Arithmetic progressions with a pseudorandom step

Abstract

Let α,σ>0 and let A and S be subsets of a finite abelian group G of densities α and σ, respectively, both independent of |G|. Without any additional restrictions, the set A need not contain a 3-term arithmetic progression whose common gap lies in S. What is then the weakest pseudorandomness assumption that if put on S would imply that A contains such a pattern?More precisely, what is the least integer k≥2 for which there exists an η=η(α,σ) such that ‖S−σ‖Uk(G)≤η implies that A contains a non-trivial 3-term arithmetic progression with a common gap in S? Here, ‖⋅‖Uk(G) denotes the kth Gowers norm.For G=Zn we observe that k must be at least 3. However for G=Fpn we show that k=2 is sufficient, where here p is an odd prime and n is sufficiently large.