Arithmetic progressions with a pseudorandom step

Let α, σ > 0 and let A and S be subsets of a finite abelian group G of densities α and σ independent of |G|, respectively. Without additional restrictions A need not contain a 3-term arithmetic progression whose common gap is in S. What is then the least integer k ≥ 2 for which there exists an η = η(α, σ) such that ‖S‖Uk(G) ≤ η implies that A contains a non-trivial 3-term arithmetic progression with a common gap in S? For G = Zn (n sufficiently large and odd) we show that k = 3, while for G = Fp (p an odd prime and n sufficiently large) we show that k = 2.