The Additive Structure of Cartesian Products Spanning Few Distinct   Distances
  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 $|A-A|\ll |A|^{2-1/8}$. 
Brandon Hanson
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