replaced with a new argument which avoids appealing to the arithmetic regularity lemma (and can handle a wider class of equations), and the results of Section \[sec:generalisation\_to\_systems\] are new to this version.

Additive Combinatorial Lemmas {#sec:import_lemmas}
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In this section, we will collect some lemmas that will be necessary for the proof of Theorem \[thm:main\_theorem\].

For any set $A \subseteq {\mathbb{Z}}$, let $\delta[A]$ be its growth under the differencing operator, $\frac{|A-A|}{|A|}$. If $A {\text{ and }}B$ are two sets of integers, let the *additive energy* between $A {\text{ and }}B, E(A, B)$, be defined by $$E(A,B) = \# \{(a_1,b_1,a_2,b_2) \in A \times B \times A \times B: a_1 + b_1 = a_2 + b_2\}.$$ It is easy to see that this satisfies the following inequalities:

\[eqn:bound\_energy\] E(A,B) &|A|\^2|B|\
E(A,B) &|A||B|\^2\
E(A,B) &|A|\^[3/2]{}|B|\^[3/2]{},

the third of which follows immediately from the first two.

We will require the following lemma, which states that, when two sets $A {\text{ and }}B$ have $\delta[A] {\text{ and }}\delta[B]$ small, and if $E(A, B)$ is large, then $|A - B|$ is also small.

\[lem:gs\_doubling\_lem\] Suppose that $A, B \subseteq {\mathbb{Z}}$ are sets with $E(A, B) {\geqslant}\eta |A|^{3/2}|B|^{3/2}$.

Then, $|A - B| {\leqslant}\frac{\delta[A]\delta[B]}{\eta} |A|^{1/2}|B|^{1/2}$.

We will also 