A family of four-variable expanders with quadratic growth

Abstract

We prove that if $g\left(x,y\right)$ is a polynomial of degree $d$ that is not a polynomial of only $y$, then for any finite set $A\subset ℝ$

$$\left|X\right|{\gg }_d|A{|}^2,\text{ where }X:=\left\{\frac{g\left(a_1,b_1\right)-g\left(a_2,b_2\right)}{b_2-b_1}:a_1,a_2,b_1,b_2\in A\right\}.$$

We will see this bound is also tight for some polynomial $g\left(x,y\right)$.