| import OAI.NumberTheory.Ostmann.Tree.DoubleMellinEnergy |
| import OAI.NumberTheory.Ostmann.Tree.SpectralParts |
|
|
| namespace OAI |
|
|
| namespace Ostmann.FiniteField |
| noncomputable section |
| open scoped BigOperators ComplexConjugate |
| variable {p : ℕ} [Fact p.Prime] |
|
|
| theorem principalEnergy_mulInv (g : ZMod p → ℂ) (σ : (ZMod p)ˣ) |
| (χ ν ψ : MulChar (ZMod p) ℂ) (a : ZMod p) : |
| principalEnergy g σ χ (ν*ψ⁻¹) a= |
| if ψ=ν then ‖twistedPairFourier g σ χ 1 a‖^2 else 0 := by |
| classical |
| by_cases he : ψ=ν |
| · subst ψ |
| simp [principalEnergy] |
| · have hα : (ν*ψ⁻¹ : MulChar (ZMod p) ℂ)≠1 := by |
| intro hmul |
| exact he (mul_inv_eq_one.mp hmul).symm |
| simp [principalEnergy,he,hα] |
|
|
| theorem principalEnergy_total (g : ZMod p → ℂ) (σ : (ZMod p)ˣ) |
| (ν : MulChar (ZMod p) ℂ) (e : ZMod p ≃ ZMod p) (hg0 : g 0=0) (hg : l2Sq g≤1) : |
| (∑ χ : MulChar (ZMod p) ℂ,∑ ψ : MulChar (ZMod p) ℂ,∑ a : ZMod p, |
| principalEnergy g σ ψ (ν*χ⁻¹) (e a)) ≤ 2*(p:ℝ)/(Fintype.card (ZMod p)ˣ:ℝ) := by |
| classical |
| rw [Finset.sum_comm] |
| have he (ψ : MulChar (ZMod p) ℂ) : |
| (∑ χ : MulChar (ZMod p) ℂ,∑ a : ZMod p,principalEnergy g σ ψ (ν*χ⁻¹) (e a)) = |
| ∑ a : ZMod p,‖twistedPairFourier g σ ψ 1 a‖^2 := by |
| rw [Finset.sum_comm] |
| simp_rw [principalEnergy_mulInv] |
| simp only [Finset.sum_ite_eq',Finset.mem_univ,ite_true] |
| exact e.sum_comp (fun a => ‖twistedPairFourier g σ ψ 1 a‖^2) |
| simp_rw [he] |
| exact (Finset.sum_le_sum (fun ψ _ => twistedPairFourier_energy g σ ψ 1)).trans |
| (pairFourthMass_total g hg0 hg) |
|
|
| theorem bad_mixed_left_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*principalEnergy h τ ψ (ν⁻¹*χ⁻¹) (-a)) ≤ |
| 2*((p:ℝ)/(Fintype.card (ZMod p)ˣ:ℝ))^3/(p:ℝ) := by |
| let B : ℝ := (p:ℝ)⁻¹*((p:ℝ)/(Fintype.card (ZMod p)ˣ:ℝ))^2 |
| calc |
| _ ≤ ∑ χ : MulChar (ZMod p) ℂ,∑ ψ : MulChar (ZMod p) ℂ,∑ a : ZMod p, |
| B*principalEnergy h τ ψ (ν⁻¹*χ⁻¹) (-a) := by |
| apply Finset.sum_le_sum |
| intro χ _ |
| apply Finset.sum_le_sum |
| intro ψ _ |
| apply Finset.sum_le_sum |
| intro a _ |
| exact mul_le_mul_of_nonneg_right (nonprincipalEnergy_le g σ χ _ a hg0 hg) |
| (principalEnergy_nonneg h τ ψ _ (-a)) |
| _ = B*(∑ χ : MulChar (ZMod p) ℂ,∑ ψ : MulChar (ZMod p) ℂ,∑ a : ZMod p, |
| principalEnergy h τ ψ (ν⁻¹*χ⁻¹) (-a)) := by simp only [Finset.mul_sum] |
| _ ≤ B*(2*(p:ℝ)/(Fintype.card (ZMod p)ˣ:ℝ)) := |
| mul_le_mul_of_nonneg_left (principalEnergy_total h τ ν⁻¹ (Equiv.neg _) hh0 hh) (by dsimp [B]; positivity) |
| _ = _ := by dsimp [B]; ring |
|
|
| theorem bad_mixed_right_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, |
| principalEnergy g σ χ (ν*ψ⁻¹) a*nonprincipalEnergy h τ ψ (ν⁻¹*χ⁻¹) (-a)) ≤ |
| 2*((p:ℝ)/(Fintype.card (ZMod p)ˣ:ℝ))^3/(p:ℝ) := by |
| have he : (∑ χ : MulChar (ZMod p) ℂ,∑ ψ : MulChar (ZMod p) ℂ,∑ a : ZMod p, |
| principalEnergy g σ χ (ν*ψ⁻¹) a*nonprincipalEnergy h τ ψ (ν⁻¹*χ⁻¹) (-a)) = |
| ∑ ψ : MulChar (ZMod p) ℂ,∑ χ : MulChar (ZMod p) ℂ,∑ a : ZMod p, |
| nonprincipalEnergy h τ ψ (ν⁻¹*χ⁻¹) a*principalEnergy g σ χ (ν*ψ⁻¹) (-a) := by |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro ψ _ |
| apply Finset.sum_congr rfl |
| intro χ _ |
| have hi := (Equiv.neg (ZMod p)).bijective.sum_comp |
| (fun a => nonprincipalEnergy h τ ψ (ν⁻¹*χ⁻¹) a*principalEnergy g σ χ (ν*ψ⁻¹) (-a)) |
| simpa only [Equiv.neg_apply,neg_neg,mul_comm] using hi |
| rw [he] |
| simpa only [inv_inv] using bad_mixed_left_bound h g τ σ ν⁻¹ hh0 hh hg0 hg |
|
|
| theorem bad_principal_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, |
| principalEnergy g σ χ (ν*ψ⁻¹) a*principalEnergy h τ ψ (ν⁻¹*χ⁻¹) (-a)) ≤ |
| (((p:ℝ)/(Fintype.card (ZMod p)ˣ:ℝ))*(correlationBound g:ℝ)^2+ |
| (Fintype.card (ZMod p)ˣ:ℝ)⁻¹)^2 * |
| ((p:ℝ)/(Fintype.card (ZMod p)ˣ:ℝ))^2 := by |
| classical |
| let B : ℝ := ((p:ℝ)/(Fintype.card (ZMod p)ˣ:ℝ))*(correlationBound g:ℝ)^2+ |
| (Fintype.card (ZMod p)ˣ:ℝ)⁻¹ |
| have he : (∑ χ : MulChar (ZMod p) ℂ,∑ ψ : MulChar (ZMod p) ℂ,∑ a : ZMod p, |
| principalEnergy g σ χ (ν*ψ⁻¹) a*principalEnergy h τ ψ (ν⁻¹*χ⁻¹) (-a)) = |
| ∑ a : ZMod p, ‖twistedPairFourier g σ ν⁻¹ 1 a‖^2*‖twistedPairFourier h τ ν 1 (-a)‖^2 := by |
| rw [sum_three_rotate] |
| apply Finset.sum_congr rfl |
| intro a _ |
| simp only [principalEnergy_mulInv,ite_mul,mul_ite,zero_mul,mul_zero] |
| simp |
| rw [he] |
| calc |
| _ ≤ ∑ a : ZMod p,B^2*‖twistedPairFourier h τ ν 1 (-a)‖^2 := by |
| apply Finset.sum_le_sum |
| intro a _ |
| exact mul_le_mul_of_nonneg_right |
| (pow_le_pow_left₀ (norm_nonneg _) (norm_twistedPairFourier_principal_le g σ ν⁻¹ a hg0 hg) 2) (sq_nonneg _) |
| _ = B^2*(∑ a : ZMod p, ‖twistedPairFourier h τ ν 1 a‖^2) := by |
| rw [← Finset.mul_sum] |
| congr 1 |
| exact (Equiv.neg (ZMod p)).bijective.sum_comp (fun a => ‖twistedPairFourier h τ ν 1 a‖^2) |
| _ ≤ B^2*pairFourthMass h ν := mul_le_mul_of_nonneg_left (twistedPairFourier_energy h τ ν 1) (sq_nonneg _) |
| _ ≤ _ := mul_le_mul_of_nonneg_left (pairFourthMass_basic_le h ν hh0 hh) (sq_nonneg _) |
|
|
| end |
| end Ostmann.FiniteField |
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| end OAI |
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