| import OAI.NumberTheory.TwoPointCorrelations.HalaszHyperbola |
| import Mathlib.Analysis.SpecialFunctions.Stirling |
|
|
| namespace OAI |
|
|
| /-! Recover an ordinary mean from its logarithmically weighted sum. |
| The absolute error is at most N for a one-bounded sequence. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset |
|
|
| lemma halasz_sum_log_factorial (N : ℕ) : |
| (∑ n ∈ Icc 1 N, Real.log (n : ℝ)) = Real.log (N.factorial : ℝ) := by |
| induction N with |
| | zero => simp |
| | succ N ih => |
| rw [sum_Icc_succ_top (by omega : 1 ≤ N + 1), ih, Nat.factorial_succ, Nat.cast_mul, |
| Real.log_mul (by positivity) (by positivity)] |
| ring |
|
|
| lemma halasz_sum_log_gap (N : ℕ) (hN : 1 ≤ N) : |
| (∑ n ∈ Icc 1 N, (Real.log (N : ℝ) - Real.log (n : ℝ))) ≤ N := by |
| rw [sum_sub_distrib, sum_const, nsmul_eq_mul, Nat.card_Icc, halasz_sum_log_factorial] |
| have hst := Stirling.le_log_factorial_stirling (Nat.ne_zero_of_lt hN) |
| have hlogN : 0 ≤ Real.log (N : ℝ) := Real.log_nonneg (by exact_mod_cast hN) |
| have hlogpi : 0 ≤ Real.log (2 * Real.pi) := Real.log_nonneg (by linarith [Real.two_le_pi]) |
| simp only [Nat.add_sub_cancel] at * |
| linarith |
|
|
| lemma halasz_logarithmic_sum_error (f : ℕ → ℂ) (hf : OneBounded f) |
| (N : ℕ) (hN : 1 ≤ N) : |
| ‖(∑ n ∈ Icc 1 N, f n) * (Real.log (N : ℝ) : ℂ) - |
| ∑ n ∈ Icc 1 N, f n * (Real.log (n : ℝ) : ℂ)‖ ≤ (N : ℝ) := by |
| let S := ∑ n ∈ Icc 1 N, f n |
| let W := ∑ n ∈ Icc 1 N, f n * (Real.log (n : ℝ) : ℂ) |
| have he : S * (Real.log (N : ℝ) : ℂ) - W = |
| ∑ n ∈ Icc 1 N, f n * ((Real.log (N : ℝ) - Real.log (n : ℝ) : ℝ) : ℂ) := by |
| dsimp [S, W] |
| rw [sum_mul, ← sum_sub_distrib] |
| apply sum_congr rfl |
| intro n _ |
| push_cast |
| ring |
| rw [he] |
| apply (norm_sum_le _ _).trans |
| apply (sum_le_sum ?_).trans (halasz_sum_log_gap N hN) |
| intro n hn |
| have hn' := mem_Icc.mp hn |
| have hlog : 0 ≤ Real.log (N : ℝ) - Real.log (n : ℝ) := |
| sub_nonneg.mpr (Real.log_le_log (by exact_mod_cast hn'.1) (by exact_mod_cast hn'.2)) |
| rw [norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg hlog] |
| exact mul_le_of_le_one_left hlog (hf n hn'.1) |
|
|
| theorem halasz_logarithmic_mean_bound (f : ℕ → ℂ) (hf : OneBounded f) |
| (N : ℕ) (hN : 1 ≤ N) : |
| Real.log (N : ℝ) * ‖∑ n ∈ Icc 1 N, f n‖ ≤ |
| ‖∑ n ∈ Icc 1 N, f n * (Real.log (n : ℝ) : ℂ)‖ + N := by |
| let S := ∑ n ∈ Icc 1 N, f n |
| let W := ∑ n ∈ Icc 1 N, f n * (Real.log (n : ℝ) : ℂ) |
| have hgap := halasz_logarithmic_sum_error f hf N hN |
| have hh := norm_add_le W (S * (Real.log (N : ℝ) : ℂ) - W) |
| have hlogN : 0 ≤ Real.log (N : ℝ) := Real.log_nonneg (by exact_mod_cast hN) |
| have hnorm : ‖S * (Real.log (N : ℝ) : ℂ)‖ = Real.log (N : ℝ) * ‖S‖ := by |
| rw [norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg hlogN, mul_comm] |
| rw [add_sub_cancel, hnorm] at hh |
| exact hh.trans (add_le_add le_rfl hgap) |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
