to show that there is *no* constant $c > 0$, such that for any system of 2 equations in 4 variables, there are large sets $S$ with at least $c |S|^2$ solutions to the system. We will prove this fact in Section \[sec:generalisation\_to\_systems\].

Notation {#notation .unnumbered}
--------

As we have already noted, $T(S)$ will be the number of solutions to $a_1x_1 + a_2x_2 + a_3x_3$ in S. We can extend this by defining $T(S_1, S_2, S_3)$ to be the number of solutions to $a_1x_1 + a_2x_2 + a_3x_3$, where $x_i \in S_i$.

We will use the notation $a \cdot S$ to denote the set $\{ax, x \in S\}$.

We will also make frequent use of the Vinogradov notation $f \ll g$ to mean that $f = O(g)$. When the $\ll$ is subscripted, we allow the implicit constant to depend on the subscripts.

The author is supported by an EPSRC grant EP/N509711/1. The author would like to thank his supervisor, Ben Green, for his continued support and encouragement, and the anonymous referee for a thorough reading of a previous version of this paper.

This version of the paper replaces a previous version [@me]. The argument used to prove Theorem \[thm:main\_theorem\] is 