Three-variable expanding polynomials and higher-dimensional distinct distances

Abstract

We determine which quadratic polynomials in three variables are expanders over an arbitrary field $\mathbb{F}$. More precisely, we prove that for a quadratic polynomial $f\in \mathbb{F}[x,y,z]$, which is not of the form $g(h(x)+k(y)+l(z))$, we have $|f(A\times B\times C)|\gg N^{3/2}$ for any sets $A,B,C\subset \mathbb{F}$ with $|A|=|B|=|C|=N$, with $N$ not too large compared to the characteristic of $\mathbb{F}$. We give several applications. We use this result for $f=(x-y)^2+z$ to obtain new lower bounds on $|A+A^2|$ and $\max\{|A+A|,|A^2+A^2|\}$, and to prove that a Cartesian product $A\times\cdots \times A\subset \mathbb{F}^d$ determines almost $|A|^2$ distinct distances if $|A|$ is not too large.