simple real factor of resolution $M$, at most $1+N/M$ upper endpoints lie in $[1, N)$. Hence the number of intervals in the join of $d$ simple real factors of resolutions $M_1$, …, $M_d$ is at most $2d + N(M_1^{-1} + \dots + M_d^{-1})$.

We now prove a weak regularity lemma for the cut norm via an energy increment argument.

\[weak regularity\] Let $q {\leqslant}N$ be positive integers and $\delta >0$. Either $N \ll (q/\delta)^{O(1)}$, or for any function $f : [N] \to [0, 1]$ there exists a local factor $\mathcal{B}$ of dimension $d \ll \delta^{-O(1)}$, resolution $\gg (\delta/q)^{O(1)}N^{1/2}$, and modulus $qq'$ for some $q'{\leqslant}O{\left( 1/ \delta \right)}^{O(d)}$ such that $$\label{proj error}
{\left\| f - \Pi_\mathcal{B} f\right\|}_{q, N} {\leqslant}\delta$$

We run an energy increment argument, initialising at stage $0$ with the trivial factor $\mathcal{B}_0 := {\left\{[N]\right\}}$. Suppose that at stage $d$ of this iteration we have a local factor $\mathcal{B}$ of resolution $ \gg (\delta/q)^{O(1)}N^{1/2}$, dimension at most $2d$, and modulus $qq'$ for some $q'{\leqslant}O(1/  \delta)^{O(d)}$. In addition, suppose that we have the energy lower bound $$\label{energy bound}
{\left\| \Pi_{\mathcal{B}} f\right\|}_{\ell^2}^2 \gg d\delta^{O(1)} N.$$

With these assumptions in place, we query if the following holds $$\label{query}
{\left\| f - \Pi_{\mathcal{B}}f\right\|}_{q, N} 