linear equation in a set of integers
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Introduction {#sec:intro}
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Let $a_1, a_2 {\text{ and }}a_3$ be fixed coprime integers, none of which is zero. We will consider the linear equation $$\label{eqn:to_solve}
a_1x_1 + a_2x_2 + a_3x_3 = 0.$$ In this paper, we are interested in the problem of finding sets with as many solutions to (\[eqn:to\_solve\]) as possible. This leads to the following definition.

Given a finite set $S {\subseteq}{\mathbb{Z}}$, define $T(S) = T_{a_1, a_2, a_3}(S)$ to be the number of triples $x_1, x_2, x_3 \in S$ satisfying (\[eqn:to\_solve\]).

The trivial upper bound on $T(S)$ is $T(S) {\leqslant}|S|^2$. This is because, for any choice of $x_1 {\text{ and }}x_2$, there is at most one choice of $x_3$ such that $a_1x_1 + a_2x_2 + a_3x_3 = 0$, namely $x_3 = \frac{-a_1x_1-a_2x_2}{a_3}$. We are interested in making $T(S)$ as large as possible, for a fixed size $|S|$.

For some choices of coefficients $a_1, a_2 {\text{ and }}a_3$, the exact maximal value of $T(S)$ is known. For example, consider the case $a_1 = a_2 = a_3 = 1$. Then, work of Hardy and Littlewood [@HLIneqNotes] and Gabriel [@Gabriel] shows that, when $|S|$ is odd, $T(S)$ is maximised when $S$ is an 