| import OAI.NumberTheory.Ostmann.Tree.Fourier |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators ComplexConjugate |
| namespace Ostmann.Characters |
| variable {p : ℕ} [NeZero p] |
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| def translatedMean (S : Finset (ZMod p)) (χ : MulChar (ZMod p) ℂ) |
| (t : ZMod p) : ℂ := (S.card : ℂ)⁻¹ * ∑ x ∈ S, χ (x-t) |
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| def translatedBias (S : Finset (ZMod p)) (χ : MulChar (ZMod p) ℂ) |
| (t : ZMod p) : ℝ := ‖translatedMean S χ t‖ |
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| def maxTranslatedBias (S : Finset (ZMod p)) (χ : MulChar (ZMod p) ℂ) : NNReal := |
| Finset.univ.sup (fun t => ‖translatedMean S χ t‖₊) |
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| 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) |
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|
| 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) |
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| 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) |
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| 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 |
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| end |
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| end OAI |
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