| import Mathlib |
| import OAI.NumberTheory.CubicGram.SieveImprovement |
|
|
| namespace OAI |
|
|
| /-! Full inner cube covers and sharp coefficient extension bounds. -/ |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
| namespace CubicFirstMoment |
|
|
| lemma complex_norm_sum_sq_le {ι : Type*} (I : Finset ι) (v : ι → ℂ) : |
| ‖∑ i ∈ I, v i‖^2 ≤ (I.card : ℝ)*∑ i ∈ I, ‖v i‖^2 := by |
| have ht := norm_sum_le I v |
| have hc := Finset.sum_mul_sq_le_sq_mul_sq I (fun _ => (1 : ℝ)) (fun i => ‖v i‖) |
| simp only [one_mul,one_pow,Finset.sum_const,nsmul_eq_mul,mul_one] at hc |
| exact (pow_le_pow_left₀ (_root_.norm_nonneg _) ht 2).trans hc |
|
|
| lemma innerCubeBlock_fixed_mass (P S : Finset Eisenstein) |
| (hP : ∀ a ∈ P, primary a) (hS : ∀ s ∈ S, primary s) |
| (r : Eisenstein) (v : Eisenstein → ℂ) : |
| (∑ a ∈ P, ‖∑ s ∈ S, v s*cubicSymbol a (r*s)‖^2) ≤ |
| finiteCubicBound S P * ∑ s ∈ S, ‖v s‖^2 := by |
| apply le_trans _ (finiteCubicBound_controls_mass S P v) |
| apply Finset.sum_le_sum |
| intro a ha |
| have he : (∑ s ∈ S, v s*cubicSymbol a (r*s)) = |
| cubicSymbol a r * ∑ s ∈ S, v s*cubicSymbol s a := by |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro s hs |
| rw [cubicSymbol_mul_upper (hP a ha),cubic_reciprocity (hP a ha) (hS s hs)] |
| ring |
| rw [he,norm_mul,mul_pow] |
| apply mul_le_of_le_one_left (sq_nonneg _) |
| exact (pow_le_pow_left₀ (_root_.norm_nonneg _) (norm_cubicSymbol_le_one (hP a ha) r) 2).trans_eq (by norm_num) |
|
|
| theorem innerCubeBlock_mass (P S T C : Finset Eisenstein) |
| (hP : ∀ a ∈ P, primary a) (hS : ∀ s ∈ S, primary s) |
| (r : Eisenstein) (v : Eisenstein → Eisenstein → Eisenstein → ℂ) |
| (D : ℝ) (_hD : 0 ≤ D) (hv : ∀ s ∈ S, ∀ t ∈ T, ∀ c ∈ C, ‖v s t c‖ ≤ D) : |
| (∑ a ∈ P, ‖∑ t ∈ T, ∑ c ∈ C, ∑ s ∈ S, |
| v s t c*cubicSymbol a (r*(s*t^2*c^3))‖^2) ≤ |
| ((T.card : ℝ)*C.card)^2 * finiteCubicBound S P * S.card * D^2 := by |
| have hrow (t : Eisenstein) (ht : t ∈ T) (c : Eisenstein) (hc : c ∈ C) : |
| (∑ a ∈ P, ‖∑ s ∈ S, v s t c*cubicSymbol a (r*(s*t^2*c^3))‖^2) ≤ |
| finiteCubicBound S P * S.card * D^2 := by |
| have he : ∀ s : Eisenstein, r*(s*t^2*c^3) = (r*t^2*c^3)*s := by intro s; ring |
| simp_rw [he] |
| apply (innerCubeBlock_fixed_mass P S hP hS _ (fun s => v s t c)).trans |
| have hh : (∑ s ∈ S, ‖v s t c‖^2) ≤ (S.card : ℝ)*D^2 := by |
| calc |
| _ ≤ ∑ s ∈ S, D^2 := Finset.sum_le_sum (fun s hs => |
| pow_le_pow_left₀ (_root_.norm_nonneg _) (hv s hs t ht c hc) 2) |
| _ = _ := by simp |
| exact (mul_le_mul_of_nonneg_left hh (finiteCubicBound_nonneg _ _)).trans_eq (by ring) |
| have htriangle (a : Eisenstein) : |
| ‖∑ t ∈ T, ∑ c ∈ C, ∑ s ∈ S, v s t c*cubicSymbol a (r*(s*t^2*c^3))‖^2 ≤ |
| ((T.card : ℝ)*C.card)*∑ t ∈ T, ∑ c ∈ C, |
| ‖∑ s ∈ S, v s t c*cubicSymbol a (r*(s*t^2*c^3))‖^2 := by |
| simpa only [Finset.product_eq_sprod,Finset.sum_product,Finset.card_product,Nat.cast_mul] using |
| complex_norm_sum_sq_le (T.product C) |
| (fun z => ∑ s ∈ S, v s z.1 z.2*cubicSymbol a (r*(s*z.1^2*z.2^3))) |
| calc |
| _ ≤ ∑ a ∈ P, ((T.card : ℝ)*C.card)*∑ t ∈ T, ∑ c ∈ C, |
| ‖∑ s ∈ S, v s t c*cubicSymbol a (r*(s*t^2*c^3))‖^2 := |
| Finset.sum_le_sum (fun a _ => htriangle a) |
| _ = ((T.card : ℝ)*C.card)*∑ t ∈ T, ∑ c ∈ C, ∑ a ∈ P, |
| ‖∑ s ∈ S, v s t c*cubicSymbol a (r*(s*t^2*c^3))‖^2 := by |
| rw [← Finset.mul_sum,Finset.sum_comm (s := P) (t := T)] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro t ht |
| exact Finset.sum_comm |
| _ ≤ ((T.card : ℝ)*C.card)*∑ t ∈ T, ∑ c ∈ C, |
| finiteCubicBound S P * S.card * D^2 := by |
| gcongr with t ht c hc |
| exact hrow t ht c hc |
| _ = _ := by simp only [Finset.sum_const,nsmul_eq_mul]; ring |
|
|
| def finiteCoverCoefficient {ι : Type*} (I : Finset ι) (f : ι → Eisenstein) |
| (H : Finset Eisenstein) (v : Eisenstein → ℂ) (x : ι) : ℂ := |
| if f x ∈ H then v (f x) / ((I.filter (fun y => f y = f x)).card : ℂ) else 0 |
|
|
| lemma finiteCoverCoefficient_norm_le {ι : Type*} (I : Finset ι) (f : ι → Eisenstein) |
| (H : Finset Eisenstein) (v : Eisenstein → ℂ) (D : ℝ) (hD : 0 ≤ D) |
| (hv : ∀ n ∈ H, ‖v n‖ ≤ D) {x : ι} (hx : x ∈ I) : |
| ‖finiteCoverCoefficient I f H v x‖ ≤ D := by |
| unfold finiteCoverCoefficient |
| split_ifs with h |
| · have hc : 1 ≤ (I.filter (fun y => f y = f x)).card := |
| Finset.one_le_card.mpr ⟨x,Finset.mem_filter.mpr ⟨hx,rfl⟩⟩ |
| rw [norm_div,Complex.norm_natCast] |
| exact (div_le_self (_root_.norm_nonneg _) (by exact_mod_cast hc)).trans (hv _ h) |
| · simpa using hD |
|
|
| lemma finiteCoverCoefficient_sum {ι : Type*} (I : Finset ι) (f : ι → Eisenstein) |
| (H : Finset Eisenstein) (hcover : H ⊆ I.image f) |
| (v k : Eisenstein → ℂ) : |
| (∑ x ∈ I, finiteCoverCoefficient I f H v x * k (f x)) = |
| ∑ n ∈ H, v n*k n := by |
| classical |
| have he : (∑ x ∈ I, finiteCoverCoefficient I f H v x * k (f x)) = |
| ∑ x ∈ I with f x ∈ H, |
| (v (f x) / ((I.filter (fun y => f y = f x)).card : ℂ))*k (f x) := by |
| rw [Finset.sum_filter] |
| apply Finset.sum_congr rfl |
| intro x hx |
| simp only [finiteCoverCoefficient,ite_mul,zero_mul] |
| rw [he,← Finset.sum_fiberwise_eq_sum_filter' I H f |
| (fun n => (v n / ((I.filter (fun y => f y = n)).card : ℂ))*k n)] |
| apply Finset.sum_congr rfl |
| intro n hn |
| have hc : (I.filter (fun y => f y = n)).card ≠ 0 := by |
| obtain ⟨x,hx,hfx⟩ := Finset.mem_image.mp (hcover hn) |
| exact Finset.card_ne_zero.mpr ⟨x,Finset.mem_filter.mpr ⟨hx,hfx⟩⟩ |
| have hc' : ((I.filter (fun y => f y = n)).card : ℂ) ≠ 0 := by exact_mod_cast hc |
| simp only [Finset.sum_const,nsmul_eq_mul] |
| field_simp |
|
|
| abbrev InnerCubeIndex := (_index : ℕ×ℕ×ℕ) × (Eisenstein×Eisenstein×Eisenstein×Eisenstein) |
|
|
| def innerCubeIndices (j : ℕ) : Finset InnerCubeIndex := |
| (cubeDyadTriples j).sigma (fun z => (nonzeroNormBall 729).product |
| ((squarefreePrimaryDyad z.1).product |
| ((squarefreePrimaryDyad z.2.1).product (frequencyDyad z.2.2)))) |
|
|
| def innerCubeValue (q : InnerCubeIndex) : Eisenstein := |
| q.2.1*(q.2.2.1*q.2.2.2.1^2*q.2.2.2.2^3) |
|
|
| lemma frequencyDyad_subset_innerCubeImage (j : ℕ) : |
| frequencyDyad j ⊆ (innerCubeIndices j).image innerCubeValue := by |
| intro n hn |
| have hh := frequencyDyad_subset_cubeCover j hn |
| obtain ⟨z,hz,hm⟩ := Finset.mem_biUnion.mp hh |
| obtain ⟨r,hr,hm⟩ := Finset.mem_biUnion.mp hm |
| obtain ⟨m,hm,he⟩ := Finset.mem_image.mp hm |
| obtain ⟨⟨s,t,c⟩,hstc,hem⟩ := Finset.mem_image.mp hm |
| obtain ⟨hs,htc⟩ := Finset.mem_product.mp hstc |
| obtain ⟨ht,hc⟩ := Finset.mem_product.mp htc |
| refine Finset.mem_image.mpr ⟨⟨z,r,s,t,c⟩,?_,?_⟩ |
| · exact Finset.mem_sigma.mpr ⟨hz,Finset.mem_product.mpr ⟨hr, |
| Finset.mem_product.mpr ⟨hs,Finset.mem_product.mpr ⟨ht,hc⟩⟩⟩⟩ |
| · dsimp [innerCubeValue] |
| rw [← he,← hem] |
|
|
| def innerCubeCoefficient (j : ℕ) (v : Eisenstein → ℂ) : InnerCubeIndex → ℂ := |
| finiteCoverCoefficient (innerCubeIndices j) innerCubeValue (frequencyDyad j) v |
|
|
| def innerCubeBlockSum (j : ℕ) (v : Eisenstein → ℂ) (a : Eisenstein) |
| (z : ℕ×ℕ×ℕ) (r : Eisenstein) : ℂ := |
| ∑ t ∈ squarefreePrimaryDyad z.2.1, ∑ c ∈ frequencyDyad z.2.2, |
| ∑ s ∈ squarefreePrimaryDyad z.1, |
| innerCubeCoefficient j v ⟨z,r,s,t,c⟩*cubicSymbol a (r*(s*t^2*c^3)) |
|
|
| lemma innerCubeCoefficient_sum (j : ℕ) (v : Eisenstein → ℂ) (a : Eisenstein) : |
| (∑ n ∈ frequencyDyad j, v n*cubicSymbol a n) = |
| ∑ z ∈ cubeDyadTriples j, ∑ r ∈ nonzeroNormBall 729, |
| innerCubeBlockSum j v a z r := by |
| rw [← finiteCoverCoefficient_sum (innerCubeIndices j) innerCubeValue (frequencyDyad j) |
| (frequencyDyad_subset_innerCubeImage j) v (cubicSymbol a)] |
| simp only [innerCubeIndices,Finset.sum_sigma,Finset.product_eq_sprod,Finset.sum_product] |
| apply Finset.sum_congr rfl |
| intro z hz |
| apply Finset.sum_congr rfl |
| intro r hr |
| dsimp only [innerCubeBlockSum,innerCubeValue,innerCubeCoefficient] |
| rw [Finset.sum_comm (s := squarefreePrimaryDyad z.1) (t := squarefreePrimaryDyad z.2.1)] |
| apply Finset.sum_congr rfl |
| intro t ht |
| exact Finset.sum_comm |
|
|
| lemma innerCubeCoefficient_block_mass (P : Finset Eisenstein) (hP : ∀ a ∈ P, primary a) |
| (j : ℕ) (v : Eisenstein → ℂ) (D : ℝ) (hD : 0 ≤ D) |
| (hv : ∀ n ∈ frequencyDyad j, ‖v n‖ ≤ D) |
| (z : ℕ×ℕ×ℕ) (hz : z ∈ cubeDyadTriples j) (r : Eisenstein) |
| (hr : r ∈ nonzeroNormBall 729) : |
| (∑ a ∈ P, ‖innerCubeBlockSum j v a z r‖^2) ≤ |
| (((squarefreePrimaryDyad z.2.1).card : ℝ)*(frequencyDyad z.2.2).card)^2 * |
| finiteCubicBound (squarefreePrimaryDyad z.1) P * |
| (squarefreePrimaryDyad z.1).card * D^2 := by |
| apply innerCubeBlock_mass P _ _ _ hP |
| (fun s hs => (mem_squarefreePrimaryDyad.mp hs).2.1) r |
| (fun s t c => innerCubeCoefficient j v ⟨z,r,s,t,c⟩) D hD |
| intro s hs t ht c hc |
| apply finiteCoverCoefficient_norm_le _ _ _ _ D hD hv |
| exact Finset.mem_sigma.mpr ⟨hz,Finset.mem_product.mpr ⟨hr, |
| Finset.mem_product.mpr ⟨hs,Finset.mem_product.mpr ⟨ht,hc⟩⟩⟩⟩ |
|
|
| lemma complex_mass_sum_sq_le {ι κ : Type*} (I : Finset ι) (P : Finset κ) |
| (F : κ → ι → ℂ) : |
| (∑ a ∈ P, ‖∑ i ∈ I, F a i‖^2) ≤ |
| (I.card : ℝ)*∑ i ∈ I, ∑ a ∈ P, ‖F a i‖^2 := by |
| calc |
| _ ≤ ∑ a ∈ P, (I.card : ℝ)*∑ i ∈ I, ‖F a i‖^2 := |
| Finset.sum_le_sum (fun a _ => complex_norm_sum_sq_le I (F a)) |
| _ = _ := by rw [← Finset.mul_sum,Finset.sum_comm] |
|
|
| theorem innerCubeCover_mass (P : Finset Eisenstein) (hP : ∀ a ∈ P, primary a) |
| (j : ℕ) (v : Eisenstein → ℂ) (D : ℝ) (hD : 0 ≤ D) |
| (hv : ∀ n ∈ frequencyDyad j, ‖v n‖ ≤ D) : |
| (∑ a ∈ P, ‖∑ n ∈ frequencyDyad j, v n*cubicSymbol a n‖^2) ≤ |
| (cubeDyadTriples j).card * ((nonzeroNormBall 729).card : ℝ)^2 * |
| ∑ z ∈ cubeDyadTriples j, |
| (((squarefreePrimaryDyad z.2.1).card : ℝ)*(frequencyDyad z.2.2).card)^2 * |
| finiteCubicBound (squarefreePrimaryDyad z.1) P * |
| (squarefreePrimaryDyad z.1).card * D^2 := by |
| have htriangle := complex_mass_sum_sq_le |
| ((cubeDyadTriples j).product (nonzeroNormBall 729)) P |
| (fun a q => innerCubeBlockSum j v a q.1 q.2) |
| simp only [Finset.product_eq_sprod,Finset.sum_product,Finset.card_product,Nat.cast_mul] |
| at htriangle |
| simp_rw [innerCubeCoefficient_sum j v] |
| apply htriangle.trans |
| calc |
| _ ≤ ((cubeDyadTriples j).card : ℝ)*(nonzeroNormBall 729).card * |
| ∑ z ∈ cubeDyadTriples j, ∑ r ∈ nonzeroNormBall 729, |
| (((squarefreePrimaryDyad z.2.1).card : ℝ)*(frequencyDyad z.2.2).card)^2 * |
| finiteCubicBound (squarefreePrimaryDyad z.1) P * |
| (squarefreePrimaryDyad z.1).card * D^2 := by |
| apply mul_le_mul_of_nonneg_left _ (by positivity) |
| exact Finset.sum_le_sum (fun z hz => Finset.sum_le_sum (fun r hr => |
| innerCubeCoefficient_block_mass P hP j v D hD hv z hz r hr)) |
| _ = _ := by simp only [Finset.sum_const,nsmul_eq_mul,← Finset.mul_sum]; ring |
|
|
| theorem innerCubeBlock_sharp {ε : ℝ} (hε : 0 < ε) : |
| ∃ C : ℝ, 0 < C ∧ ∀ (P : Finset Eisenstein) (N : ℝ), 1 ≤ N → |
| (∀ a ∈ P, primary a ∧ Squarefree a ∧ norm a ≤ N) → |
| ∀ (j : ℕ) (z : ℕ×ℕ×ℕ), z ∈ cubeDyadTriples j → |
| (((squarefreePrimaryDyad z.2.1).card : ℝ)*(frequencyDyad z.2.2).card)^2 * |
| finiteCubicBound (squarefreePrimaryDyad z.1) P * |
| (squarefreePrimaryDyad z.1).card ≤ |
| C*(N*(128*(2 : ℝ)^j))^ε*(128*(2 : ℝ)^j)* |
| (N+128*(2 : ℝ)^j+(N*(128*(2 : ℝ)^j))^(2/3 : ℝ)) := by |
| obtain ⟨C,hC,hsharp⟩ := finiteCubicBound_sharp ε hε |
| refine ⟨18^5*C,by positivity,?_⟩ |
| intro P N hN hP j z hz |
| let S : ℝ := 2*(2 : ℝ)^z.1 |
| let T : ℝ := 2*(2 : ℝ)^z.2.1 |
| let K : ℝ := 2*(2 : ℝ)^z.2.2 |
| let J : ℝ := 128*(2 : ℝ)^j |
| have htwo (i : ℕ) : (1 : ℝ) ≤ 2*2^i := by |
| have hh : (1 : ℝ) ≤ 2^i := one_le_pow₀ (by norm_num) |
| linarith |
| have hS : 1 ≤ S := htwo _ |
| have hT : 1 ≤ T := htwo _ |
| have hK : 1 ≤ K := htwo _ |
| have hp : S*T^2*K^3 ≤ J := cube_scaled_sizes hz |
| have hSJ : S ≤ J := by |
| apply le_trans _ hp |
| exact (le_mul_of_one_le_right (by positivity : 0 ≤ S) (one_le_pow₀ hT)).trans |
| (le_mul_of_one_le_right (by positivity) (one_le_pow₀ hK)) |
| have hsf : ∀ s ∈ squarefreePrimaryDyad z.1, |
| primary s ∧ Squarefree s ∧ norm s ≤ S := by |
| intro s hs |
| exact ⟨(mem_squarefreePrimaryDyad.mp hs).2.1,(mem_squarefreePrimaryDyad.mp hs).2.2, |
| (frequencyDyad_norm (mem_squarefreePrimaryDyad.mp hs).1).2⟩ |
| have hb : finiteCubicBound (squarefreePrimaryDyad z.1) P ≤ |
| C*(N*J)^ε*(N+J+(N*J)^(2/3 : ℝ)) := by |
| apply (hsharp _ _ N S hN hS hsf hP).trans |
| gcongr |
| have hcardS : ((squarefreePrimaryDyad z.1).card : ℝ) ≤ 18*S := by |
| dsimp [S]; nlinarith [squarefreePrimaryDyad_card_le z.1] |
| have hcardT : ((squarefreePrimaryDyad z.2.1).card : ℝ) ≤ 18*T := by |
| dsimp [T]; nlinarith [squarefreePrimaryDyad_card_le z.2.1] |
| have hcardK : ((frequencyDyad z.2.2).card : ℝ) ≤ 18*K := by |
| dsimp [K]; nlinarith [frequencyDyad_card_le z.2.2] |
| have hf : (((squarefreePrimaryDyad z.2.1).card : ℝ)*(frequencyDyad z.2.2).card)^2 * |
| (squarefreePrimaryDyad z.1).card ≤ 18^5*J := by |
| calc |
| _ ≤ ((18*T)*(18*K))^2*(18*S) := by gcongr |
| _ = 18^5*(S*T^2*K^2) := by ring |
| _ ≤ 18^5*(S*T^2*K^3) := by |
| gcongr |
| norm_num |
| _ ≤ _ := mul_le_mul_of_nonneg_left hp (by norm_num) |
| calc |
| _ = ((((squarefreePrimaryDyad z.2.1).card : ℝ)*(frequencyDyad z.2.2).card)^2 * |
| (squarefreePrimaryDyad z.1).card)*finiteCubicBound (squarefreePrimaryDyad z.1) P := by ring |
| _ ≤ (18^5*J)*(C*(N*J)^ε*(N+J+(N*J)^(2/3 : ℝ))) := |
| mul_le_mul hf hb (finiteCubicBound_nonneg _ _) (by positivity) |
| _ = _ := by dsimp [J]; ring |
|
|
| theorem cubic_inner_extension_scaled {ε : ℝ} (hε : 0 < ε) : |
| ∃ C : ℝ, 0 < C ∧ ∀ (P : Finset Eisenstein) (N : ℝ), 1 ≤ N → |
| (∀ a ∈ P, primary a ∧ Squarefree a ∧ norm a ≤ N) → |
| ∀ (j : ℕ) (v : Eisenstein → ℂ) (D : ℝ), 0 ≤ D → |
| (∀ n ∈ frequencyDyad j, ‖v n‖ ≤ D) → |
| (∑ a ∈ P, ‖∑ n ∈ frequencyDyad j, v n*cubicSymbol a n‖^2) ≤ |
| C*(N*(128*(2 : ℝ)^j))^ε*(128*(2 : ℝ)^j)* |
| (N+128*(2 : ℝ)^j+(N*(128*(2 : ℝ)^j))^(2/3 : ℝ))*D^2 := by |
| obtain ⟨B,hB,hblock⟩ := innerCubeBlock_sharp (show 0 < ε/2 by linarith) |
| obtain ⟨E,hE,hpoly⟩ := dyadic_polynomial_small_power 6 (show 0 < ε/2 by linarith) |
| let R : ℝ := (nonzeroNormBall 729).card |
| refine ⟨(R+1)^2*B*E,by dsimp [R]; positivity,?_⟩ |
| intro P N hN hP j v D hD hv |
| let J : ℝ := 128*(2 : ℝ)^j |
| let F : ℝ := N+J+(N*J)^(2/3 : ℝ) |
| have hJpos : 0 < J := by dsimp [J]; positivity |
| have hNpos : 0 < N := zero_lt_one.trans_le hN |
| have hF : 0 ≤ F := by dsimp [F]; positivity |
| have hR : 0 ≤ R := by dsimp [R]; positivity |
| have hR' : R ≤ R+1 := by linarith |
| have hcount : ((cubeDyadTriples j).card : ℝ)^2 ≤ E*J^(ε/2) := by |
| have hc : ((cubeDyadTriples j).card : ℝ) ≤ ((j : ℝ)+2)^3 := by |
| exact_mod_cast cubeDyadTriples_card j |
| calc |
| _ ≤ (((j : ℝ)+2)^3)^2 := by gcongr |
| _ = ((j : ℝ)+2)^6 := by ring |
| _ ≤ E*((2 : ℝ)^j)^(ε/2) := hpoly j |
| _ ≤ _ := by |
| gcongr |
| dsimp [J] |
| have hh : (0 : ℝ) ≤ 2^j := by positivity |
| linarith |
| have heps : J^(ε/2)*(N*J)^(ε/2) ≤ (N*J)^ε := by |
| calc |
| _ ≤ (N*J)^(ε/2)*(N*J)^(ε/2) := by |
| gcongr |
| exact le_mul_of_one_le_left hJpos.le hN |
| _ = _ := by rw [← Real.rpow_add (by positivity)]; congr 1; ring |
| have hh := innerCubeCover_mass P (fun a ha => (hP a ha).1) j v D hD hv |
| apply hh.trans |
| calc |
| _ ≤ (cubeDyadTriples j).card * R^2 * |
| ∑ z ∈ cubeDyadTriples j, B*(N*J)^(ε/2)*J*F*D^2 := by |
| apply mul_le_mul_of_nonneg_left _ (by positivity) |
| apply Finset.sum_le_sum |
| intro z hz |
| exact mul_le_mul_of_nonneg_right (hblock P N hN hP j z hz) (sq_nonneg D) |
| _ = ((cubeDyadTriples j).card : ℝ)^2 * R^2 * B*(N*J)^(ε/2)*J*F*D^2 := by |
| simp only [Finset.sum_const,nsmul_eq_mul]; ring |
| _ ≤ (E*J^(ε/2)) * (R+1)^2 * B*(N*J)^(ε/2)*J*F*D^2 := by |
| gcongr |
| _ = ((R+1)^2*B*E)* (J^(ε/2)*(N*J)^(ε/2))*J*F*D^2 := by ring |
| _ ≤ ((R+1)^2*B*E)* (N*J)^ε*J*F*D^2 := by gcongr |
| _ = _ := rfl |
|
|
| theorem cubic_inner_extension {ε : ℝ} (hε : 0 < ε) : |
| ∃ C : ℝ, 0 < C ∧ ∀ (P : Finset Eisenstein) (N : ℝ), 1 ≤ N → |
| (∀ a ∈ P, primary a ∧ Squarefree a ∧ norm a ≤ N) → |
| ∀ (j : ℕ) (v : Eisenstein → ℂ) (D : ℝ), 0 ≤ D → |
| (∀ n ∈ frequencyDyad j, ‖v n‖ ≤ D) → |
| (∑ a ∈ P, ‖∑ n ∈ frequencyDyad j, v n*cubicSymbol a n‖^2) ≤ |
| C*(N*(2 : ℝ)^j)^ε*(2 : ℝ)^j* |
| (N+(2 : ℝ)^j+(N*(2 : ℝ)^j)^(2/3 : ℝ))*D^2 := by |
| obtain ⟨C,hC,h⟩ := cubic_inner_extension_scaled hε |
| refine ⟨C*128^ε*128^2,by positivity,?_⟩ |
| intro P N hN hP j v D hD hv |
| have hN0 : 0 ≤ N := zero_le_one.trans hN |
| have hJ0 : 0 < (2 : ℝ)^j := by positivity |
| have hp : (128 : ℝ)^(2/3 : ℝ) ≤ 128 := by |
| exact (Real.rpow_le_rpow_of_exponent_le (by norm_num) (by norm_num : (2/3 : ℝ) ≤ 1)).trans_eq (Real.rpow_one _) |
| have hm : (N*(128*(2 : ℝ)^j))^(2/3 : ℝ) ≤ |
| 128*((N*(2 : ℝ)^j)^(2/3 : ℝ)) := by |
| rw [show N*(128*(2 : ℝ)^j) = 128*(N*(2 : ℝ)^j) by ring, |
| Real.mul_rpow (by norm_num) (by positivity)] |
| exact mul_le_mul_of_nonneg_right hp (by positivity) |
| have hb : N+128*(2 : ℝ)^j+(N*(128*(2 : ℝ)^j))^(2/3 : ℝ) ≤ |
| 128*(N+(2 : ℝ)^j+(N*(2 : ℝ)^j)^(2/3 : ℝ)) := by nlinarith |
| have he : (N*(128*(2 : ℝ)^j))^ε = 128^ε*(N*(2 : ℝ)^j)^ε := by |
| rw [show N*(128*(2 : ℝ)^j) = 128*(N*(2 : ℝ)^j) by ring, |
| Real.mul_rpow (by norm_num) (by positivity)] |
| apply (h P N hN hP j v D hD hv).trans |
| rw [he] |
| calc |
| _ ≤ C*(128^ε*(N*(2 : ℝ)^j)^ε)*(128*(2 : ℝ)^j)* |
| (128*(N+(2 : ℝ)^j+(N*(2 : ℝ)^j)^(2/3 : ℝ)))*D^2 := by gcongr |
| _ = _ := by ring |
|
|
| end CubicFirstMoment |
| end |
| end |
|
|
| end OAI |
|
|
