bound follows for $\mathcal{B}'$, allowing us to proceed to the next stage of our iteration.

Since the function $f$ is $1$-bounded, the projection $\Pi_\mathcal{B}f$ is also 1-bounded, hence the energy is always bounded above by $N$. It follows that this energy increment must terminate at stage $d$ for some $d \ll \delta^{-O(1)}$, yielding the lemma.

The density increment lemma {#increment proof sec}
===========================

In this section we prove Lemma \[increment lemma\], modelling our argument on that given by Green and Tao [@GreenTaoNewII Corollary 5.8]. We first record, for the sake of convenience, the following immediate consequence of the triangle inequality.

\[L1 control\] Suppose that $N {\geqslant}q$. Then for any $f_0, f_1, f_2 : [N]\to {\mathbb{C}}$ we have $$|\Lambda_{q, N}(f_0, f_1, f_2)| {\leqslant}N^{-1}{\left\| f_i\right\|}_{\ell^1} \prod_{j \neq i} {\left\| f_j\right\|}_\infty .$$

We prove the result for $i = 1$, the other cases being similar. A reparametrisation gives $$\begin{aligned}
{\left| \Lambda_{q, N}(f_0, f_1, f_2)\right|} &= {\left|  {\mathbb{E}}_{x\in[N]} f_1(x) {\mathbb{E}}_{y \in [M]} f_0(x-y)f_2(x+qy^2 - y)\right|}\\
 & {\leqslant}{\mathbb{E}}_{x\in[N]} |f_1(x)| {\mathbb{E}}_{y \in [M]} |f_0(x-y)||f_2(x+qy^2 - y)|.\end{aligned}$$

We are now in a position to prove Lemma \[increment lemma\], and thereby complete our proof of Theorem \[main\].

Let $A$ satisfy the assumptions of Lemma \[increment lemma\]. Increasing $\delta$ 