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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 | import OAI.NumberTheory.Ostmann.Tree.BadNonprincipalConvolution
import OAI.NumberTheory.Ostmann.Tree.BadPrincipalConvolution
import OAI.NumberTheory.Ostmann.Tree.CorrelationSymmetry
namespace OAI
namespace Ostmann.FiniteField
noncomputable section
open scoped BigOperators ComplexConjugate
variable {p : ℕ} [Fact p.Prime]
theorem bad_convolution_precise (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,
‖twistedPairFourier g σ χ (ν*ψ⁻¹) a‖^2*
‖twistedPairFourier h τ ψ (ν⁻¹*χ⁻¹) (-a)‖^2) ≤
((p:ℝ)/(Fintype.card (ZMod p)ˣ:ℝ))^3*
(2*(correlationBound g:ℝ)^2+2*(correlationBound h:ℝ)^2+7/(p:ℝ))+
(((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
have hn := bad_nonprincipal_convolution_bound g h σ τ ν hg0 hg hh0 hh
have hl := bad_mixed_left_bound g h σ τ ν hg0 hg hh0 hh
have hr := bad_mixed_right_bound g h σ τ ν hg0 hg hh0 hh
have hp := bad_principal_bound g h σ τ ν hg0 hg hh0 hh
simp_rw [spectralEnergy_decomposition,add_mul,mul_add,Finset.sum_add_distrib]
simp only [div_eq_mul_inv] at hn hl hr hp ⊢
nlinarith only [hn,hl,hr,hp]
theorem prime_unit_ratio_le_two : (p:ℝ)/(Fintype.card (ZMod p)ˣ:ℝ)≤2 := by
have hp : (2:ℝ)≤p := by exact_mod_cast (Fact.out : p.Prime).two_le
have hN : 0<(Fintype.card (ZMod p)ˣ:ℝ) := by exact_mod_cast Fintype.card_pos
have he : (Fintype.card (ZMod p)ˣ:ℝ)=(p:ℝ)-1 := by
rw [ZMod.card_units,Nat.cast_sub (Fact.out : p.Prime).one_lt.le,Nat.cast_one]
apply (div_le_iff₀ hN).mpr
rw [he]
linarith
theorem bad_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,
‖twistedPairFourier g σ χ (ν*ψ⁻¹) a‖^2*
‖twistedPairFourier h τ ψ (ν⁻¹*χ⁻¹) (-a)‖^2) ≤
100*((correlationBound g:ℝ)^2+(correlationBound h:ℝ)^2+(p:ℝ)⁻¹) := by
let C : ℝ := (p:ℝ)/(Fintype.card (ZMod p)ˣ:ℝ)
let x : ℝ := (correlationBound g:ℝ)^2
let y : ℝ := (correlationBound h:ℝ)^2
let t : ℝ := (p:ℝ)⁻¹
have hC0 : 0≤C := by dsimp [C]; positivity
have hC2 : C≤2 := prime_unit_ratio_le_two
have hx0 : 0≤x := sq_nonneg _
have hy0 : 0≤y := sq_nonneg _
have hx1 : x≤1 := by
exact (pow_le_pow_left₀ (NNReal.coe_nonneg _) (correlationBound_le_one g hg) 2).trans_eq (by norm_num)
have ht0 : 0≤t := by dsimp [t]; positivity
have ht1 : t≤1 := by
dsimp [t]
exact inv_le_one_of_one_le₀ (by exact_mod_cast (Fact.out : p.Prime).one_lt.le)
have hC3 : C^3≤8 := (pow_le_pow_left₀ hC0 hC2 3).trans_eq (by norm_num)
have hC4 : C^4≤16 := (pow_le_pow_left₀ hC0 hC2 4).trans_eq (by norm_num)
have hs : (x+t)^2≤2*x+2*t := by
have hx := mul_le_mul_of_nonneg_left hx1 hx0
have ht := mul_le_mul_of_nonneg_left ht1 ht0
nlinarith only [hx,ht,sq_nonneg (x-t)]
have hprec := bad_convolution_precise g h σ τ ν hg0 hg hh0 hh
have hp : (p:ℝ)≠0 := by exact_mod_cast (Fact.out : p.Prime).ne_zero
have hform :
C^3*(2*x+2*y+7/(p:ℝ))+
(C*x+(Fintype.card (ZMod p)ˣ:ℝ)⁻¹)^2*C^2 =
C^3*(2*x+2*y+7*t)+C^4*(x+t)^2 := by
dsimp [C,t]
field_simp
change _≤C^3*(2*x+2*y+7/(p:ℝ))+(C*x+(Fintype.card (ZMod p)ˣ:ℝ)⁻¹)^2*C^2 at hprec
rw [hform] at hprec
have hfirst := mul_le_mul_of_nonneg_right hC3 (show 0≤2*x+2*y+7*t by positivity)
have hsecond := mul_le_mul hC4 hs (sq_nonneg _) (by norm_num : (0:ℝ)≤16)
change _≤100*(x+y+t)
nlinarith only [hprec,hfirst,hsecond,hx0,hy0,ht0]
end
end Ostmann.FiniteField
end OAI
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