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 whennear to this lower bound, a point set $\mathcal P$ of the form $A\times A$ mustsatisfy $|A-A|\ll |A|^{2-1/8}$.
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机译：Guth和Katz证明，平面中的任何点集$ \ mathcal P $都会确定$ \ Omega（| \ mathcal P | / \ log | \ mathcal P |）$的不同距离。我们表明，当接近该下限时，点集$ \数学P $的形式为$ A \ times A $必须满足$ | A-A | \ ll | A | ^ {2-1 / 8} $。
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