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a72760f | 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 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 | 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
end OAI
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