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8769036 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 | 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
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