| import OAI.NumberTheory.Ostmann.Tree.Mellin |
| import OAI.NumberTheory.Ostmann.Tree.QuarticWeighted |
|
|
| namespace OAI |
|
|
| namespace Ostmann.FiniteField |
| noncomputable section |
| open scoped BigOperators ComplexConjugate |
| variable {F : Type*} [Field F] [Fintype F] [DecidableEq F] |
| local instance characterFourthFintype : Fintype (MulChar F ℂ) := Fintype.ofFinite _ |
|
|
| theorem sum_mulChar_field_kernel (x y : F) : |
| (∑ χ : MulChar F ℂ, χ x*conj (χ y)) = |
| if x=y ∧ x≠0 then (Fintype.card Fˣ:ℂ) else 0 := by |
| classical |
| by_cases hx : x=0 |
| · simp [hx,MulChar.map_zero] |
| by_cases hy : y=0 |
| · simp [hy,MulChar.map_zero] |
| let u : Fˣ := Units.mk0 y hy |
| let v : Fˣ := Units.mk0 x hx |
| calc |
| _ = ∑ χ : MulChar F ℂ, conj (χ u)*χ v := by |
| apply Finset.sum_congr rfl |
| intro χ _ |
| exact mul_comm _ _ |
| _ = if u=v then (Fintype.card Fˣ:ℂ) else 0 := sum_mulChar_kernel u v |
| _ = _ := by |
| by_cases he : x=y |
| · have huv : u=v := Units.ext he.symm |
| simp [he,huv,hy] |
| · have huv : u≠v := fun h => he (congrArg (fun t : Fˣ => (t:F)) h).symm |
| simp [he,huv] |
|
|
| omit [Field F] [DecidableEq F] in |
| theorem norm_sum_fourth_expansion (z : F → ℂ) : |
| (‖∑ b, z b‖^4:ℂ) = ∑ b : Quartet F, z b.1.1*z b.1.2*conj (z b.2.1)*conj (z b.2.2) := by |
| calc |
| _ = ((‖∑ b,z b‖:ℂ)^2)^2 := by ring |
| _ = ((∑ b,z b)*conj (∑ b,z b))^2 := by rw [Complex.mul_conj'] |
| _ = _ := by |
| simp only [Fintype.sum_prod_type,map_sum,← Finset.mul_sum,← Finset.sum_mul] |
| ring |
|
|
| def weightedCharacterSum (w : F → ℝ) (χ : MulChar F ℂ) (a : F) : ℂ := |
| ∑ b, (w b:ℂ)*χ (a-b) |
| |
| theorem weightedCharacterSum_fourth_identity (w : F → ℝ) (a : F) : |
| (∑ χ : MulChar F ℂ, ‖weightedCharacterSum w χ a‖^4) = |
| (Fintype.card Fˣ:ℝ) * ∑ b : Quartet F, |
| if quartetCollision a b ∧ (a-b.1.1)*(a-b.1.2)≠0 then quartetWeight w b else 0 := by |
| classical |
| apply Complex.ofReal_injective |
| push_cast |
| have he (χ : MulChar F ℂ) : |
| (‖weightedCharacterSum w χ a‖:ℂ)^4 = ∑ b : Quartet F, |
| (quartetWeight w b:ℂ) * (χ ((a-b.1.1)*(a-b.1.2))*conj (χ ((a-b.2.1)*(a-b.2.2)))) := by |
| rw [weightedCharacterSum,norm_sum_fourth_expansion] |
| apply Finset.sum_congr rfl |
| intro b _ |
| simp only [quartetWeight,map_mul,Complex.conj_ofReal] |
| push_cast |
| ring |
| simp_rw [he] |
| rw [Finset.sum_comm,Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro b _ |
| rw [← Finset.mul_sum,sum_mulChar_field_kernel] |
| split_ifs < |
| ring |
|
|
| theorem weightedCharacterSum_fourth_bound (w : F → ℝ) (hw : ∀ x,0≤w x) : |
| (∑ χ : MulChar F ℂ, ∑ a : F, ‖weightedCharacterSum w χ a‖^4) ≤ |
| (Fintype.card Fˣ:ℝ) * |
| (2*(Fintype.card F:ℝ)*(∑ x,w x^2)^2+(∑ x,w x)^4) := by |
| classical |
| rw [Finset.sum_comm] |
| simp_rw [weightedCharacterSum_fourth_identity] |
| rw [← Finset.mul_sum] |
| apply mul_le_mul_of_nonneg_left _ (by positivity) |
| apply le_trans _ (quartetCollision_weighted_bound w hw) |
| apply Finset.sum_le_sum |
| intro a _ |
| apply Finset.sum_le_sum |
| intro b _ |
| have hW := quartetWeight_nonneg w hw b |
| split_ifs < |
|
|
| end |
| end Ostmann.FiniteField |
|
|
| end OAI |
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|
