On the size of Kakeya sets in finite fields
On the size of Kakeya sets in finite fields
复制标题
关于有限域中挂谷集的大小
DOI：
10.1090/s0894-0347-08-00607-3
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发表时间：
2008
影响因子：
3.9
通讯作者：
Zeev Dvir
中科院分区：
文献类型：
--
作者：
Zeev Dvir
研究有限域上的Kakeya集的动机是试图更好地理解W1中关于Kakeya集的更复杂的问题。一个Kakeya集KC Rn是一个紧集，它在每个方向上都包含一个单位长度的线段。著名的挂谷猜想指出，这样的集合必须有豪斯多夫（或闵可夫斯基）维数等于n。这个猜想的重要性部分是由于它与调和分析、数论和偏微分方程中的许多问题有联系。这个猜想被证明为n = 2 [Dav 71]，并且对于更大的n值是开放的（我们请读者参考调查论文[Wol 99，BouOO，TaoOl]以获得更多信息）。研究有限域Kakeya集最早是由Wolff [Wol 99]提出的。有人问[Wol 99]是否存在一个下界的形式Cn · qn的大小，这样的集在Fn。在[Wol 99]中出现的下界形式为Cn c（n+2）/2。在[RogO 1，BKT 04，MT 04，Tao 08]中，对于一般n和特定的小n值（例如，n = 3，4），该结合得到进一步改善。对于一般的n，当前最好的下界是在[RogOl，MT 04]中获得的Cn ·q4 n/7的下界（基于[KT 99]的结果）。用于证明该界限的主要技术是一个与A+ r B形式的不同和集的大小相关的加性数论引理，其中A和B是Fn中的固定集，r的范围是F中的几个不同值（在Kakeya集的背景下使用加性数论的想法是由于Bourgain [Bou 99]）。第2节证明的下一个定理给出了挂谷集大小的近最优界。粗略地说，证明如下观察，任何次数q 2齐次多项式在F[#i，. . .，xn]可以从它在任何Kakeya集合K c¥n上的值“重构”。这意味着K的大小至少是q 2次多项式空间的维度，即<q71 '1（当q很大时）。
The motivation for studying Kakeya sets over finite fields is to try to better understand the more complicated questions regarding Kakeya sets in W1. A Kakeya set K C Rn is a compact set containing a line segment of unit length in every direction. The famous Kakeya Conjecture states that such sets must have Hausdorff (or Minkowski) dimension equal to n. The importance of this conjecture is partially due to the connections it has to many problems in harmonic analysis, number theory and PDE. This conjecture was proved for n = 2 [Dav71] and is open for larger values of n (we refer the reader to the survey papers [Wol99, BouOO, TaoOl] for more information). It was first suggested by Wolff [Wol99] to study finite field Kakeya sets. It was asked in [Wol99] whether there exists a lower bound of the form Cn • qn on the size of such sets in Fn. The lower bound appearing in [Wol99] was of the form Cn c(n+2)/2. This bound was further improved in [RogOl, BKT04, MT04, Tao08] both for general n and for specific small values of n (e.g. for n = 3, 4). For general n, the most current best lower bound is the one obtained in [RogOl, MT04] (based on results from [KT99]) of Cn • q4n/7 . The main technique used to show this bound is an additive number theoretic lemma relating the sizes of different sum sets of the form A+rB, where A and B are fixed sets in Fn and r ranges over several different values in F (the idea to use additive number theory in the context of Kakeya sets is due to Bourgain [Bou99]). The next theorem, proven in Section 2, gives a near-optimal bound on the size of Kakeya sets. Roughly speaking, the proof follows by observing that any degree q 2 homogeneous polynomial in F[#i, . . . , xn] can be 'reconstructed' from its value on any Kakeya set K c¥n. This implies that the size of K is at least the dimension of the space of polynomials of degree q 2, which is « q71'1 (when q is large).