its dual seminorm $\|\cdot\|^*$ is defined by $$\label{dual norm}
\|f\|^{*}:=\sup_{\|g\|{\leqslant}1}|\langle f,g\rangle|.$$ Hence, $$\label{dual ineq}
{\left| {\left\langlef,g\right\rangle}\right|} {\leqslant}{\left\| f\right\|}^* {\left\| g\right\|}.$$

For a complex-valued function $f$ and positive-valued function $g$, write $f \ll g$ or $f = O(g)$ if there exists a constant $C$ such that $|f(x)| \le C g(x)$ for all $x$. We write $f = \Omega(g)$ if $f \gg g$.

### Local conventions

Up to normalisation, all of the above are well-used in the literature. Next, we list notation specific to our paper. We have tried to minimise this in order to aid the casual reader.

The quantity $(N/q)^{1/2}$ appears repeatedly, where $N$ and $q$ are integers fixed throughout the majority of our paper. We therefore adopt the convention that $$\label{M def}
M:= {\left\lfloor \sqrt{N/q} \right\rfloor}.$$ Assuming this, define the *counting operator* on the functions $f_i : {\mathbb{Z}}\to {\mathbb{C}}$ by $$\label{counting op}
\Lambda_{q, N}(f_0, f_1, f_2) := {\mathbb{E}}_{x \in [N]} {\mathbb{E}}_{y \in [M]} f_0(x)f_1(x+y) f_2(x+qy^2).$$ When the $f_0=f_1=f_2=f$, we simply write $\Lambda_{q, N}(f)$ for $\Lambda_{q,N}(f_0,f_1,f_2)$.

For a real parameter $H {\geqslant}1$, we use $\mu_H : {\mathbb{Z}}\to [0,1]$ to represent the following normalised Fejér kernel $$\label{fejer}
\mu_H(h) := {\frac{1}{{\left\lfloor H \right\rfloor}}} {\left( 1 - \frac{|h|}{{\left\lfloor H \right\rfloor}} \right)}_+ = \frac{(1_{[H]} 