A family of four-variable expanders with quadratic growth

Abstract

We prove that if $g(x,y)$ is a polynomial of constant degree $d$ that $y_2-y_1$ does not divide $g(x_1,y_1)-g(x_2,y_2)$, then for any finite set $A \subset \mathbb{R}$ \[ |X| \gg_d |A|^2, \quad \text{where} \ X:=\left\{\frac{g(a_1,b_1)-g(a_2,b_2)}{b_2-b_1} :\, a_1,a_2,b_1,b_2 \in A \right\}. \] We will see this bound is also tight for some polynomial $g(x,y)$.