in more than 3 variables {#sec:generalisation_to_k_var}
==================================

A fairly natural extension of Theorem \[thm:main\_theorem\] is to ask if a similar result holds for $k$-variable equations $$\label{eqn:to_solve_k_var}
a_1x_1 + \dots + a_kx_k = 0.$$ As before, let $T(S)$ be the number of solutions to (\[eqn:to\_solve\_k\_var\]) in $S$. Similarly, let $T(S_1, \dots, S_k)$ denote the number of solutions with $x_i$ taken from $S_i$. We have a trivial upper bound for $T(S)$, namely that $T(S) {\leqslant}|S|^{k-1}$.

Before presenting our analogous example to Proposition \[prop:twelfth\], we require some notation and definitions. Let $I_x : {\mathbb{R}}\rightarrow {\mathbb{R}}$ denote the indicator function of a (real) interval of length $x$ centred at the origin, so $I_x(y) = 1$ if and only if $|y| {\leqslant}\frac{x}{2}$, and $I_x(y) = 0$ otherwise.

\[def:defn\_of\_sigma\] For an integer $k {\geqslant}3$, define $$\label{eqn:interval_formula_for_sigma}
\sigma_k = (\underbrace{I_1 \ast \dots \ast I_1}_k)(0).$$

In the introduction, we gave the following formula for $\sigma_k$: $$\tag{\ref{eqn:definition_of_sigma}}
\sigma_k = \int_{-\infty}^{\infty} \left(\frac{\sin \pi x}{\pi x} \right)^k dx.$$ The equivalence of these forms follows from taking a Fourier transform and applying the convolution identity; the details can be seen in [@BorweinIntegral].

See also [@nathanson], where it can be shown that $\sigma_{2h}$ is the leading coefficient of the polynomial $\Psi_{h}(n)$.

$\sigma_k$ obeys 