| import OAI.NumberTheory.TwoPoint.Halasz.HalaszCollisionPartition |
| import OAI.NumberTheory.TwoPoint.Halasz.HalaszZeroRepresentation |
|
|
| namespace OAI |
|
|
| /-! Counting allowed differences by the zero-representation inequality. -/ |
| namespace TwoPointCorrelations |
|
|
| open Finset |
| open scoped Classical |
|
|
| lemma halasz_difference_set_count {ι : Type*} [Fintype ι] {k : ℕ} |
| (f : ι → Fin k → ℤ) (W : Finset (Fin k → ℤ)) : |
| (univ.filter (fun xy : ι×ι => f xy.1-f xy.2∈W)).card≤ |
| W.card*halaszRepresentationCount f 0 := by |
| have hsub : univ.filter (fun xy : ι×ι => f xy.1-f xy.2∈W) ⊆ |
| W.biUnion (fun w => univ.filter (fun xy : ι×ι => f xy.1-f xy.2=w)) := by |
| intro xy hxy |
| exact mem_biUnion.mpr ⟨f xy.1-f xy.2,(mem_filter.mp hxy).2, |
| mem_filter.mpr ⟨mem_univ _,rfl⟩⟩ |
| calc |
| _ ≤ _ := card_le_card hsub |
| _ ≤ ∑ w∈W,(univ.filter (fun xy : ι×ι => f xy.1-f xy.2=w)).card := card_biUnion_le |
| _ = ∑ w∈W,halaszRepresentationCount f w := rfl |
| _ ≤ ∑ _w∈W,halaszRepresentationCount f 0 := |
| sum_le_sum (fun w _ => halasz_zero_representation_dominates f w) |
| _ = _ := by simp |
|
|
| lemma halasz_vinogradov_difference_count (s k M : ℕ) (W : Finset (Fin k → ℤ)) : |
| (univ.filter (fun xy : (Fin s → Fin M)×(Fin s → Fin M) => |
| halaszVinogradovFrequency k xy.1-halaszVinogradovFrequency k xy.2∈W)).card≤ |
| W.card*halaszVinogradovCount s k M := by |
| have h := halasz_difference_set_count (halaszVinogradovFrequency (s := s) (N := M) k) W |
| rw [← halasz_fiber_energy_representation,halasz_frequency_energy] at h |
| exact h |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
