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2b3d0c0 | 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 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 | import OAI.NumberTheory.Ostmann.Tree.Fourier
namespace OAI
noncomputable section
open scoped BigOperators ComplexConjugate
namespace Ostmann.Characters
variable {p : ℕ} [NeZero p]
def translatedMean (S : Finset (ZMod p)) (χ : MulChar (ZMod p) ℂ)
(t : ZMod p) : ℂ := (S.card : ℂ)⁻¹ * ∑ x ∈ S, χ (x-t)
def translatedBias (S : Finset (ZMod p)) (χ : MulChar (ZMod p) ℂ)
(t : ZMod p) : ℝ := ‖translatedMean S χ t‖
def maxTranslatedBias (S : Finset (ZMod p)) (χ : MulChar (ZMod p) ℂ) : NNReal :=
Finset.univ.sup (fun t => ‖translatedMean S χ t‖₊)
theorem translatedBias_le_max (S : Finset (ZMod p))
(χ : MulChar (ZMod p) ℂ) (t : ZMod p) :
translatedBias S χ t ≤ (maxTranslatedBias S χ : ℝ) := by
exact_mod_cast Finset.le_sup (f := fun t => ‖translatedMean S χ t‖₊)
(Finset.mem_univ t)
omit [NeZero p] in
theorem square_ne_one (χ : MulChar (ZMod p) ℂ) (hχ : 2 < orderOf χ) : χ^2 ≠ 1 := by
intro h
exact (not_le_of_gt hχ) (orderOf_le_of_pow_eq_one (by decide) h)
theorem square_sum_zero (χ : MulChar (ZMod p) ℂ) (hχ : 2 < orderOf χ) :
∑ x : ZMod p, (χ^2) x = 0 :=
MulChar.sum_eq_zero_of_ne_one (square_ne_one χ hχ)
theorem sum_affine_eq_zero (χ : MulChar (ZMod p) ℂ) (hχ : χ ≠ 1)
(u : (ZMod p)ˣ) (a : ZMod p) : ∑ x : ZMod p, χ (↑u*x+a) = 0 := by
calc
_ = ∑ x : ZMod p, χ x :=
Fintype.sum_bijective _ ((Equiv.addRight a).bijective.comp u.mulLeft_bijective) _ _ (fun _ => rfl)
_ = 0 := MulChar.sum_eq_zero_of_ne_one hχ
end Ostmann.Characters
end
end OAI
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