require a weak form of a structure theorem due to Green and Sisask.

\[thm:green\_sisask\_structure\] Let ${\varepsilon}_1 \in (0,1/2)$ be a parameter. Then there are choices of (large) integers $K_1 = K_1({\varepsilon}_1) {\text{ and }}K_2 = K_2({\varepsilon}_1)$ with the following property. For any set $S \subseteq {\mathbb{Z}}$, there is a decomposition of $S$ as a disjoint union $S_1 \amalg \dots \amalg S_n \amalg S_0$ such that

1.  $|S_i| {\geqslant}|S|/K_1$ for $i=1,\dots,n$;

2.  $\delta[S_i] {\leqslant}K_2$ for $i = 1,\dots,n$;

3.  $E(S_0,S) {\leqslant}{\varepsilon}_1 |S|^3$.

Observe that property (1) guarantees that $n {\leqslant}K_1$.

The quantity $T(S_1, S_2, S_3)$ is related to the additive energy via the following lemma.

\[lem:easy\_TNRG\_lemma\] Suppose that $S_1, S_2 {\text{ and }}S_3 \subseteq {\mathbb{Z}}$ are finite sets. Then $$T(S_1, S_2, S_3)^2 {\leqslant}E(a_1\cdot S_1, a_2\cdot S_2)|S_3|.$$

For any $t \in {\mathbb{Z}}$, let $\mu(t)$ denote the number of ways of writing $t = a_1x_1 + a_2x_2$, for $x_i \in S_i$. Thus, by definition, $$E(a_1\cdot S_1, a_2\cdot S_2) = \sum_{t} \mu(t)^2.$$ Now, we see that $$T(S_1, S_2, S_3)^2 = \left(\sum_{t \in -a_3 \cdot S_3} \mu(t)\right)^2 {\leqslant}\left(\sum_t \mu(t)^2\right)|S_3| = E(a_1\cdot S_1, a_2\cdot S_2) |S_3|,$$ the inequality following from Cauchy-Schwarz. This completes the proof of Lemma \[lem:easy\_TNRG\_lemma\].

The following two facts are standard results in 