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import OAI.NumberTheory.CubicGauss.Cutoffs
namespace OAI
/-! Cube extraction for cubic operators and weighted sums over cube factors. -/
noncomputable section
open scoped BigOperators
open Module Complex UniqueFactorizationMonoid
attribute [local instance] Classical.propDecidable
namespace CubicFirstMoment
open HeckeTheta
lemma norm_pow_eq (a : Eisenstein) (n : ℕ) : norm (a^n) = (norm a)^n := by
induction n with
| zero => simp
| succ n ih => rw [pow_succ,norm_mul_eq,ih,pow_succ]
def cubicRowEnergy (S : Finset Eisenstein) (u : Eisenstein → ℂ) (m : Eisenstein) : ℝ :=
‖∑ b ∈ S, u b * cubicSymbol b m‖^2
lemma cubicRowEnergy_nonneg (S : Finset Eisenstein) (u : Eisenstein → ℂ) (m : Eisenstein) :
0 ≤ cubicRowEnergy S u m := sq_nonneg _
lemma cubic_cube_row_twist (S : Finset Eisenstein) (hS : ∀ b ∈ S,primary b)
(u : Eisenstein → ℂ) (w c t s : Eisenstein) :
cubicRowEnergy S u (w*s*t^2*c^3) =
cubicRowEnergy S (fun b => u b * cubicSymbol b (w*t^2*c^3)) s := by
unfold cubicRowEnergy
congr 2
apply Finset.sum_congr rfl
intro b hb
rw [show w*s*t^2*c^3 = (w*t^2*c^3)*s by ring,cubicSymbol_mul_upper (hS b hb)]
ring
lemma cubic_cube_row_twist_conj (S : Finset Eisenstein) (hS : ∀ b ∈ S,primary b)
(u : Eisenstein → ℂ) (w c t s : Eisenstein) :
cubicRowEnergy S u (w*t*s^2*c^3) =
cubicRowEnergy S (fun b => star (u b * cubicSymbol b (w*t*c^3))) s := by
unfold cubicRowEnergy
have he : (∑ b ∈ S, u b * cubicSymbol b (w*t*s^2*c^3)) =
star (∑ b ∈ S, star (u b * cubicSymbol b (w*t*c^3)) * cubicSymbol b s) := by
rw [star_sum]
apply Finset.sum_congr rfl
intro b hb
rw [star_mul,star_star,show w*t*s^2*c^3 = (w*t*c^3)*s^2 by ring,
cubicSymbol_mul_upper (hS b hb),cubicSymbol_pow_upper (hS b hb),cubicSymbol_sq_eq_star (hS b hb)]
ring
rw [he,norm_star]
lemma cubic_twist_energy_le (S : Finset Eisenstein) (hS : ∀ b ∈ S,primary b)
(u : Eisenstein → ℂ) (m : Eisenstein) :
∑ b ∈ S, ‖u b * cubicSymbol b m‖^2 ≤ ∑ b ∈ S, ‖u b‖^2 := by
apply Finset.sum_le_sum
intro b hb
rw [norm_mul,mul_pow]
exact mul_le_of_le_one_right (sq_nonneg _) (by
simpa using pow_le_pow_left₀ (_root_.norm_nonneg _) (norm_cubicSymbol_le_one (hS b hb) m) 2)
lemma cubic_cube_box_bound (S : Finset Eisenstein) (hS : ∀ b ∈ S,primary b)
(u : Eisenstein → ℂ) (w c t : Eisenstein) (Y K : ℝ) (hK : 0 ≤ K)
(hop : ∀ v : Eisenstein → ℂ, ∑ s ∈ squarefreePrimaryBall Y,
cubicRowEnergy S v s ≤ K * ∑ b ∈ S, ‖v b‖^2) :
(∑ s ∈ squarefreePrimaryBall Y, cubicRowEnergy S u (w*s*t^2*c^3)) +
(∑ s ∈ squarefreePrimaryBall Y, cubicRowEnergy S u (w*t*s^2*c^3)) ≤
2*K*∑ b ∈ S, ‖u b‖^2 := by
simp_rw [cubic_cube_row_twist S hS u w c t,
cubic_cube_row_twist_conj S hS u w c t]
have h1 := (hop (fun b => u b * cubicSymbol b (w*t^2*c^3))).trans
(mul_le_mul_of_nonneg_left (cubic_twist_energy_le S hS u _) hK)
have h2 := hop (fun b => star (u b * cubicSymbol b (w*t*c^3)))
simp only [norm_star] at h2
have h2 := h2.trans (mul_le_mul_of_nonneg_left (cubic_twist_energy_le S hS u _) hK)
linarith
abbrev CubeIndex := Σ _w : Eisenstein, Σ _c : Eisenstein, Σ _t : Eisenstein, Eisenstein
def cubeIndices (X : ℝ) : Finset CubeIndex :=
(nonzeroNormBall 9).sigma fun _w =>
(nonzeroNormBall (X^(1/3:ℝ))).sigma fun c =>
(squarefreePrimaryBall (X^(1/3:ℝ)/norm c)).sigma fun t =>
squarefreePrimaryBall (X / ((norm t)^2*(norm c)^3))
def cubeMap₁ (v : CubeIndex) : Eisenstein := v.1 * v.2.2.2 * v.2.2.1^2 * v.2.1^3
def cubeMap₂ (v : CubeIndex) : Eisenstein := v.1 * v.2.2.1 * v.2.2.2^2 * v.2.1^3
lemma cube_parameter_bounds {X a b c : ℝ} (hX : 0 ≤ X) (_ : 1 ≤ a) (hb : 1 ≤ b)
(hc : 1 ≤ c) (hab : b ≤ a) (hprod : a*b^2*c^3 ≤ X) :
c ≤ X^(1/3:ℝ) ∧ b ≤ X^(1/3:ℝ)/c ∧ a ≤ X/(b^2*c^3) := by
have hc0 : 0 < c := lt_of_lt_of_le zero_lt_one hc
have hb0 : 0 < b := lt_of_lt_of_le zero_lt_one hb
have hd : 0 < b^2*c^3 := by positivity
have hbc : (b*c)^3 ≤ X := by
calc
_ = b*b^2*c^3 := by ring
_ ≤ a*b^2*c^3 := by gcongr
_ ≤ X := hprod
have hrt : b*c ≤ X^(1/3:ℝ) := by
rw [one_div]
apply (Real.le_rpow_inv_iff_of_pos (by positivity : 0 ≤ b*c) hX (by norm_num : (0:ℝ)<3)).mpr
simpa only [show (3:ℝ) = (3:ℕ) by norm_num,Real.rpow_natCast] using hbc
exact ⟨(le_mul_of_one_le_left hc0.le hb).trans hrt,
(le_div_iff₀ hc0).mpr hrt,(le_div_iff₀ hd).mpr (by simpa [mul_assoc] using hprod)⟩
lemma nonzero_ball_subset_cube_images {X : ℝ} (hX : 1 ≤ X) :
nonzeroNormBall X ⊆ (cubeIndices X).image cubeMap₁ ∪ (cubeIndices X).image cubeMap₂ := by
intro n hn
obtain ⟨hnX,hn0⟩ := mem_nonzeroNormBall.mp hn
obtain ⟨u,i,s,t,c,hi,hs,ht,hss,htt,hst,hc,he⟩ := full_cube_decomposition_unit hn0
let w : Eisenstein := u * lambdaE^i
have hw0 : w ≠ 0 := mul_ne_zero u.ne_zero (pow_ne_zero _ lambdaE_prime.ne_zero)
have hnw : norm w = (3:ℝ)^i := by
rw [norm_mul_eq,norm_of_isUnit u.isUnit,norm_pow_eq,norm_lambdaE,one_mul]
have nw : w ∈ nonzeroNormBall 9 := mem_nonzeroNormBall.mpr ⟨by
rw [hnw]; interval_cases i <;> norm_num,hw0⟩
have hnprod : norm s*(norm t)^2*(norm c)^3 ≤ X := by
have hnorm : norm n = norm w * (norm s*(norm t)^2*(norm c)^3) := by
rw [he]
simp only [norm_mul_eq,norm_pow_eq,w]
ring
have hle : norm s*(norm t)^2*(norm c)^3 ≤ norm n := by
rw [hnorm]
exact le_mul_of_one_le_left
(mul_nonneg (mul_nonneg (norm_nonneg s) (sq_nonneg _)) (pow_nonneg (norm_nonneg c) 3))
(one_le_norm hw0)
exact hle.trans hnX
have hs1 := one_le_norm (primary_ne_zero hs)
have ht1 := one_le_norm (primary_ne_zero ht)
have hc1 := one_le_norm hc
by_cases hts : norm t ≤ norm s
· obtain ⟨hcX,htX,hsX⟩ := cube_parameter_bounds (by linarith) hs1 ht1 hc1 hts hnprod
apply Finset.mem_union_left
apply Finset.mem_image.mpr
refine ⟨⟨w,c,t,s⟩,?_,he.symm⟩
simp only [cubeIndices,Finset.mem_sigma]
exact ⟨nw,mem_nonzeroNormBall.mpr ⟨hcX,hc⟩,
mem_squarefreePrimaryBall.mpr ⟨ht,htt,htX⟩,
mem_squarefreePrimaryBall.mpr ⟨hs,hss,hsX⟩⟩
· have hst' : norm s ≤ norm t := (lt_of_not_ge hts).le
have hprod' : norm t*(norm s)^2*(norm c)^3 ≤ X := by
apply le_trans _ hnprod
have : norm s*norm t*(norm s) ≤ norm s*norm t*(norm t) := by gcongr
nlinarith [mul_le_mul_of_nonneg_right this (pow_nonneg (norm_nonneg c) 3)]
obtain ⟨hcX,hsX,htX⟩ := cube_parameter_bounds (by linarith) ht1 hs1 hc1 hst' hprod'
apply Finset.mem_union_right
apply Finset.mem_image.mpr
refine ⟨⟨w,c,s,t⟩,?_,he.symm⟩
simp only [cubeIndices,Finset.mem_sigma]
exact ⟨nw,mem_nonzeroNormBall.mpr ⟨hcX,hc⟩,
mem_squarefreePrimaryBall.mpr ⟨hs,hss,hsX⟩,
mem_squarefreePrimaryBall.mpr ⟨ht,htt,htX⟩⟩
lemma cubic_full_cube_sum_le {X : ℝ} (hX : 1 ≤ X)
(S : Finset Eisenstein) (u : Eisenstein → ℂ) :
∑ m ∈ nonzeroNormBall X, cubicRowEnergy S u m ≤
∑ w ∈ nonzeroNormBall 9, ∑ c ∈ nonzeroNormBall (X^(1/3:ℝ)),
∑ t ∈ squarefreePrimaryBall (X^(1/3:ℝ)/norm c),
((∑ s ∈ squarefreePrimaryBall (X/((norm t)^2*(norm c)^3)),
cubicRowEnergy S u (w*s*t^2*c^3)) +
(∑ s ∈ squarefreePrimaryBall (X/((norm t)^2*(norm c)^3)),
cubicRowEnergy S u (w*t*s^2*c^3))) := by
let f := cubicRowEnergy S u
let I := cubeIndices X
have hn (m : Eisenstein) : 0 ≤ f m := cubicRowEnergy_nonneg S u m
have h1 := Finset.sum_le_sum_of_subset_of_nonneg
(nonzero_ball_subset_cube_images hX) (fun m _ _ => hn m)
have h2 : ∑ m ∈ I.image cubeMap₁ ∪ I.image cubeMap₂, f m ≤
(∑ m ∈ I.image cubeMap₁, f m) + (∑ m ∈ I.image cubeMap₂, f m) := by
rw [← Finset.sum_union_inter]
exact le_add_of_nonneg_right (Finset.sum_nonneg (fun m _ => hn m))
have h3 := add_le_add
(Finset.sum_image_le_of_nonneg (s := I) (g := cubeMap₁) (fun m _ => hn m))
(Finset.sum_image_le_of_nonneg (s := I) (g := cubeMap₂) (fun m _ => hn m))
apply (h1.trans h2).trans
convert h3 using 1
simp only [I,cubeIndices,Finset.sum_sigma,cubeMap₁,cubeMap₂,Finset.sum_add_distrib,f]
theorem cubic_full_cube_bound {X : ℝ} (hX : 1 ≤ X)
(S : Finset Eisenstein) (hS : ∀ b ∈ S,primary b) (u : Eisenstein → ℂ)
(K : ℝ → ℝ) (hK : ∀ Y, 0 ≤ K Y)
(hop : ∀ (Y : ℝ) (v : Eisenstein → ℂ),
∑ s ∈ squarefreePrimaryBall Y, cubicRowEnergy S v s ≤
K Y * ∑ b ∈ S, ‖v b‖^2) :
∑ m ∈ nonzeroNormBall X, cubicRowEnergy S u m ≤
2 * ((nonzeroNormBall 9).card : ℝ) *
(∑ c ∈ nonzeroNormBall (X^(1/3:ℝ)),
∑ t ∈ squarefreePrimaryBall (X^(1/3:ℝ)/norm c),
K (X/((norm t)^2*(norm c)^3))) * ∑ b ∈ S, ‖u b‖^2 := by
apply (cubic_full_cube_sum_le hX S u).trans
have hb := Finset.sum_le_sum (s := nonzeroNormBall 9) (fun w _ =>
Finset.sum_le_sum (s := nonzeroNormBall (X^(1/3:ℝ))) (fun c _ =>
Finset.sum_le_sum (s := squarefreePrimaryBall (X^(1/3:ℝ)/norm c)) (fun t _ =>
cubic_cube_box_bound S hS u w c t (X/((norm t)^2*(norm c)^3)) _ (hK _) (hop _))))
convert hb using 1
simp only [← Finset.mul_sum,← Finset.sum_mul,Finset.sum_const, nsmul_eq_mul]
ring
end CubicFirstMoment
namespace CubicFirstMoment
def latticeZeta (r : ℝ) : ℝ := ∑' a : Eisenstein,(norm a)^(-r)
lemma latticeZeta_nonneg (r : ℝ) : 0 ≤ latticeZeta r :=
tsum_nonneg fun a => Real.rpow_nonneg (norm_nonneg a) _
lemma latticeZeta_pos {r : ℝ} (hr : 1<r) : 0<latticeZeta r := by
have hs := HeckeTheta.lattice_norm_rpow_summable r hr
have hle := hs.le_tsum (1:Eisenstein) (fun a _ => Real.rpow_nonneg (norm_nonneg a) _)
have he : (norm (1:Eisenstein))^(-r) = 1 := by simp
change 0 < ∑' a : Eisenstein,(norm a)^(-r)
linarith
lemma norm_rpow_finset_le (S : Finset Eisenstein) {r : ℝ} (hr : 1<r) :
∑ a ∈ S,(norm a)^(-r) ≤ latticeZeta r :=
Summable.sum_le_tsum S (fun a _ => Real.rpow_nonneg (norm_nonneg a) _)
(HeckeTheta.lattice_norm_rpow_summable r hr)
lemma quotient_norm_rpow {X a b p : ℝ} (hX : 0≤X) (ha : 0<a) (hb : 0<b) :
(X/(a^2*b^3))^p = X^p*a^(-(2*p))*b^(-(3*p)) := by
rw [Real.div_rpow hX (by positivity),Real.mul_rpow (by positivity) (by positivity)]
have he2 : (a^2)^p = a^(2*p) := by
rw [← Real.rpow_natCast,← Real.rpow_mul ha.le]
norm_num
have he3 : (b^3)^p = b^(3*p) := by
rw [← Real.rpow_natCast,← Real.rpow_mul hb.le]
norm_num
rw [he2,he3,Real.rpow_neg ha.le,Real.rpow_neg hb.le]
simp only [div_eq_mul_inv,mul_inv_rev]
ring
lemma cube_rpow_sum_bound {X p : ℝ} (hX : 0≤X) (hp : (2/3:ℝ)≤p) :
(∑ c ∈ nonzeroNormBall (X^(1/3:ℝ)),
∑ t ∈ squarefreePrimaryBall (X^(1/3:ℝ)/norm c),
(X/((norm t)^2*(norm c)^3))^p) ≤ X^p*(latticeZeta (4/3))^2 := by
have hZ := latticeZeta_nonneg (4/3)
calc
_ ≤ ∑ c ∈ nonzeroNormBall (X^(1/3:ℝ)),
∑ t ∈ squarefreePrimaryBall (X^(1/3:ℝ)/norm c),
X^p*(norm t)^(-(4/3:ℝ))*(norm c)^(-(4/3:ℝ)) := by
apply Finset.sum_le_sum
intro c hc
have hc0 := (mem_nonzeroNormBall.mp hc).2
have hc1 := one_le_norm hc0
apply Finset.sum_le_sum
intro t ht
have ht0 := primary_ne_zero (mem_squarefreePrimaryBall.mp ht).1
have ht1 := one_le_norm ht0
rw [quotient_norm_rpow hX (norm_pos_of_ne_zero ht0) (norm_pos_of_ne_zero hc0)]
gcongr <;> linarith
_ ≤ ∑ c ∈ nonzeroNormBall (X^(1/3:ℝ)),
X^p*latticeZeta (4/3)*(norm c)^(-(4/3:ℝ)) := by
apply Finset.sum_le_sum
intro c hc
rw [← Finset.sum_mul,← Finset.mul_sum]
exact mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_left (norm_rpow_finset_le _ (by norm_num : (1:ℝ)<4/3))
(Real.rpow_nonneg hX _)) (Real.rpow_nonneg (norm_nonneg c) _)
_ ≤ _ := by
rw [← Finset.mul_sum]
apply (mul_le_mul_of_nonneg_left (norm_rpow_finset_le _ (by norm_num : (1:ℝ)<4/3))
(mul_nonneg (Real.rpow_nonneg hX _) hZ)).trans_eq
ring
lemma squarefreePrimaryBall_card_le {Y : ℝ} (hY : 0≤Y) :
((squarefreePrimaryBall Y).card:ℝ) ≤ 18*Y := by
apply le_trans _ (nonzeroNormBall_card_le hY)
exact_mod_cast Finset.card_le_card (Finset.filter_subset _ _)
lemma norm_reciprocal_sum_bound {R δ : ℝ} (hR : 0<R) (hδ : 0<δ) :
∑ c ∈ nonzeroNormBall R,1/norm c ≤ R^δ*latticeZeta (1+δ) := by
calc
_ ≤ ∑ c ∈ nonzeroNormBall R,R^δ*(norm c)^(-(1+δ)) := by
apply Finset.sum_le_sum
intro c hc
obtain ⟨hcR,hc0⟩ := mem_nonzeroNormBall.mp hc
have hcpos := norm_pos_of_ne_zero hc0
have he : 1/norm c = (norm c)^δ*(norm c)^(-(1+δ)) := by
rw [← Real.rpow_add hcpos]
rw [show δ+ -(1+δ) = (-1:ℝ) by ring,Real.rpow_neg_one]
exact one_div _
rw [he]
exact mul_le_mul_of_nonneg_right (Real.rpow_le_rpow hcpos.le hcR hδ.le)
(Real.rpow_nonneg hcpos.le _)
_ ≤ _ := by
rw [← Finset.mul_sum]
exact mul_le_mul_of_nonneg_left (norm_rpow_finset_le _ (by linarith)) (Real.rpow_nonneg hR.le _)
lemma cube_pair_count_bound {X δ : ℝ} (hX : 1≤X) (hδ : 0<δ) :
(∑ c ∈ nonzeroNormBall (X^(1/3:ℝ)),
((squarefreePrimaryBall (X^(1/3:ℝ)/norm c)).card:ℝ)) ≤
18*latticeZeta (1+δ)*X^(1/3+δ) := by
have hX0 : 0<X := by linarith
have hR : 0<X^(1/3:ℝ) := Real.rpow_pos_of_pos hX0 _
have hRX : X^(1/3:ℝ) ≤ X := Real.rpow_le_self_of_one_le hX (by norm_num)
calc
_ ≤ ∑ c ∈ nonzeroNormBall (X^(1/3:ℝ)),18*X^(1/3:ℝ)*(1/norm c) := by
apply Finset.sum_le_sum
intro c hc
have hc0 := norm_pos_of_ne_zero (mem_nonzeroNormBall.mp hc).2
simpa [div_eq_mul_inv,mul_assoc] using squarefreePrimaryBall_card_le
(div_nonneg hR.le hc0.le)
_ = 18*X^(1/3:ℝ)*∑ c ∈ nonzeroNormBall (X^(1/3:ℝ)),1/norm c := by rw [Finset.mul_sum]
_ ≤ 18*X^(1/3:ℝ)*((X^(1/3:ℝ))^δ*latticeZeta (1+δ)) :=
mul_le_mul_of_nonneg_left (norm_reciprocal_sum_bound hR hδ) (by positivity)
_ ≤ 18*X^(1/3:ℝ)*(X^δ*latticeZeta (1+δ)) := by
have hZ := latticeZeta_nonneg (1+δ)
gcongr
_ = _ := by rw [Real.rpow_add hX0]; ring
end CubicFirstMoment
end
end OAI
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