The Additive Structure of Cartesian Products Spanning Few Distinct Distances

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

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