  Gowers and Wolf have conjectured that, given a set of linear forms
$\{\psi_i\}_{i=1}^t$ each mapping $\mathbb{Z}^D$ to $\mathbb{Z}$, if $s$ is an
integer such that the functions $\psi_1^{s+1},\ldots, \psi_t^{s+1}$ are
linearly independent, then averages of the form $\mathbb{E}_{\boldsymbol{x}}
\prod_{i=1}^t f(\psi_i(\boldsymbol{x}))$ may be controlled by the Gowers
$U^{s+1}$-norm of $f$. We prove (a stronger version of) this conjecture.
