which states that $$T(S_1, S_2, S_3) {\leqslant}\frac{2(s_1s_2 + s_2s_3 + s_3s_1) - (s_1^2 + s_2^2 + s_3^2) + 1}{4}.$$ (\[eqn:uniform\_bound\]) follows in this case via an easy application of the Cauchy-Schwarz inequality.

Finally, for arbitrary coefficients $a_1, a_2 {\text{ and }}a_3$, observe that $$\begin{aligned}
T_{a_1, a_2, a_3}(S_1, S_2, S_3) &= T_{1,1,1}(a_1 \cdot S_1, a_2 \cdot S_2, a_3 \cdot S_3) \\
&{\leqslant}\frac{1}{4}\left(s_1s_2 + s_2s_3 + s_1s_3 + 1\right).\end{aligned}$$ This completes the proof of Lemma \[lem:uniform\_bound\_symmetric\].

Finally, we will require the following theorem of Bukh:

\[thm:bukh\_sod\_thm\] Given two coprime integers $\lambda_1 {\text{ and }}\lambda_2$, we have that for any $S \subseteq {\mathbb{Z}}$, $$|\lambda_1 \cdot S + \lambda_2 \cdot S| {\geqslant}(|\lambda_1| + |\lambda_2|)|S| - o_{\lambda_1, \lambda_2}(|S|)$$

Proof of Theorem \[thm:main\_theorem\] {#sec:proof_of_thm}
======================================

In this section, we will use the lemmas of Section \[sec:import\_lemmas\] to prove Theorem \[thm:main\_theorem\]. We must prove that, given a suitable choice of $a_1, a_2 {\text{ and }}a_3$, all sufficiently large sets $S$ have $T(S) {\leqslant}\left(\frac{1}{12} + {\varepsilon}\right)|S|^2$.

Let ${\varepsilon}> 0$. Given our choice of ${\varepsilon}$, we must choose the values of the coefficients $a_1, a_2 {\text{ and }}a_3$; we will do so later. Suppose that $S$ is a sufficiently large set. We will immediately apply the structure theorem, Theorem 