| import OAI.NumberTheory.TwoPoint.Halasz.HalaszRepeatedCoordinates |
| import OAI.NumberTheory.TwoPoint.Halasz.HalaszEnergyRestriction |
| import OAI.NumberTheory.TwoPoint.Halasz.HalaszCollisionAbsorption |
|
|
| namespace OAI |
|
|
| /-! The complete-system moment is controlled by the tuples whose |
| chosen long coordinates are pairwise distinct. -/ |
| namespace TwoPointCorrelations |
|
|
| open Finset |
| open scoped Classical |
|
|
| noncomputable def halaszGoodLongTuples {n N ℓ : ℕ} (L : Fin ℓ → Fin n) : |
| Finset (Fin n → Fin N) := univ.filter (fun x => Function.Injective (x ∘ L)) |
|
|
| noncomputable def halaszBadLongCount {n k N ℓ : ℕ} (L : Fin ℓ → Fin (n+2)) : ℕ := |
| (univ.filter (fun xy : (Fin (n+2) → Fin N) × (Fin (n+2) → Fin N) => |
| ¬Function.Injective (xy.1 ∘ L) ∧ |
| halaszVinogradovFrequency k xy.1=halaszVinogradovFrequency k xy.2)).card |
| |
| theorem halasz_bad_long_count_bound {n k N ℓ : ℕ} (L : Fin ℓ → Fin (n+2)) |
| (hL : Function.Injective L) : |
| halaszBadLongCount (k := k) (N := N) L ≤ ℓ^2*halaszRepeatedCount n k N := by |
| let P := (univ : Finset (Fin ℓ × Fin ℓ)).filter (fun ij => ij.1≠ij.2) |
| let S := fun ij : Fin ℓ × Fin ℓ => halaszPairCollision n k N (L ij.1) (L ij.2) |
| have hsub : (univ.filter (fun xy : (Fin (n+2) → Fin N) × (Fin (n+2) → Fin N) => |
| ¬Function.Injective (xy.1 ∘ L) ∧ |
| halaszVinogradovFrequency k xy.1=halaszVinogradovFrequency k xy.2)) ⊆ |
| P.biUnion S := by |
| intro xy hxy |
| obtain ⟨hbad,hfreq⟩ := (mem_filter.mp hxy).2 |
| obtain ⟨i,j,hij,hne⟩ := Function.not_injective_iff.mp hbad |
| apply mem_biUnion.mpr |
| refine ⟨(i,j),mem_filter.mpr ⟨mem_univ _,hne⟩,?_⟩ |
| exact mem_filter.mpr ⟨mem_univ _,hij,hfreq⟩ |
| have hP : P.card≤ℓ^2 := by |
| calc |
| P.card ≤ (univ : Finset (Fin ℓ × Fin ℓ)).card := card_filter_le _ _ |
| _ = ℓ^2 := by simp [pow_two] |
| calc |
| halaszBadLongCount (k := k) (N := N) L ≤ (P.biUnion S).card := card_le_card hsub |
| _ ≤ ∑ ij∈P, (S ij).card := card_biUnion_le |
| _ ≤ ∑ _ij∈P, halaszRepeatedCount n k N := by |
| apply sum_le_sum |
| intro ij hij |
| exact halasz_pair_collision_bound _ _ (fun h => (mem_filter.mp hij).2 (hL h)) |
| _ = P.card*halaszRepeatedCount n k N := by simp |
| _ ≤ ℓ^2*halaszRepeatedCount n k N := Nat.mul_le_mul_right _ hP |
| |
| lemma halasz_frequency_energy (n k N : ℕ) : |
| halaszFiberEnergy (univ : Finset (Fin n → Fin N)) (halaszVinogradovFrequency k) = |
| halaszVinogradovCount n k N := by |
| rw [← halasz_nat_power_energy n k N] |
| apply halasz_fiber_energy_congr |
| intro x _ y _ |
| constructor |
| · intro h |
| funext j |
| have hj := congrFun h j |
| simp only [halaszVinogradovFrequency,← Nat.cast_sum] at hj |
| exact_mod_cast hj |
| · intro h |
| funext j |
| simp only [halaszVinogradovFrequency,← Nat.cast_sum] |
| exact_mod_cast congrFun h j |
| |
| theorem halasz_long_collision_partition {n k N ℓ : ℕ} (L : Fin ℓ → Fin (n+2)) |
| (hL : Function.Injective L) : |
| halaszVinogradovCount (n+2) k N ≤ 2*(ℓ^2*halaszRepeatedCount n k N) + |
| halaszFiberEnergy (halaszGoodLongTuples (N := N) L) (halaszVinogradovFrequency k) := by |
| have h := halasz_energy_restriction (univ : Finset (Fin (n+2) → Fin N)) |
| (halaszGoodLongTuples (N := N) L) (halaszVinogradovFrequency k) |
| rw [halasz_frequency_energy] at h |
| apply h.trans |
| apply Nat.add_le_add_right |
| apply Nat.mul_le_mul_left 2 |
| convert halasz_bad_long_count_bound (k := k) (N := N) L hL using 1 |
| unfold halaszCrossCount halaszBadLongCount |
| congr 1 |
| ext xy |
| simp only [mem_filter,mem_product,mem_sdiff,mem_univ,true_and, |
| halaszGoodLongTuples,and_true] |
| |
| end TwoPointCorrelations |
| |
| end OAI |
| |
