  Guth and Katz proved that any point set $\mathcal P$ in the plane determines
$\Omega(|\mathcal P|/\log|\mathcal P|)$ distinct distances. We show that when
near to this lower bound, a point set $\mathcal P$ of the form $A\times A$ must
satisfy $|A-A|\ll |A|^{2-1/8}$.
