| import OAI.NumberTheory.TwoPoint.Fourier.MajorArcDivisors |
| import OAI.NumberTheory.TwoPoint.Fourier.MajorArcDivisorBounds |
|
|
| namespace OAI |
|
|
| /-! Rational short sums with the square-root character loss. The |
| reciprocal gcd volume is retained before summing the strata. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset |
| open scoped Classical |
|
|
| noncomputable def majorArcDivisorWindow {ι : Type*} (J : Finset ι) |
| (P : ι → Finset ℕ) (F : ℕ → ℂ) (q : ℕ) (hq : 0 < q) (r : ℤ) |
| (d : {d : ℕ // d ∈ q.divisors}) (v H : ℕ) : ℂ := by |
| letI : NeZero (q/d.val) := ⟨(major_arc_divisor_quotient_pos hq d).ne'⟩ |
| exact F d.val * ∑ χ : DirichletCharacter ℂ (q/d.val), |
| mrtAdditiveCharacterCoefficient (q/d.val) r χ * |
| shortExponentialSum (dilationSequence d.val (mrtTypicalCoefficient J P (twistByCharacter F χ))) H 0 v |
|
|
| lemma major_arc_divisor_window {ι : Type*} (J : Finset ι) |
| (P : ι → Finset ℕ) (F : ℕ → ℂ) (q : ℕ) (hq : 0 < q) (r : ℤ) |
| (d : {d : ℕ // d ∈ q.divisors}) (v H : ℕ) : |
| (∑ n ∈ Icc (v+1) (v+H), majorArcDivisorTerm J P F q hq r d n) = |
| majorArcDivisorWindow J P F q hq r d v H := by |
| let : NeZero (q/d.val) := ⟨(major_arc_divisor_quotient_pos hq d).ne'⟩ |
| unfold majorArcDivisorTerm majorArcDivisorWindow |
| rw [←mul_sum,sum_comm] |
| congr 1 |
| apply sum_congr rfl |
| intro χ _ |
| rw [shortExponentialSum_at_nat] |
| simp [additiveCharacter,mul_sum] |
|
|
| lemma major_arc_divisor_window_average {ι : Type*} (J : Finset ι) |
| (P : ι → Finset ℕ) (F : ℕ → ℂ) (hF : OneBounded F) |
| (q : ℕ) (hq : 0 < q) (r : ℤ) (d : {d : ℕ // d ∈ q.divisors}) (X H : ℕ) |
| (A : ℝ) (hA : 0 ≤ A) |
| (hshort : ∀ χ : DirichletCharacter ℂ (q/d.val), |
| (d.val:ℝ)*shortExponentialIntegral (mrtTypicalCoefficient J P (twistByCharacter F χ)) |
| (X/d.val+1) (H/d.val+1) 0 ≤ A/(d.val:ℝ)) : |
| (∑ v ∈ range X, ‖majorArcDivisorWindow J P F q hq r d v H‖) ≤ |
| Real.sqrt ((q/d.val:ℕ):ℝ)*(A/(d.val:ℝ)+X) := by |
| let : NeZero (q/d.val) := ⟨(major_arc_divisor_quotient_pos hq d).ne'⟩ |
| have hd : 0 < d.val := Nat.pos_of_dvd_of_pos (Nat.mem_divisors.mp d.property).1 hq |
| have hb : ∀ v : ℕ, ‖majorArcDivisorWindow J P F q hq r d v H‖ ≤ |
| ∑ χ : DirichletCharacter ℂ (q/d.val), ‖mrtAdditiveCharacterCoefficient (q/d.val) r χ‖* |
| ‖shortExponentialSum (dilationSequence d.val |
| (mrtTypicalCoefficient J P (twistByCharacter F χ))) H 0 v‖ := by |
| intro v |
| unfold majorArcDivisorWindow |
| rw [norm_mul] |
| apply (mul_le_of_le_one_left (norm_nonneg _) (hF d.val hd)).trans |
| exact (norm_sum_le _ _).trans_eq (sum_congr rfl (fun χ _ => norm_mul _ _)) |
| calc |
| _ ≤ ∑ v ∈ range X, ∑ χ : DirichletCharacter ℂ (q/d.val), |
| ‖mrtAdditiveCharacterCoefficient (q/d.val) r χ‖* |
| ‖shortExponentialSum (dilationSequence d.val |
| (mrtTypicalCoefficient J P (twistByCharacter F χ))) H 0 v‖ := sum_le_sum (fun v _ => hb v) |
| _ = ∑ χ : DirichletCharacter ℂ (q/d.val), ‖mrtAdditiveCharacterCoefficient (q/d.val) r χ‖* |
| (∑ v ∈ range X, ‖shortExponentialSum (dilationSequence d.val |
| (mrtTypicalCoefficient J P (twistByCharacter F χ))) H 0 v‖) := by |
| rw [sum_comm] |
| simp only [mul_sum] |
| _ ≤ ∑ χ : DirichletCharacter ℂ (q/d.val), |
| ‖mrtAdditiveCharacterCoefficient (q/d.val) r χ‖*(A/(d.val:ℝ)+X) := by |
| apply sum_le_sum |
| intro χ _ |
| apply mul_le_mul_of_nonneg_left _ (norm_nonneg _) |
| exact (major_arc_dilation_average _ |
| (mrtTypicalCoefficient_oneBounded J P _ (hF.twistByCharacter χ)) d.val X H hd).trans |
| (add_le_add (hshort χ) le_rfl) |
| _ = (∑ χ : DirichletCharacter ℂ (q/d.val), |
| ‖mrtAdditiveCharacterCoefficient (q/d.val) r χ‖)*(A/(d.val:ℝ)+X) := (sum_mul _ _ _).symm |
| _ ≤ _ := mul_le_mul_of_nonneg_right (mrt_additive_character_l1 (q/d.val) r) (by positivity) |
|
|
| theorem major_arc_typical_rational_sharp {ι : Type*} (J : Finset ι) |
| (P : ι → Finset ℕ) (hP : ∀ j ∈ J, ∀ p ∈ P j, p.Prime) |
| (F : ℕ → ℂ) (hFm : ∀ a b, 0 < a → 0 < b → F (a*b)=F a*F b) (hF : OneBounded F) |
| (q : ℕ) (hq : 0 < q) (r : ℤ) (X H : ℕ) (A : ℝ) (hA : 0 ≤ A) |
| (havoid : ∀ d ∈ q.divisors, mrtPrimeAvoids (J.biUnion P) d) |
| (hshort : ∀ d : {d : ℕ // d ∈ q.divisors}, ∀ χ : DirichletCharacter ℂ (q/d.val), |
| (d.val:ℝ)*shortExponentialIntegral (mrtTypicalCoefficient J P (twistByCharacter F χ)) |
| (X/d.val+1) (H/d.val+1) 0 ≤ A/(d.val:ℝ)) : |
| shortExponentialIntegral (mrtTypicalCoefficient J P F) X H ((r:ℝ)/q) ≤ |
| Real.sqrt (q:ℝ)*((1+Real.log q)*A+(q:ℝ)*X) := by |
| let : NeZero q := ⟨hq.ne'⟩ |
| have hw (v : ℕ) : shortExponentialSum (mrtTypicalCoefficient J P F) H ((r:ℝ)/q) v = |
| ∑ d : {d : ℕ // d ∈ q.divisors}, majorArcDivisorWindow J P F q hq r d v H := by |
| rw [shortExponentialSum_at_nat] |
| have he (n : ℕ) (hn : n ∈ Icc (v+1) (v+H)) := |
| major_arc_divisor_expansion J P hP F hFm q hq r havoid n |
| (show 0 < n by have := (mem_Icc.mp hn).1; omega) |
| calc |
| _ = ∑ n∈Icc (v+1) (v+H), ∑ d : {d : ℕ // d∈q.divisors}, |
| majorArcDivisorTerm J P F q hq r d n := sum_congr rfl he |
| _ = ∑ d : {d : ℕ // d∈q.divisors}, ∑ n∈Icc (v+1) (v+H), |
| majorArcDivisorTerm J P F q hq r d n := sum_comm |
| _ = _ := sum_congr rfl (fun d _ => major_arc_divisor_window J P F q hq r d v H) |
| rw [shortExponentialIntegral_eq_sum] |
| calc |
| _ ≤ ∑ v ∈ range X, ∑ d : {d : ℕ // d ∈ q.divisors}, ‖majorArcDivisorWindow J P F q hq r d v H‖ := by |
| apply sum_le_sum |
| intro v _ |
| rw [hw] |
| exact norm_sum_le _ _ |
| _ = ∑ d : {d : ℕ // d ∈ q.divisors}, ∑ v ∈ range X, ‖majorArcDivisorWindow J P F q hq r d v H‖ := sum_comm |
| _ ≤ ∑ d : {d : ℕ // d ∈ q.divisors}, Real.sqrt ((q/d.val:ℕ):ℝ)*(A/(d.val:ℝ)+X) := |
| sum_le_sum (fun d _ => major_arc_divisor_window_average J P F hF q hq r d X H A hA (hshort d)) |
| _ = ∑ d ∈ q.divisors, Real.sqrt ((q/d:ℕ):ℝ)*(A/(d:ℝ)+X) := by |
| exact Finset.sum_coe_sort q.divisors (fun d : ℕ => Real.sqrt ((q/d:ℕ):ℝ)*(A/(d:ℝ)+(X:ℝ))) |
| _ ≤ _ := major_arc_divisor_weighted q hq A X hA (Nat.cast_nonneg X) |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
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|
