have $$\begin{aligned}
\gamma_{1,1,\pm 1} &= \frac{3}{4} \label{eqn:trivial_case}\\ 
\gamma_{1,-2,\pm 1} &= \frac{1}{2} \label{eqn:abc_is_2}\end{aligned}$$ $\gamma_{1,-2,1}$ is [@GreenSisask Theorem 1.2], and $\gamma_{1,-2,-1} = \frac{1}{2}$ is [@LevPinchasi Theorem 2]. The same holds in the third non-equivalent case with $|a_1a_2a_3| = 2$, namely $\gamma_{1, 2, 1} = \frac{1}{2}$. Even the value of $\gamma_{1,2,-3}$ is not known, although the author conjectures that it is $\frac{1}{3}$, which is the value calculated for $S = [-M/2, M/2]$.

The main theorem of this paper is a converse, of sorts, to Proposition \[prop:twelfth\]. In particular, we will prove the following.

\[thm:main\_theorem\] The constant $\frac{1}{12}$ in the statement of Proposition \[prop:twelfth\] is optimal, in the following sense. For any ${\varepsilon}> 0$, there exists a choice of $a_1, a_2 {\text{ and }}a_3$ for which $\gamma_{a_1, a_2, a_3} {\leqslant}\frac{1}{12} + {\varepsilon}$.

In view of this theorem, (\[eqn:gamma\_uniform\_bound\]) gives the best possible bounds on $\gamma_{a_1, a_2, a_3}$ that are independent of the coefficients $a_i$.

The plan for this paper is as follows. In Section \[sec:import\_lemmas\], we will record some additive combinatorial lemmas that we will need in order to establish Theorem \[thm:main\_theorem\]. In Section \[sec:proof\_of\_thm\], we will use these lemmas to prove Theorem \[thm:main\_theorem\].

One might also ask about generalising Theorem \[thm:main\_theorem\] to other 