On Sets Defining Few Ordinary Lines
On Sets Defining Few Ordinary Lines
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关于定义很少的普通线的集合
DOI：
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发表时间：
2012
影响因子：
0.8
通讯作者：
T. Tao
中科院分区：
文献类型：
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作者：
B. Green;T. Tao
设$$P$$P是平面上的一组$n个点，而不是所有的点都在一条直线上。我们证明，如果$$n是大的，那么至少有$n/2普通线，也就是说，线通过两个点的$P。这证实了，对于大$n，狄拉克和Motzkin的猜想。事实上，我们描述了这个问题的精确极值，以及对于某个绝对常数$$C$$C，具有少于$$n-C$$n-C普通线的所有集合。我们还解决了，对于大的$$n，“果园种植问题”，它要求通过$$P的3个点的线的最大数量。这些结果的基础是一个结构定理，该定理指出，如果$$P最多有$$Kn$$普通线，那么除了O（K）点之外，$$P$$P都位于三次曲线上，如果$$n$$n是足够大的依赖于$$K$$K。
Let $$P$$P be a set of $$n$$n points in the plane, not all on a line. We show that if $$n$$n is large then there are at least $$n/2$$n/2ordinary lines, that is to say lines passing through exactly two points of $$P$$P. This confirms, for large $$n$$n, a conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for this problem, as well as all sets having fewer than $$n-C$$n-C ordinary lines for some absolute constant $$C$$C. We also solve, for large $$n$$n, the “orchard-planting problem”, which asks for the maximum number of lines through exactly 3 points of $$P$$P. Underlying these results is a structure theorem which states that if $$P$$P has at most $$Kn$$Kn ordinary lines then all but O(K) points of $$P$$P lie on a cubic curve, if $$n$$n is sufficiently large depending on $$K$$K.