Maximising the number of solutions to linear equations

Abstract

We study the asymptotically maximal possible number of integer solutions to the linear equation $ax+by+cz = 0$ with a fixed choice of $a, b, c \in \mathbb{Z}$ and variables $x, y, z \in S$ for some finite set $S\subset \mathbb{Z}$, as $|S|\to +\infty$. Define $\gamma_{a, b, c}$ to be the largest constant for which there are arbitrary large finite sets $S\subset \mathbb{Z}$ such that the number of solutions to $ax+by+cz=0$ with $x,y,z\in S$ is $\gamma_{a,b,c}|S|^2-o(|S|^2)$. We prove structural results for general $a, b, c$ and moreover, we show that $5/13\le \gamma_{1,1,-3}\le 1/2-\delta$ for some constant $\delta>0$. In addition we show that the limit as $a \rightarrow \infty$ of $\gamma_{1,1,-a}$ is equal to precisely $1/5$.