On a conjecture of Gowers and Wolf

Gowers and Wolf have conjectured that, given a set of linear forms $\{ψ_i\}_{i=1}^t$ each mapping $\mathbb{Z}^D$ to $\mathbb{Z}$, if $s$ is an integer such that the functions $ψ_1^{s+1},\ldots, ψ_t^{s+1}$ are linearly independent, then averages of the form $\mathbb{E}_{\boldsymbol{x}} \prod_{i=1}^t f(ψ_i(\boldsymbol{x}))$ may be controlled by the Gowers $U^{s+1}$-norm of $f$. We prove (a stronger version of) this conjecture.

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