关于定义少数普通平面的集合
On sets defining few ordinary planes
摘要：
令$S$是实三维空间中的一组$n$点，不存在三个共线且不全共面。我们证明，如果事件的平面的数目正好有三个点的$S$是小于$Kn^2$的一些$K=o(n^{\frac{1}{7}})$，然后，$n$足够大，但最多$O(K)$点的$S$包含在两个二次曲面的交集。此外，我们证明，有一个恒定的$c$，使如果与$S$正好三个点的入射平面的数量小于$\frac{1}{2}n^2-cn$，然后，对于$n$足够大，$S$是一个棱镜，一个反棱镜，一个棱镜与一个点删除或一个反棱镜删除。作为主要结果的推论，我们推出下列定理。设$S$是实平面中的一组$n$点。如果对于某些$K=o(n^{\frac{1}{7}})$，与$S$的正好三个点关联的圆的数目小于$Kn^2$，则对于足够大的$n$，除了$S$的至多$O(K)$点之外，所有的$O(K)$点都包含在至多四次的曲线中。
英文摘要：
Let $S$ be a set of $n$ points in real three-dimensional space, no three collinear and not all co-planar. We prove that if the number of planes incident with exactly three points of $S$ is less than $Kn^2$ for some $K=o(n^{\frac{1}{7}})$ then, for $n$ sufficiently large, all but at most $O(K)$ points of $S$ are contained in the intersection of two quadrics. Furthermore, we prove that there is a constant $c$ such that if the number of planes incident with exactly three points of $S$ is less than $\frac{1}{2}n^2-cn$ then, for $n$ sufficiently large, $S$ is either a prism, an anti-prism, a prism with a point removed or an anti-prism with a point removed. As a corollary to the main result, we deduce the following theorem. Let $S$ be a set of $n$ points in the real plane. If the number of circles incident with exactly three points of $S$ is less than $Kn^2$ for some $K=o(n^{\frac{1}{7}})$ then, for $n$ sufficiently large, all but at most $O(K)$ points of $S$ are contained in a curve of degree at most four.
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