sets $S$ with as many affine copies of $\{0, 1, 3\}$, or solutions to $x + 2y = 3z$, as possible. In this case, such a result would necessarily be less clean; for instance, there are more solutions to $x + 2y = 3z$ in $\{0,1,3\}$ than in $\{0,1,2\}$.

Indeed, in general, much less is known. For a lower bound on the maximal value of $T(S)$, a fairly good bound is given by the following example.

\[prop:twelfth\] Regardless of the values of $a_1, a_2 {\text{ and }}a_3$, there are choices of $S$ with $|S|$ arbitrarily large, for which $T(S) {\geqslant}\frac{1}{12}|S|^2 + O(|S|)$.

The idea behind the construction is to split $S$ into three pieces $S_1, S_2 {\text{ and }}S_3$, of roughly equal size, for which there are many solutions to $a_1x_1 + a_2x_2 + a_3x_3 = 0$ with each $x_i$ taken from $S_i$. Let $M$ be a large integer, which we assume to be divisible by 6. We will define $$\begin{aligned}
S_1 &= a_2a_3[-M/6, M/6] \\
S_2 &= a_1a_3[-M/6, M/6] \\
S_3 &= a_1a_2[-M/6, M/6], \end{aligned}$$ where $[-M/6, M/6]$ refers to the set of integers with absolute value no greater than $M/6$, and set $S = S_1 \cup S_2 \cup 