Lemma 2.18. Assuming that:

  • p≥3 prime

  • A⊆𝔽pn of density α>0

  • sup⁡t≠0|𝟙A^(t)|≤𝜀

Then the number of 3-term arithmetic progressions in A differs from α3(pn)2 by at most 𝜀(pn)2.