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

Abstract

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 $\left(\frac{1}{12}+o(1)\right)|S{|}^2$. 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 $\left(\frac{1}{12}+\epsilon \right)|S{|}^2$ solutions. For equations in $k\ge 3$ 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 $a_1{x}_1+\cdots +a_k{x}_k=0$. Moreover, there are choices of coefficients $a_1,\dots ,a_k$ for which any large set $S$ must have no more than $(\frac{{\sigma }_k}{k^{k-1}}+\epsilon )|S{|}^{k-1}$ solutions, for any $\epsilon >0$.

18 pages. Corrections in light of comments from referee

References in corpus (2)

• Upper tails for arithmetic progressions in a random set
• Maximising the number of solutions to a linear equation in a set of integers

Cited by in corpus (2)

• Maximising the number of solutions to a linear equation in a set of integers
• The maximal number of $3$-term arithmetic progressions in finite sets in different geometries