  A very useful fact in additive combinatorics is that analytic expressions
that can be used to count the number of structures of various kinds in subsets
of Abelian groups are robust under quasirandom perturbations, and moreover that
quasirandomness can often be measured by means of certain easily described
norms, known as uniformity norms. However, determining which uniformity norms
work for which structures turns out to be a surprisingly hard question. In
[GW09a] and [GW09b, GW09c] we gave a complete answer to this question for
groups of the form $G=\mathbb{F}_p^n$, provided $p$ is not too small. In
$\mathbb{Z}_N$, substantial extra difficulties arise, of which the most
important is that an "inverse theorem" even for the uniformity norm
$\|.\|_{U^3}$ requires a more sophisticated (local) formulation. When $N$ is
prime, $\mathbb{Z}_N$ is not rich in subgroups, so one must use regular Bohr
neighbourhoods instead. In this paper, we prove the first non-trivial case of
the main conjecture from [GW09a].
