A family of four-variable expanders with quadratic growth

TL;DR

This paper establishes quadratic growth bounds for a family of four-variable polynomial expanders, showing that certain polynomial difference ratios grow at least quadratically with the size of the input set.

Contribution

It proves a quadratic lower bound for the size of a specific polynomial difference set under certain conditions, extending understanding of polynomial expanders.

Findings

The set X has size at least proportional to |A|^2.

The quadratic growth bound is tight for some polynomials.

Conditions on g(x,y) determine the expansion behavior.

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 $g(x,y)$.