start by restricting which triples $(i, j, k)$ can have the property that there are many solutions with $x_1 \in S_i, x_2 \in S_j {\text{ and }}x_3 \in S_k$. For instance, the fact that $\delta[S_1]$ is small, together with an assumption that $a_1$ and $a_2$ are coprime and $|a_1 + a_2|$ is large, will imply that there cannot be too many solutions with $x_1, x_2 {\text{ and }}x_3$ all in $S_1$.

In particular, this will give us a fairly rigid structure on the collection of triples $S_i, S_j, S_k$ such that $T(S_i, S_j, S_k)$ can give a non-trivial contribution to $T(S, S, S)$. In order to quantify this structure, we will draw a labelled digraph $G$ whose vertices correspond to the $S_i$ with $i {\geqslant}1$. We will draw an edge from $S_i$ to $S_j$ with label $\frac{a_1}{a_2}$ if and only if $T(S_i, S_j, S) {\geqslant}\frac{{\varepsilon}}{24 K_1^2} |S|^2$, where $K_1$ is as in the statement of Theorem \[thm:green\_sisask\_structure\]. Similarly, we will draw an edge with label $\frac{a_3}{a_2}$ if $T(S, S_j, S_i) {\geqslant}\frac{{\varepsilon}}{24 K_1^2} |S|^2$, and similarly for the other four possible labels.

In particular, observe that if there is an edge from $S_i$ to $S_j$ with label $x$, then there will 