| import Mathlib |
| import OAI.NumberTheory.CubicGram.CubicOperator |
|
|
| namespace OAI |
|
|
| /-! Primary cube decomposition and finite operator block bounds. -/ |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators ContDiff |
| open Set Filter MeasureTheory Topology |
| attribute [local instance] Classical.propDecidable |
| namespace CubicFirstMoment |
|
|
| lemma primary_of_primary_mul_left {a b : Eisenstein} (ha : primary a) |
| (hab : primary (a*b)) : primary b := by |
| rw [primary_iff_residue_one] at ha hab ⊢ |
| simpa only [map_mul, ha, one_mul] using hab |
|
|
| theorem primary_cube_decomposition {n : Eisenstein} (hn : primary n) : |
| ∃ s t c : Eisenstein, primary s ∧ primary t ∧ primary c ∧ |
| Squarefree s ∧ Squarefree t ∧ IsCoprime s t ∧ n = s*t^2*c^3 := by |
| apply primary_induction (b := n) ?_ ?_ hn |
| · refine ⟨1,1,1,?_,?_,?_,squarefree_one,squarefree_one,isCoprime_one_left,?_⟩ <;> |
| simp [primary] |
| · intro p b hp hb ih |
| obtain ⟨s,t,c,hs,ht,hc,hss,hst,hcop,rfl⟩ := ih |
| by_cases hps : p ∣ s |
| · obtain ⟨s',rfl⟩ := hps |
| have hs' := primary_of_primary_mul_left hp.1 hs |
| have hps' : IsCoprime p s' := |
| isRelPrime_iff_isCoprime.mp (squarefree_mul_iff.mp hss).1 |
| have hpt : IsCoprime p t := hcop.of_mul_left_left |
| refine ⟨s',p*t,c,hs',primary_mul hp.1 ht,hc,hss.of_mul_right, |
| squarefree_mul_iff.mpr ⟨hpt.isRelPrime,hp.2.squarefree,hst⟩, |
| hps'.symm.mul_right hcop.of_mul_left_right,?_⟩ |
| ring |
| · by_cases hpt : p ∣ t |
| · obtain ⟨t',rfl⟩ := hpt |
| refine ⟨s,t',p*c,hs,primary_of_primary_mul_left hp.1 ht,primary_mul hp.1 hc, |
| hss,hst.of_mul_right,hcop.of_mul_right_right,?_⟩ |
| ring |
| · have hps' := hp.2.coprime_iff_not_dvd.mpr hps |
| have hpt' := hp.2.coprime_iff_not_dvd.mpr hpt |
| refine ⟨p*s,t,c,primary_mul hp.1 hs,ht,hc, |
| squarefree_mul_iff.mpr ⟨hps'.isRelPrime,hp.2.squarefree,hss⟩, |
| hst,hpt'.mul_left hcop,?_⟩ |
| ring |
| |
| lemma cubicSymbol_pow_upper {p : Eisenstein} (hp : primary p) (a : Eisenstein) (k : ℕ) : |
| cubicSymbol p (a^k) = cubicSymbol p a ^ k := by |
| induction k with |
| | zero => |
| simp only [pow_zero] |
| apply primary_induction (b := p) ?_ ?_ hp |
| · exact cubicSymbol_one_lower 1 |
| · intro q b hq hb ih |
| rw [cubicSymbol_mul_lower hq.2.ne_zero (primary_ne_zero hb), |
| cubicSymbol_prime hq, cubicSymbolAtPrime_one hq, ih, one_mul] |
| | succ k ih => rw [pow_succ, cubicSymbol_mul_upper hp, ih, pow_succ] |
| |
| lemma finiteCubicBound_twist_controls_mass (P S : Finset Eisenstein) |
| (u z : Eisenstein → ℂ) (hz : ∀ b ∈ P, ‖z b‖ ≤ 1) : |
| (∑ s ∈ S, ‖∑ b ∈ P, u b*z b*cubicSymbol b s‖^2) ≤ |
| finiteCubicBound P S * ∑ b ∈ P, ‖u b‖^2 := by |
| apply (finiteCubicBound_controls_mass P S (fun b => u b*z b)).trans |
| apply mul_le_mul_of_nonneg_left _ (finiteCubicBound_nonneg _ _) |
| 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 _) (hz b hb) 2) |
| |
| lemma finiteCubicBound_twist_conjugate_mass (P S : Finset Eisenstein) |
| (u z : Eisenstein → ℂ) (hz : ∀ b ∈ P, ‖z b‖ ≤ 1) : |
| (∑ s ∈ S, ‖∑ b ∈ P, u b*z b*star (cubicSymbol b s)‖^2) ≤ |
| finiteCubicBound P S * ∑ b ∈ P, ‖u b‖^2 := by |
| have h := finiteCubicBound_twist_controls_mass P S (fun b => star (u b)) |
| (fun b => star (z b)) (fun b hb => by simpa using hz b hb) |
| have he (s : Eisenstein) : |
| (∑ b ∈ P, u b*z b*star (cubicSymbol b s)) = |
| star (∑ b ∈ P, star (u b)*star (z b)*cubicSymbol b s) := by |
| simp [star_sum, star_mul, mul_comm, mul_assoc] |
| simpa only [he, norm_star] using h |
| |
| end CubicFirstMoment |
| |
| namespace CubicFirstMoment |
| |
| def cubeBlockMass (P S T C : Finset Eisenstein) (u : Eisenstein → ℂ) : ℝ := |
| ∑ s ∈ S, ∑ t ∈ T, ∑ c ∈ C, |
| ‖∑ b ∈ P, u b * cubicSymbol b (s*t^2*c^3)‖^2 |
| |
| lemma cubeBlockMass_le_first (P S T C : Finset Eisenstein) |
| (hP : ∀ b ∈ P, primary b) (u : Eisenstein → ℂ) : |
| cubeBlockMass P S T C u ≤ |
| (T.card : ℝ) * C.card * finiteCubicBound P S * ∑ b ∈ P, ‖u b‖^2 := by |
| have hrow (t c : Eisenstein) : |
| (∑ s ∈ S, ‖∑ b ∈ P, u b * cubicSymbol b (s*t^2*c^3)‖^2) ≤ |
| finiteCubicBound P S * ∑ b ∈ P, ‖u b‖^2 := by |
| have h := finiteCubicBound_twist_controls_mass P S u |
| (fun b => cubicSymbol b (t^2*c^3)) (fun b hb => norm_cubicSymbol_le_one (hP b hb) _) |
| convert h using 1 |
| apply Finset.sum_congr rfl |
| intro s hs |
| congr 2 |
| apply Finset.sum_congr rfl |
| intro b hb |
| rw [mul_assoc s, cubicSymbol_mul_upper (hP b hb)] |
| ring |
| unfold cubeBlockMass |
| rw [Finset.sum_comm (s := S) (t := T)] |
| simp_rw [Finset.sum_comm (s := S) (t := C)] |
| calc |
| _ ≤ ∑ t ∈ T, ∑ c ∈ C, finiteCubicBound P S * ∑ b ∈ P, ‖u b‖^2 := |
| Finset.sum_le_sum (fun t _ => Finset.sum_le_sum (fun c _ => hrow t c)) |
| _ = _ := by simp only [Finset.sum_const, nsmul_eq_mul]; ring |
| |
| lemma cubeBlockMass_le_second (P S T C : Finset Eisenstein) |
| (hP : ∀ b ∈ P, primary b) (u : Eisenstein → ℂ) : |
| cubeBlockMass P S T C u ≤ |
| (S.card : ℝ) * C.card * finiteCubicBound P T * ∑ b ∈ P, ‖u b‖^2 := by |
| have hrow (s c : Eisenstein) : |
| (∑ t ∈ T, ‖∑ b ∈ P, u b * cubicSymbol b (s*t^2*c^3)‖^2) ≤ |
| finiteCubicBound P T * ∑ b ∈ P, ‖u b‖^2 := by |
| have h := finiteCubicBound_twist_conjugate_mass P T u |
| (fun b => cubicSymbol b (s*c^3)) (fun b hb => norm_cubicSymbol_le_one (hP b hb) _) |
| convert h using 1 |
| apply Finset.sum_congr rfl |
| intro t ht |
| congr 2 |
| apply Finset.sum_congr rfl |
| intro b hb |
| rw [cubicSymbol_mul_upper (hP b hb), cubicSymbol_mul_upper (hP b hb), |
| cubicSymbol_mul_upper (hP b hb), cubicSymbol_pow_upper (hP b hb), |
| cubicSymbol_sq_eq_star (hP b hb)] |
| ring |
| unfold cubeBlockMass |
| simp_rw [Finset.sum_comm (s := T) (t := C)] |
| calc |
| _ ≤ ∑ s ∈ S, ∑ c ∈ C, finiteCubicBound P T * ∑ b ∈ P, ‖u b‖^2 := |
| Finset.sum_le_sum (fun s _ => Finset.sum_le_sum (fun c _ => hrow s c)) |
| _ = _ := by simp only [Finset.sum_const, nsmul_eq_mul]; ring |
| |
| theorem cubeBlockMass_le_min (P S T C : Finset Eisenstein) |
| (hP : ∀ b ∈ P, primary b) (u : Eisenstein → ℂ) : |
| cubeBlockMass P S T C u ≤ |
| min ((T.card : ℝ) * C.card * finiteCubicBound P S) |
| ((S.card : ℝ) * C.card * finiteCubicBound P T) * ∑ b ∈ P, ‖u b‖^2 := by |
| rw [min_mul_of_nonneg _ _ (Finset.sum_nonneg (fun _ _ => sq_nonneg _))] |
| exact le_min (cubeBlockMass_le_first P S T C hP u) (cubeBlockMass_le_second P S T C hP u) |
| |
| end CubicFirstMoment |
| |
| namespace CubicFirstMoment |
| |
| def cubeBlockSupport (S T C : Finset Eisenstein) : Finset Eisenstein := |
| (S.product (T.product C)).image (fun z => z.1*z.2.1^2*z.2.2^3) |
| |
| lemma cubeBlockSupport_mass_le (P S T C : Finset Eisenstein) (u : Eisenstein → ℂ) : |
| (∑ n ∈ cubeBlockSupport S T C, ‖∑ b ∈ P, u b*cubicSymbol b n‖^2) ≤ |
| cubeBlockMass P S T C u := by |
| calc |
| _ ≤ ∑ z ∈ S.product (T.product C), |
| ‖∑ b ∈ P, u b*cubicSymbol b (z.1*z.2.1^2*z.2.2^3)‖^2 := |
| Finset.sum_image_le_of_nonneg |
| (s := S.product (T.product C)) (g := fun z => z.1*z.2.1^2*z.2.2^3) |
| (f := fun n => ‖∑ b ∈ P, u b*cubicSymbol b n‖^2) (fun _ _ => sq_nonneg _) |
| _ = _ := by |
| unfold cubeBlockMass |
| exact (Finset.sum_product S (T.product C) _).trans |
| (Finset.sum_congr rfl (fun s _ => Finset.sum_product T C _)) |
| |
| theorem finiteCubicBound_cubeBlock (P S T C : Finset Eisenstein) |
| (hP : ∀ b ∈ P, primary b) : |
| finiteCubicBound P (cubeBlockSupport S T C) ≤ |
| min ((T.card : ℝ) * C.card * finiteCubicBound P S) |
| ((S.card : ℝ) * C.card * finiteCubicBound P T) := by |
| apply (finiteCubicBound_le_iff _ _ (le_min |
| (mul_nonneg (by positivity) (finiteCubicBound_nonneg P S)) |
| (mul_nonneg (by positivity) (finiteCubicBound_nonneg P T)))).mpr |
| intro u |
| exact (cubeBlockSupport_mass_le P S T C u).trans (cubeBlockMass_le_min P S T C hP u) |
| |
| end CubicFirstMoment |
| end |
| end |
| |
| end OAI |
| |
