Expanding polynomials on sets with few products

Abstract

In this note, we prove that if $A$ is a finite set of real numbers such that $|AA|=K|A|$, then for every polynomial $f\in \mathbb{R}[x,y]$ we have that $|f(A,A)|={\Omega }_{K,\mathrm{deg}f}(|A{|}^2)$, unless $f$ is of the form $f(x,y)=g(M(x,y))$ for some monomial $M$ and some univariate polynomial $g$.