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 $a\to \infty$ of ${\gamma }_{1,1,-a}$ is equal to precisely $1/5$.

31 pages