Every set of points $mathcal{P}$ determines $Omega(|mathcal{P}| / log |mathcal{P}|)$ distances. A close version of this was initially conjectured by Erd?s 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 $A imes A$ must satisfy $|A - A| ll |A|^{2-rac{2}{7}} log^{rac{1}{7}} |A|$. This improves recent results of Hanson and Roche-Newton.
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机译：每组点$ mathcal {p} $ commentines $ omega（| mathcal {p} | / log | mathcal {p} |）$距离。最初由ERD的一个接近版本在1946年被ERD猜想，而是通过Guth和Katz证明。我们展示了当近界限附近，一个点设置$ mathcal {p} $ a times a $必须满足$ | a - a | ll | a | ^ {2- frac {2} {7}} log ^ { frac {1} {7} | a | $。这改善了汉森和罗氏牛顿的最近结果。
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