Linear forms condition says that these functions are morally the functions that are close to $1$ in appropriate $U^k$ norm. What I mean is that $U^k$ norms are a special kind of linear forms, and so linear forms condition implies proximity to $1$ in $U^k$, on one hand. On the other hand,if one controls $\nu-1$ in $U^t$ norm for sufficiently large $t=t(k)$, then by Cauchy-Schwarz argument one can control arbitrary linear forms.
[EDIT: The rest of the answer is result of my misunderstanding. See the comments.] There is an unpublished work of David Conlon and Timothy Gowers on Szemerédi-type results in random sets, in which, if I understood correctly what David explained to me, they show as a special case that the control in an appropriate $U^t$ norm is enough. (In particular the correlation condition is no longer necessary, and was an artifact of the original proof.)
So, the answer to your question is that the theory of these functions is essentially the theory of functions with small $U^k$ norm.
