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8769036 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 | 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
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