Kakeya问题离散模拟的代数方法
Algebraic Methods in Discrete Analogs of the Kakeya Problem
摘要：
我们证明了关节猜想，表明对于任何${\Bbb R}^3$中的$N$线，最多有$O(N^{3 \over 2})$点，在这3条线相交非共面。我们还证明了一个猜想Bourgain显示，给定$N^2$线在${\Bbb R}^3$，使没有$N$线位于同一平面，使每条线相交一组$P$的点至少$N$点，那么基数的一组点是$Ω(N^3)$。我们的两个证明都是对Dvir关于有限域Kakeya问题的论点的改编。
英文摘要：
We prove the joints conjecture, showing that for any $N$ lines in ${\Bbb R}^3$, there are at most $O(N^{3 \over 2})$ points at which 3 lines intersect non-coplanarly. We also prove a conjecture of Bourgain showing that given $N^2$ lines in ${\Bbb R}^3$ so that no $N$ lines lie in the same plane and so that each line intersects a set $P$ of points in at least $N$ points then the cardinality of the set of points is $Ω(N^3)$. Both our proofs are adaptations of Dvir's argument for the finite field Kakeya problem.
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