| import OAI.NumberTheory.TwoPoint.Fourier.MinorArcBilinearSieve |
| import Mathlib.NumberTheory.DiophantineApproximation.Basic |
|
|
| namespace OAI |
|
|
| /-! A reduced rational approximation for the major/minor arc split. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| lemma minor_arc_rational_approximation (α : ℝ) (Q : ℕ) (hQ : 0 < Q) : |
| ∃ (r : ℤ) (q : ℕ), 0 < q ∧ q ≤ Q ∧ IsCoprime (q : ℤ) r ∧ |
| |α - (r : ℝ) / q| ≤ 1 / (((Q : ℝ) + 1) * q) ∧ |
| |α - (r : ℝ) / q| ≤ 1 / (q : ℝ) ^ 2 := by |
| obtain ⟨a, ha, hden⟩ := Real.exists_rat_abs_sub_le_and_den_le α hQ |
| have hq0 : (0 : ℝ) < a.den := by exact_mod_cast a.pos |
| have happrox : |α - (a.num : ℝ) / a.den| ≤ 1 / (((Q : ℝ) + 1) * a.den) := by |
| simpa only [Rat.cast_def] using ha |
| refine ⟨a.num, a.den, a.pos, hden, ?_, happrox, ?_⟩ |
| · apply Int.isCoprime_iff_nat_coprime.mpr |
| simpa only [Int.natAbs_natCast] using a.reduced.symm |
| · apply happrox.trans |
| apply one_div_le_one_div_of_le (sq_pos_of_pos hq0) |
| have hden' : (a.den : ℝ) ≤ Q := by exact_mod_cast hden |
| nlinarith |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
