settings. For instance, given a system of $m$ linear equations in $k$ variables (where we assume that $m {\leqslant}k - 2$), can we prove an analogue of Theorem \[thm:main\_theorem\]?

If $m = 1$, then an analogue of Proposition \[prop:twelfth\] holds for any value of $k {\geqslant}3$. Set $$\label{eqn:definition_of_sigma}
\sigma_k = \int_{-\infty}^{\infty} \left(\frac{\sin \pi x}{\pi x} \right)^k dx.$$ Then, for any choice of coefficients $a_1, \dots, a_k$, there are sets $S$ with at least $\frac{\sigma_k}{k^{k-1}}|S|^{k-1} + O(|S|^{k-2})$ solutions to $a_1x_1 + \dots + a_kx_k = 0$. We will discuss (\[eqn:definition\_of\_sigma\]) further in Section \[sec:generalisation\_to\_k\_var\].

Furthermore, the corresponding analogue of Theorem \[thm:main\_theorem\] holds. For any ${\varepsilon}> 0$, there are choices of coefficients $a_1, \dots, a_k$ for which any large set $S$ must have no more than $\left(\frac{\sigma_k}{k^{k-1}} + \varepsilon\right)|S|^{k-1}$ solutions. For instance, for any small positive ${\varepsilon}$ we can find coefficients $a_1, a_2, a_3 {\text{ and }}a_4$ with the property that $
T(S) {\leqslant}\left( \frac{1}{96} + {\varepsilon}\right) |S|^3,
$ where $T(S)$ counts the number of solutions to $a_1x_1 + a_2x_2 + a_3x_3 + a_4x_4 = 0.$ We will discuss this in Section \[sec:generalisation\_to\_k\_var\].

On the other hand, the opposite is true in the case that $m > 1$. Indeed, it is possible 