that either $
|\Lambda_{q, N}(f, g_1, g_2)| {\geqslant}\delta
$ or $
|\Lambda_{q, N}(g_1,f, g_2)| {\geqslant}\delta.
$ In the latter case, we may apply [@PelusePrendivilleQuantitative Theorem 7.1] to deduce that there exist positive integers $q'\ll\delta^{-O(1)}$ and $N' \gg (\delta/q)^{O(1)}N^{1/2}$ such that $$\sum_{x}{\left| \sum_{y\in[N']}f(x+qq'y)\right|}\gg \delta^{O(1)}NN'.$$ In the former case, the reader may check that the argument of [@PelusePrendivilleQuantitative Theorem 7.1] delivers the same conclusion[^2].

To ease notation, write $Q := qq'$. Partitioning the integers into arithmetic progressions of length $N'$ and common difference $Q$ gives $$\begin{gathered}
\delta^{O(1)} NN'  \ll \sum_{z \in [N']}\sum_{u\in [Q]} \sum_{x\in{\mathbb{Z}}} {\left| \sum_{y \in [N']} f( Qz +QN'x +u + Qy)\right|}\\  {\leqslant}N'\max_z\sum_{u\in [Q]} \sum_{x\in {\mathbb{Z}}} {\left| \sum_{y \in [N']} f(Qz + QN'x +u  +Qy)\right|}.\end{gathered}$$ Defining $ \psi_z(u,x)$ to be the conjugate phase of the inner sum, we deduce the existence of $z$ for which $$\begin{aligned}
\delta^{O(1)} N  \ll \sum_{u \in [Q]}\sum_x\sum_{y \in [N']}f(Qz + QN'x +u  +Qy)\psi_z(u,x).\end{aligned}$$ The result follows on noting that every integer has a unique representation of the form $QN'x +u  +Qy$ with $u \in [Q]$, $x \in {\mathbb{Z}}$ and $y \in [N']$. Hence the map $$Qz + QN'x +u  +Qy \mapsto \psi_z(u,x)$$ is a local function of resolution $QN'$ and modulus $Q$.

Now we can 