform power-sum estimate, using centered cells past the frequency. -/ | |
| namespace TwoPointCorrelations | |
| open MeasureTheory Finset | |
| lemma halasz_phase_cells_telescope (t : ℝ) (A N : ℕ) (hA : 1 ≤ A) (hAN : A ≤ N) : | |
| (∑ n ∈ Ioc A N, ∫ x in ((n : ℝ) - 1 / 2)..((n : ℝ) + 1 / 2), | |
| halaszPowerPhase t x) = | |
| ∫ x in ((A : ℝ) + 1 / 2)..((N : ℝ) + 1 / 2), halaszPowerPhase t x := by | |
| induction N, hAN using Nat.le_induction with | |
| | base => simp | |
| | succ N hAN ih => | |
| rw [sum_Ioc_succ_top hAN, ih] | |
| have he : ((N + 1 : ℕ) : ℝ) - 1 / 2 = (N : ℝ) + 1 / 2 := by push_cast; ring | |
| rw [he] | |
| apply intervalIntegral.integral_add_adjacent_intervals | |
| · exact halasz_power_phase_intervalIntegrable t ((A : ℝ) + 1 / 2) | |
| ((N : ℝ) + 1 / 2) (by positivity) (by positivity) | |
| · exact halasz_power_phase_intervalIntegrable t ((N : ℝ) + 1 / 2) | |
| (((N + 1 : ℕ) : ℝ) + 1 / 2) (by positivity) (by positivity) | |
| lemma halasz_phase_cell_sum_error (t : ℝ) (A N : ℕ) (hA : 1 ≤ A) | |
| (hAN : A ≤ N) (ht : |t| ≤ (A : ℝ)) : | |
| ‖(∑ n ∈ Ioc A N, halaszPowerPhase t n) - | |
| ∫ x in ((A : ℝ) + 1 / 2)..((N : ℝ) + 1 / 2), halaszPowerPhase t x‖ ≤ | |
| (t ^ 2 + |t|) / A := by | |
| rw [← halasz_phase_cells_telescope t A N hA hAN, ← sum_sub_distrib] | |
| calc | |
| _ ≤ ∑ n ∈ Ioc A N, | |
| ‖halaszPowerPhase t n - | |
| ∫ x in ((n : ℝ) - 1 / 2)..((n : ℝ) + 1 / 2), halaszPowerPhase t x‖ := | |
| norm_sum_le _ _ | |
| _ ≤ ∑ n ∈ Ioc A N, (t ^ 2 + |t|) / (n : ℝ) ^ 2 := by | |
| apply sum_le_sum | |
| intro n hn | |
| have hnA := (mem_Ioc.mp hn).1 | |
| have hn1 : (1 : ℝ) ≤ n := by exact_mod_cast hA.trans hnA.le | |
| have htn : |t| ≤ (n : ℝ) := ht.trans (by exact_mod_cast hnA.le) | |
| have hh := halasz_power_phase_cell_integral t n hn1 htn | |
| rw [intervalIntegral.integral_comp_add_left] at hh | |
| have he : (n : ℝ) + (-1 / 2) = (n : ℝ) - 1 / 2 := by ring | |
| rw [he] at hh | |
| simpa only [norm_sub_rev] using hh | |
| _ = (t ^ 2 + |t|) * ∑ n ∈ Ioc A N, ((n : ℝ) ^ 2)⁻¹ := by | |
| simp_rw [div_eq_mul_inv, mul_sum] | |
| _ ≤ (t ^ 2 + |t|) * ((A : ℝ)⁻¹ - (N : ℝ)⁻¹) := | |
| mul_le_mul_of_nonneg_left (sum_Ioc_inv_sq_le_sub (by omega : A ≠ 0) hAN) (by positivity) | |
| _ ≤ (t ^ 2 + |t|) / A := by | |
| rw [div_eq_mul_inv] | |
| exact mul_le_mul_of_nonneg_left (sub_le_self _ (by positivity)) (by positivity) | |
| lemma halasz_power_phase_prefix_norm (t : ℝ) (N : ℕ) : | |
| ‖∑ n ∈ Ioc 0 N, halaszPowerPhase t n‖ ≤ N := by | |
| calc | |
| _ ≤ ∑ n ∈ Ioc 0 N, ‖halaszPowerPhase t n‖ := norm_sum_le _ _ | |
| _ = _ := by simp | |
| theorem halasz_power_phase_sum_integral (t : ℝ) (N : ℕ) : | |
| ‖(∑ n ∈ Ioc 0 N, halaszPowerPhase t n) - | |
| ∫ x in (0 : ℝ)..(N : ℝ), halaszPowerPhase t x‖ ≤ 8 * (1 + |t|) := by | |
| let A : ℕ := ⌈1 + |t|⌉₊ | |
| have hAlo : 1 + |t| ≤ (A : ℝ) := Nat.le_ceil _ | |
| have hAhi : (A : ℝ) < 2 + |t| := by | |
| have hh : (A : ℝ) < 1 + |t| + 1 := Nat.ceil_lt_add_one (by positivity) | |
| linarith | |
| have hA1 : 1 ≤ A := by exact_mod_cast (show (1 : ℝ) ≤ A by linarith [abs_nonneg t]) | |
| have hi (a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) := | |
| halasz_power_phase_intervalIntegrable t a b ha hb | |
| have hb : ∀ M : ℕ, | |
| ‖(∑ n ∈ Ioc 0 M, halaszPowerPhase t n) - | |
| ∫ x in (0 : ℝ)..(M : ℝ), halaszPowerPhase t x‖ ≤ 2 * M := by | |
| intro M | |
| calc | |
| _ ≤ ‖∑ n ∈ Ioc 0 M, halaszPowerPhase t n‖ + | |
| ‖∫ x in (0 : ℝ)..(M : ℝ), halaszPowerPhase t x‖ := norm_sub_le _ _ | |
| _ ≤ (M : ℝ) + M := add_le_add (halasz_power_phase_prefix_norm t M) | |
| (by simpa using halasz_power_phase_integral_norm t 0 M) | |
| _ = _ := by ring | |
| by_cases hNA : N ≤ A | |
| · have hNr : (N : ℝ) ≤ A := by exact_mod_cast hNA | |
| exact (hb N).trans (by linarith [abs_nonneg t]) | |
| · have hAN : A ≤ N := by omega | |
| have hs := sum_Ioc_consecutive (fun n : ℕ => halaszPowerPhase t n) | |
| (Nat.zero_le A) hAN | |
| have hj := intervalIntegral.integral_add_adjacent_intervals | |
| (hi 0 ((A : ℝ) + 1 / 2) (by norm_num) (by positivity)) | |
| (hi ((A : ℝ) + 1 / 2) ((N : ℝ) + 1 / 2) (by positivity) (by positivity)) | |
| have hk := intervalIntegral.integral_add_adjacent_intervals | |
| (hi 0 N (by norm_num) (by positivity)) | |
| (hi N ((N : ℝ) + 1 / 2) (by positivity) (by positivity)) | |
| have he : (∑ n ∈ Ioc 0 N, halaszPowerPhase t n) - | |
| ∫ x in (0 : ℝ)..(N : ℝ), halaszPowerPhase t x = | |
| ((∑ n ∈ Ioc 0 A, halaszPowerPhase t n) - | |
| ∫ x in (0 : ℝ)..((A : ℝ) + 1 / 2), halaszPowerPhase t x) + | |
| ((∑ n ∈ Ioc A N, halaszPowerPhase t n) - | |
| ∫ x in ((A : ℝ) + 1 / 2)..((N : ℝ) + 1 / 2), halaszPowerPhase t x) + | |
| ∫ x in (N : ℝ)..((N : ℝ) + 1 / 2), halaszPowerPhase t x := by | |
| rw [← hs] | |
| linear_combination hj - hk | |
| have hhead : ‖(∑ n ∈ Ioc 0 A, halaszPowerPhase t n) - | |
| ∫ x in (0 : ℝ)..((A : ℝ) + 1 / 2), halaszPowerPhase t x‖ ≤ 2 * A + 1 / 2 := by | |
| calc | |
| _ ≤ ‖∑ n ∈ Ioc 0 A, halaszPowerPhase t n‖ + | |
| ‖∫ x in (0 : ℝ)..((A : ℝ) + 1 / 2), halaszPowerPhase t x‖ := norm_sub_le _ _ | |
| _ ≤ (A : ℝ) + ((A : ℝ) + 1 / 2) := add_le_add (halasz_power_phase_prefix_norm t A) | |
| (by | |
| have hh := halasz_power_phase_integral_norm t 0 ((A : ℝ) + 1 / 2) | |
| rw [sub_zero, abs_of_nonneg (by positivity)] at hh | |
| exact hh) | |
| _ = _ := by ring | |
| have htail := halasz_phase_cell_sum_error t A N hA1 hAN (by linarith) | |
| have hend : ‖∫ x in (N : ℝ)..((N : ℝ) + 1 / 2), halaszPowerPhase t x‖ ≤ 1 / 2 := by | |
| simpa using halasz_power_phase_integral_norm t N ((N : ℝ) + 1 / 2) | |
| have hratio : (t ^ 2 + |t|) / (A : ℝ) ≤ |t| := by | |
| apply (div_le_iff₀ (by exact_mod_cast (show 0 < A by omega))).mpr | |
| have hm := mul_le_mul_of_nonneg_left hAlo (abs_nonneg t) | |
| nlinarith [sq_abs t] | |
| rw [he] | |
| exact ((norm_add_le _ _).trans (add_le_add | |
| ((norm_add_le _ _).trans (add_le_add hhead htail)) hend)).trans | |
| (by linarith [abs_nonneg t]) | |
| theorem halasz_power_phase_sum (t : ℝ) (N : ℕ) : | |
| ‖(∑ n ∈ Icc 1 N, halaszPowerPhase t n) - | |
| (N : ℂ) * halaszPowerPhase t N / (1 + (-t : ℂ) * Complex.I)‖ ≤ | |
| 8 * (1 + |t|) := by | |
| have hset : Icc 1 N = Ioc 0 N := by ext n; simp; omega | |
| have hh := halasz_power_phase_sum_integral t N | |
| rw [halasz_power_phase_integral t 0 N (by norm_num) (by positivity)] at hh | |
| simpa [hset] using hh | |
| theorem halasz_power_phase_sum_real (t x : ℝ) (hx : 0 ≤ x) : | |
| ‖(∑ n ∈ Icc 1 ⌊x⌋₊, halaszPowerPhase t n) - | |
| (x : ℂ) * halaszPowerPhase t x / (1 + (-t : ℂ) * Complex.I)‖ ≤ | |
| 9 * (1 + |t|) := by | |
| have hN : (⌊x⌋₊ : ℝ) ≤ x := Nat.floor_le hx | |
| have hN' : x - (⌊x⌋₊ : ℝ) < 1 := by linarith [Nat.lt_floor_add_one x] | |
| have hj := intervalIntegral.integral_add_adjacent_intervals | |
| (halasz_power_phase_intervalIntegrable t 0 ⌊x⌋₊ (by norm_num) (by positivity)) | |
| (halasz_power_phase_intervalIntegrable t ⌊x⌋₊ x (by positivity) hx) | |
| have he : (∑ n ∈ Icc 1 ⌊x⌋₊, halaszPowerPhase t n) - | |
| ∫ y in (0 : ℝ)..x, halaszPowerPhase t y = | |
| ((∑ n ∈ Icc 1 ⌊x⌋₊, halaszPowerPhase t n) - | |
| ∫ y in (0 : ℝ)..(⌊x⌋₊ : ℝ), halaszPowerPhase t y) - | |
| ∫ y in (⌊x⌋₊ : ℝ)..x, halaszPowerPhase t y := by | |
| rw [← hj] | |
| ring | |
| have hset : Icc 1 ⌊x⌋₊ = Ioc 0 ⌊x⌋₊ := by ext n; simp; omega | |
| have hh := halasz_power_phase_sum_integral t ⌊x⌋₊ | |
| rw [← hset] at hh | |
| have ht : ‖∫ y in (⌊x⌋₊ : ℝ)..x, halaszPowerPhase t y‖ ≤ 1 := by | |
| apply (halasz_power_phase_integral_norm t ⌊x⌋₊ x).trans | |
| rw [abs_of_nonneg (sub_nonneg.mpr hN)] | |
| exact hN'.le | |
| have h := ((norm_sub_le _ _).trans (add_le_add hh ht)).trans | |
| (show 8 * (1 + |t|) + 1 ≤ 9 * (1 + |t|) by linarith [abs_nonneg t]) | |
| rw [← he, halasz_power_phase_integral t 0 x (by norm_num) hx] at h | |
| simpa using h | |
| end TwoPointCorrelations | |
| end OAI | |