| import OAI.NumberTheory.TwoPointCorrelations.HalaszPhaseTaylor |
| import Mathlib.Analysis.SpecialFunctions.Integrals.Basic |
| import Mathlib.Analysis.PSeries |
|
|
| namespace OAI |
|
|
| /-! Unit cells for the elementary power-sum approximation. Centering the |
| cells cancels the linear phase error, leaving a summable quadratic error. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open MeasureTheory Finset |
|
|
| noncomputable def halaszPowerPhase (t x : ℝ) : ℂ := |
| Complex.exp (((-t * Real.log x : ℝ) : ℂ) * Complex.I) |
|
|
| @[simp] lemma halasz_power_phase_norm (t x : ℝ) : ‖halaszPowerPhase t x‖ = 1 := by |
| simp [halaszPowerPhase, Complex.norm_exp, Complex.mul_re] |
|
|
| lemma halasz_power_phase_add (t x y : ℝ) (hx : 0 < x) (hxy : 0 < x + y) : |
| halaszPowerPhase t (x + y) = halaszPowerPhase t x * |
| Complex.exp (((-t * Real.log (1 + y / x) : ℝ) : ℂ) * Complex.I) := by |
| have he : x + y = x * (1 + y / x) := by field_simp |
| have hp : 0 < 1 + y / x := by rw [← mul_pos_iff_of_pos_left hx, ← he]; exact hxy |
| rw [halaszPowerPhase, he, Real.log_mul hx.ne' hp.ne', mul_add, |
| Complex.ofReal_add, add_mul, Complex.exp_add] |
| rfl |
|
|
| lemma halasz_power_phase_cell_error (t x y : ℝ) (hx : 1 ≤ x) |
| (ht : |t| ≤ x) (hy : |y| ≤ 1 / 2) : |
| ‖halaszPowerPhase t (x + y) - |
| halaszPowerPhase t x * (1 + (((-t * (y / x) : ℝ) : ℂ) * Complex.I))‖ ≤ |
| (t ^ 2 + |t|) / x ^ 2 := by |
| have hx0 : 0 < x := by linarith |
| have hxy : 0 < x + y := by linarith [(abs_le.mp hy).1] |
| have hratio : |y / x| ≤ 1 / 2 := by |
| rw [abs_div, abs_of_pos hx0] |
| apply (div_le_iff₀ hx0).mpr |
| linarith |
| have htratio : |t * (y / x)| ≤ 1 / 2 := by |
| rw [abs_mul, abs_div, abs_of_pos hx0] |
| calc |
| _ ≤ x * ((1 / 2) / x) := mul_le_mul ht |
| (div_le_div_of_nonneg_right hy hx0.le) (by positivity) hx0.le |
| _ = 1 / 2 := by field_simp |
| have hh := halasz_phase_linear_error t (y / x) hratio htratio |
| have he : halaszPowerPhase t (x + y) - |
| halaszPowerPhase t x * (1 + (((-t * (y / x) : ℝ) : ℂ) * Complex.I)) = |
| halaszPowerPhase t x * |
| (Complex.exp (((-t * Real.log (1 + y / x) : ℝ) : ℂ) * Complex.I) - 1 - |
| (((-t * (y / x) : ℝ) : ℂ) * Complex.I)) := by |
| rw [halasz_power_phase_add t x y hx0 hxy] |
| ring |
| rw [he, norm_mul, halasz_power_phase_norm, one_mul] |
| apply hh.trans |
| have hy2 : y ^ 2 ≤ 1 / 4 := by |
| have hh := pow_le_pow_left₀ (abs_nonneg y) hy 2 |
| norm_num [sq_abs] at hh |
| exact hh |
| rw [div_pow] |
| calc |
| _ ≤ 4 * (t ^ 2 + |t|) * ((1 / 4) / x ^ 2) := by gcongr |
| _ = _ := by ring |
|
|
| lemma halasz_power_phase_continuousOn (t a b : ℝ) (ha : 0 < a) (hab : a ≤ b) : |
| ContinuousOn (halaszPowerPhase t) (Set.uIcc a b) := by |
| have hne : ∀ x ∈ Set.uIcc a b, x ≠ 0 := by |
| intro x hx |
| rw [Set.uIcc_of_le hab] at hx |
| exact (ha.trans_le hx.1).ne' |
| unfold halaszPowerPhase |
| fun_prop (disch := assumption) |
|
|
| lemma halasz_power_phase_cell_integral (t x : ℝ) (hx : 1 ≤ x) (ht : |t| ≤ x) : |
| ‖(∫ y in (-1 / 2 : ℝ)..(1 / 2 : ℝ), halaszPowerPhase t (x + y)) - |
| halaszPowerPhase t x‖ ≤ (t ^ 2 + |t|) / x ^ 2 := by |
| let lin : ℝ → ℂ := fun y => |
| halaszPowerPhase t x * (1 + (((-t * (y / x) : ℝ) : ℂ) * Complex.I)) |
| have hlin : Continuous lin := by dsimp [lin]; fun_prop |
| have hf : ContinuousOn (fun y => halaszPowerPhase t (x + y)) |
| (Set.uIcc (-1 / 2 : ℝ) (1 / 2 : ℝ)) := by |
| have hne : ∀ y ∈ Set.uIcc (-1 / 2 : ℝ) (1 / 2 : ℝ), x + y ≠ 0 := by |
| intro y hy |
| rw [Set.uIcc_of_le (by norm_num)] at hy |
| have hp : 0 < x + y := by linarith [hy.1] |
| exact hp.ne' |
| unfold halaszPowerPhase |
| fun_prop (disch := assumption) |
| have hli : (∫ y in (-1 / 2 : ℝ)..(1 / 2 : ℝ), lin y) = halaszPowerPhase t x := by |
| dsimp [lin] |
| rw [intervalIntegral.integral_const_mul, intervalIntegral.integral_add] |
| · rw [intervalIntegral.integral_const, intervalIntegral.integral_mul_const, |
| intervalIntegral.integral_ofReal, intervalIntegral.integral_const_mul, |
| intervalIntegral.integral_div, integral_id] |
| norm_num |
| · exact continuous_const.intervalIntegrable _ _ |
| · exact (by fun_prop : Continuous (fun y : ℝ => |
| (((-t * (y / x) : ℝ) : ℂ) * Complex.I))).intervalIntegrable _ _ |
| have hh := intervalIntegral.norm_integral_le_of_norm_le_const |
| (a := (-1 / 2 : ℝ)) (b := (1 / 2 : ℝ)) |
| (f := fun y => halaszPowerPhase t (x + y) - lin y) |
| (C := (t ^ 2 + |t|) / x ^ 2) (fun y hy => by |
| apply halasz_power_phase_cell_error t x y hx ht |
| rw [Set.uIoc_of_le (by norm_num)] at hy |
| exact abs_le.mpr ⟨by linarith [hy.1], hy.2⟩) |
| rw [intervalIntegral.integral_sub hf.intervalIntegrable (hlin.intervalIntegrable _ _), hli] at hh |
| norm_num at hh ⊢ |
| exact hh |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
