| 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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|
