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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
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