| import OAI.NumberTheory.TwoPoint.Fourier.MinorArcTypicalSaving | |
| namespace OAI | |
| /-! The dyadic index count contributes only a logarithm of a logarithm. -/ | |
| namespace TwoPointCorrelations | |
| lemma minor_arc_log_index_bound (H : ℕ) (hH : 1 ≤ H) : | |
| (Nat.log 2 H : ℝ) ≤ 2 * Real.log (H : ℝ) := by | |
| have hH0 : H ≠ 0 := by omega | |
| have hp : (2 : ℝ) ^ Nat.log 2 H ≤ H := by | |
| exact_mod_cast Nat.pow_log_le_self 2 hH0 | |
| have hl := Real.log_le_log (by positivity : (0 : ℝ) < 2 ^ Nat.log 2 H) hp | |
| rw [Real.log_pow] at hl | |
| have hhalf : (1 / 2 : ℝ) ≤ Real.log 2 := by | |
| have hh := Real.one_sub_inv_le_log_of_pos (by norm_num : (0 : ℝ) < 2) | |
| norm_num at hh | |
| exact hh | |
| have hm := mul_le_mul_of_nonneg_left hhalf (Nat.cast_nonneg (Nat.log 2 H) : | |
| (0 : ℝ) ≤ Nat.log 2 H) | |
| nlinarith | |
| lemma minor_arc_dyadic_log_log_bound (H : ℕ) (hH : 2 ≤ H) | |
| (hlogH : 1 ≤ Real.log (H : ℝ)) (hloglogH : 1 ≤ Real.log (Real.log (H : ℝ))) : | |
| 1 + Real.log (Nat.log 2 H : ℝ) ≤ 3 * Real.log (Real.log (H : ℝ)) := by | |
| have hindex : 0 < Nat.log 2 H := lt_of_lt_of_le (by norm_num) | |
| (minor_arc_dyadic_index_pos hH) | |
| have hindex0 : (0 : ℝ) < Nat.log 2 H := by exact_mod_cast hindex | |
| have hlog0 : 0 < Real.log (H : ℝ) := by linarith | |
| have hh := Real.log_le_log hindex0 (minor_arc_log_index_bound H (by omega)) | |
| rw [Real.log_mul (by norm_num : (2 : ℝ) ≠ 0) hlog0.ne'] at hh | |
| have htwo := Real.log_le_sub_one_of_pos (by norm_num : (0 : ℝ) < 2) | |
| linarith | |
| end TwoPointCorrelations | |
| end OAI | |