N}(f, g_1,g_2)|,\ |\Lambda_{q, N}(g_1, f,g_2)|,\ |\Lambda_{q, N}(g_1,g_2, f)|\},$$ where the supremum is taken over all 1-bounded functions $g_i : [N] \to {\mathbb{C}}$. We note that, in spite of our nomenclature, this is not a norm, but a seminorm. One could remedy this by summing over $y {\geqslant}0$ in the counting operator

Initially, the cut norm is too restrictive for us, so we begin by working with the weaker quantity $$\label{flat q norm eq}
{\left\| f\right\|}^\flat_{q,N} := \sup\{|\Lambda_{q, N}(f, g_1,g_2)|, |\Lambda_{q, N}(g_1, f,g_2)| :  |g_i| {\leqslant}1 \text{ and } {\mathrm{supp}}(g_i) \subset [N] \},$$ which we refer to as the *partial cut norm*.

The following lemma is simply a rephrasing of [@PelusePrendivilleQuantitative Theorem 7.1], which is the technical heart of that paper. See Definition \[local function def\] for the meaning of ‘local function’.

\[partial inverse theorem\] Let $q {\leqslant}N$ be positive integers, $\delta>0$, and $f:{\mathbb{Z}}\to{\mathbb{C}}$ be a $1$-bounded function with support in $[N]$. Suppose that $${\left\| f\right\|}^\flat_{q,N}{\geqslant}\delta .$$ Then either $N \ll (q/\delta)^{O(1)}$ or there exists a 1-bounded local function $\phi$ of resolution $\gg (\delta/q)^{O(1)}N^{1/2}$, modulus $qq'$ for some $q'\ll  \delta^{-O(1)}$, and such that $$\sum_{x\in[N]} f(x)\phi(x) \gg \delta^{O(1)} N.$$

By compactness, there exist 1-bounded functions $g_1, g_2 : [N] \to {\mathbb{C}}$ such 