Here we use the notion of pseudo-randomness (or variant of) used in the proof of the Green-Tao theorem. In particular, we say a measure $\nu : \mathbb{Z}_N \rightarrow \mathbb{R}^+$ is pseudo-random if $\mathbb{E}(\nu) = 1 + o(1)$, and $\nu$ satisfies a suitable linear forms condition and a correlation relation (see: http://arxiv.org/abs/math/0404188). In the proof of the Green-Tao theorem, they proved that the measure defined by definition 9.3 is in fact pseudo-random and thus, allows one to conclude their main theorem which implies that the primes contain arbitrarily long arithmetic progressions.
My question is, has there been any other interesting examples of pseudo-random measures? I am particularly interested when the underlying set being considered is not concrete like the primes. In particular, I am interested in sets $A$ in some additive group such that the set $kA$ is large.
