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

TL;DR

This paper determines the maximum number of solutions to linear equations in integer sets of a given size, establishing exact bounds and optimal constructions for equations with three or more variables.

Contribution

It provides tight bounds on the maximum number of solutions for linear equations in integer sets, including explicit constructions and optimality results for equations with three or more variables.

Findings

Maximum solutions for 3-variable equations is approximately |S|^2/12.

Optimal bounds are achieved by specific coefficient choices.

Generalization to k-variable equations with explicit solution bounds.

Abstract

Given a linear equation of the form $a_1x_1 + a_2x_2 + a_3x_3 = 0$ 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 $\left(\frac{1}{12} + o(1)\right)|S|^2$. Furthermore, we show that this is optimal, in the following sense. For any $\varepsilon > 0,$ 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 $k \geq 3$ variables, we also show an analogous result. Set $\sigma_k = \int_{-\infty}^{\infty} (\frac{\sin \pi x}{\pi x})^k dx.$ Then, for any choice of constants $a_1, \dots, a_k$, there are sets $S$ with at least $(\frac{\sigma_k}{k^{k-1}} + o(1))|S|^{k-1}$ solutions to $a_1x_1 + \dots + a_kx_k = 0$. Moreover, there are choices of coefficients $a_1, \dots, a_k$ for which any large set $S$ must have no more than $(\frac{\sigma_k}{k^{k-1}} + \varepsilon)|S|^{k-1}$ solutions, for any $\varepsilon > 0$.