On Cartesian Products which Determine Few Distinct Distances

Every set of points $\mathcal{P}$ determines $\Omega(|$\mathcal{P}$| / \log |$\mathcal{P}$|)$ distances. A close version of this was initially conjectured by Erd\H{o}s 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 $\mathrm{\mid }A-A\mathrm{\mid }\ll \mathrm{\mid }A{\mathrm{\mid }}^{2-\frac{2}{7}}{\log }^{\frac{1}{7}}\mathrm{\mid }A\mathrm{\mid }$. This improves recent results of Hanson and Roche-Newton.