| import OAI.NumberTheory.Ostmann.Tree.CrossedMoments | |
| import OAI.NumberTheory.Ostmann.Tree.SpectralParts | |
| namespace OAI | |
| namespace Ostmann.FiniteField | |
| noncomputable section | |
| open scoped BigOperators ComplexConjugate | |
| variable {p : ℕ} [Fact p.Prime] | |
| theorem bad_nonprincipal_convolution_bound (g h : ZMod p → ℂ) (σ τ : (ZMod p)ˣ) | |
| (ν : MulChar (ZMod p) ℂ) (hg0 : g 0=0) (hg : l2Sq g≤1) | |
| (hh0 : h 0=0) (hh : l2Sq h≤1) : | |
| (∑ χ : MulChar (ZMod p) ℂ,∑ ψ : MulChar (ZMod p) ℂ,∑ a : ZMod p, | |
| nonprincipalEnergy g σ χ (ν*ψ⁻¹) a* | |
| nonprincipalEnergy h τ ψ (ν⁻¹*χ⁻¹) (-a)) ≤ | |
| ((p:ℝ)/(Fintype.card (ZMod p)ˣ:ℝ))^3* | |
| (2*(correlationBound g:ℝ)^2+2*(correlationBound h:ℝ)^2+3/(p:ℝ)) := by | |
| have hc := crossed_moment_bound (nonprincipalEnergy g σ) (nonprincipalEnergy h τ) | |
| ν (Equiv.neg (ZMod p)) | |
| change 2*(∑ χ : MulChar (ZMod p) ℂ,∑ ψ : MulChar (ZMod p) ℂ,∑ a : ZMod p, | |
| nonprincipalEnergy g σ χ (ν*ψ⁻¹) a*nonprincipalEnergy h τ ψ (ν⁻¹*χ⁻¹) (-a)) ≤ _ at hc | |
| have hL := nonprincipalEnergy_fourth g σ hg0 hg | |
| have hR := nonprincipalEnergy_fourth h τ hh0 hh | |
| nlinarith only [hc,hL,hR] | |
| end | |
| end Ostmann.FiniteField | |
| end OAI | |