| import OAI.NumberTheory.TwoPoint.Fourier.MajorArcRational |
| import OAI.NumberTheory.TwoPoint.Fourier.MajorArcPerturbation |
|
|
| namespace OAI |
|
|
| /-! The full major-arc reduction to the short means supplied by MRT's |
| Dirichlet-polynomial argument. All residues, divided scales, shorter lengths, |
| and rounding errors remain explicit. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset |
| open scoped Classical |
|
|
| lemma major_arc_character_distance (F : ℕ → ℂ) {q : ℕ} |
| (χ : DirichletCharacter ℂ q) (t : ℝ) (N : ℕ) : |
| squaredDistance (twistByCharacter F χ) (mrtArchimedeanTwist t) N = |
| squaredDistance F (characterTwist (conjugateCharacter χ) t) N := by |
| unfold squaredDistance |
| apply sum_congr rfl |
| intro p _ |
| simp only [twistByCharacter, mrtArchimedeanTwist, characterTwist, |
| conjugateCharacter_apply, map_mul, Complex.conj_conj, mul_assoc] |
|
|
| lemma MRTDistanceLowerBound.major_arc_character {F : ℕ → ℂ} {X H : ℕ} {M : ℝ} |
| (hd : MRTDistanceLowerBound F X H M) (q : ℕ) (hq : 0 < q) |
| (hqmax : (q : ℝ) ≤ mrtModulusCutoff X H) |
| (χ : DirichletCharacter ℂ q) (t : ℝ) (ht : |t| ≤ (X : ℝ)) : |
| M ≤ squaredDistance (twistByCharacter F χ) (mrtArchimedeanTwist t) X := by |
| rw [major_arc_character_distance] |
| exact hd q hq hqmax (conjugateCharacter χ) t ht |
|
|
| theorem major_arc_reduction {ι : Type*} (J : Finset ι) |
| (P : ι → Finset ℕ) (hP : ∀ j ∈ J, ∀ p ∈ P j, p.Prime) |
| (F : ℕ → ℂ) |
| (hF : ∀ a b, 0 < a → 0 < b → F (a * b) = F a * F b) |
| (hFb : OneBounded F) (q X H : ℕ) [NeZero q] (hH : 0 < H) |
| (r : ℤ) (β W K : ℝ) (hK : 0 ≤ K) |
| (hβ : |β| ≤ W / ((H : ℝ) * q)) |
| (havoid : ∀ b : ZMod q, mrtPrimeAvoids (J.biUnion P) (b.val.gcd q)) |
| (hshort : ∀ (h : ℕ), 1 ≤ h → h ≤ H → |
| ∀ (b : ZMod q) (χ : DirichletCharacter ℂ (q / b.val.gcd q)), |
| (b.val.gcd q : ℝ) * shortExponentialIntegral |
| (mrtTypicalCoefficient J P (twistByCharacter F χ)) |
| (X / b.val.gcd q + 1) (h / b.val.gcd q + 1) 0 + X ≤ K) : |
| shortExponentialIntegral (mrtTypicalCoefficient J P F) X H |
| ((r : ℝ) / q + β) ≤ ((q : ℝ) + 2 * Real.pi * W) * K := by |
| have hprefix (h : ℕ) (hh : 1 ≤ h) (hhH : h ≤ H) : |
| shortExponentialIntegral (mrtTypicalCoefficient J P F) X h ((r : ℝ) / q) ≤ |
| (q : ℝ) * K := |
| major_arc_typical_rational_bound J P hP F hF hFb q X h r K havoid (hshort h hh hhH) |
| have hHr : (0 : ℝ) < H := by exact_mod_cast hH |
| have hqr : (0 : ℝ) < q := by exact_mod_cast NeZero.pos q |
| have hb : |β| * ((H : ℝ) * q) ≤ W := |
| (le_div_iff₀ (mul_pos hHr hqr)).mp hβ |
| calc |
| _ ≤ (1 + 2 * Real.pi * |β| * H) * ((q : ℝ) * K) := |
| major_arc_integral_perturbation_uniform _ X H hH _ β _ |
| (mul_nonneg hqr.le hK) hprefix |
| _ = (q : ℝ) * K + (2 * Real.pi * K) * (|β| * ((H : ℝ) * q)) := by ring |
| _ ≤ (q : ℝ) * K + (2 * Real.pi * K) * W := |
| add_le_add le_rfl (mul_le_mul_of_nonneg_left hb (by positivity)) |
| _ = _ := by ring |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
