Maximising the number of solutions to a linear equation in a set of integers

Given a linear equation of the form $a_1{x}_1+a_2{x}_2+a_3{x}_3=0$ with integer coefficients $a_i$, we are interested in maximising the number of solutions to this equation in a set $S\subseteq \mathbb{Z}$, for sets $S$ of a given size. We prove that, for any choice of constants $a_1,a_2$ and $a_3$, the maximum number of solutions is at least $(\frac{1}{12}+o(1))\mathrm{\mid }S{\mathrm{\mid }}^2$. Furthermore, we show that this is optimal, in the following sense. For any $\epsilon >0,$ there are choices of $a_1,a_2$ and $a_3,$ for which any large set $S$ of integers has at most $\left(\frac{1}{12} + \varepsilon\right)|S|^2$ solutions. For equations in $k\ge 3$ variables, we also show an analogous result. Set ${\sigma }_k={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(\frac{\sin \pi x}{\pi x}{)}^{k}dx\mathrm{.}$ Then, for any choice of constants $a_1,\dots ,a_k$, there are sets $S$ with at least $(\frac{{\sigma }_k}{k^{k-1}}+o(1))\mathrm{\mid }S{\mathrm{\mid }}^{k-1}$ solutions to $a_1x_1 + \dots + a_kx_k = 0$\mathrm{.}Moreover,therearechoicesofcoefficients$a_1, \dots, a_k$ for which any large set $S$ must have no more than $(\frac{\sigma_k}{k^{k-1}} + \varepsilon)|S|^{k-1}$solutions,forany$\varepsilon > 0$.