On Cartesian Products which Determine Few Distinct Distances
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 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 …