On Cartesian Products which Determine Few Distinct Distances

Abstract

Every set of points $\mathcal{P}$ determines $\Omega (|\mathcal{P}|/\log |\mathcal{P}|)$ distances. A close version of this was initially conjectured by Erdős in 1946 and rather recently proved by Guth and Katz. We show that when near this lower bound, a point set $\mathcal{P}$ of the form $A\times A$ must satisfy $|A-A|\ll |A{|}^{2-\frac{2}{7}}{\log }^{\frac{1}{7}}|A|$. This improves recent results of Hanson and Roche-Newton.