prove an inverse theorem for the cut norm itself.

\[inverse theorem\] 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 $${\left\| f\right\|}_{q,N}{\geqslant}\delta .$$ Then either $N \ll (q/\delta)^{O(1)}$ or there exist 1-bounded local functions $\phi_1$ and $\phi_2$, of resolution $\gg (\delta/q)^{O(1)}N^{1/2}$ and moduli $qq_1$ and $qq_2$, respectively, for some $q_1,q_2\ll  \delta^{-O(1)}$ such that $$\label{cut norm density increment}
{\left| \sum_{x\in[N]} f(x)\phi_1(x)\phi_2(x)\right|} \gg \delta^{O(1)} N.$$

A key tool in proving Lemma \[inverse theorem\] is the following decomposition result, which relies on the finite-dimensional Hahn–Banach theorem.

\[HB decomp\] Let $\|\cdot\|$ be a seminorm on the space of complex-valued functions with support in $[N]$. For any such function $f$ and ${\varepsilon}> 0$, there exists a decomposition $f=f_{str}+f_{unf}$ such that $${\left\| f_{str}\right\|}^* {\leqslant}{\varepsilon}^{-1}{\left\| f\right\|}_{2} \quad \text{and} \quad {\left\| f_{unf}\right\|}{\leqslant}{\varepsilon}{\left\| f\right\|}_{2}.$$

This can be found in the discussion following [@GowersDecompositions Proposition 3.6]. Although the statement therein is for norms, and not seminorms, one can check that the (simple) argument remains valid in this greater generality[^3].

Using the dual norm decomposition afforded by Lemma \[HB decomp\], we can gain control of every function in the counting operator.

By the definition of the cut norm and Lemma \[partial 