On the Erdos distinct distance problem in the plane
On the Erdos distinct distance problem in the plane
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发表时间：
2010-11
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通讯作者：
L. Guth;N. Katz
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作者：
L. Guth;N. Katz
在本文中，我们证明了${\bfR}^2 $中的一组$N $点至少有$c {N\over\log N}$不同的距离，从而获得了Erd\"os问题的尖指数。我们遵循Elekes和Sharir的设置，本着埃尔兰根计划的精神，使我们能够在平面的刚体运动组中研究这个问题。这将问题转化为空间中的点-线关联问题。我们在证明中引入了两个新的概念。为了控制多条直线相交的点，我们使用多项式火腿三明治定理创建了一个细胞分解。这就形成了一个二分法：或者大多数点都在细胞的内部，在这种情况下，我们立即得到尖锐的结果，或者这些点位于细胞的壁上，在这种情况下，它们位于一个非常低次数的多项式的零集合中，我们可以应用代数方法。为了控制只有两条线是入射的点，我们使用牧师乔治鲑鱼的拐点多项式得出结论，大多数的线位于直纹表面。然后利用直纹曲面的几何学来完成证明。
In this paper, we prove that a set of $N$ points in ${\bf R}^2$ has at least $c{N \over \log N}$ distinct distances, thus obtaining the sharp exponent in a problem of Erd\"os. We follow the set-up of Elekes and Sharir which, in the spirit of the Erlangen program, allows us to study the problem in the group of rigid motions of the plane. This converts the problem to one of point-line incidences in space. We introduce two new ideas in our proof. In order to control points where many lines are incident, we create a cell decompostion using the polynomial ham sandwich theorem. This creates a dichotomy: either most of the points are in the interiors of the cells, in which case we immediately get sharp results, or alternatively the points lie on the walls of the cells, in which case they are in the zero set of a polynomial of suprisingly low degree, and we may apply the algebraic method. In order to control points where only two lines are incident, we use the flecnode polynomial of the Rev. George Salmon to conclude that most of the lines lie on a ruled surface. Then we use the geometry of ruled surfaces to complete the proof.