多重性方法的扩展及其在Kakeya集和合并中的应用
Extensions to the Method of Multiplicities, with applications to Kakeya Sets and Mergers
摘要：
我们扩展了“多重性方法”，得到了以下结果，在组合学和随机性提取的兴趣。(A)我们证明了$\F_q^n$中的每个Kakeya集（在每个方向上包含一条线的点集）的大小必须至少为$q^n/2^n$。与之前在$n$中偏离指数因子的边界相比，对于每个$n$（作为$q \to \infty$），该边界紧到$2 + o(1)$因子内。(B)我们给出了改进的随机性提取器和“随机性合并器”。合并器是种子函数，其将$\{0,1\}^N$中的$Λ$（可能相关的）随机变量和短随机种子作为输入，并输出$\{0,1\}^N$中的单个随机变量，当$Λ$输入变量之一均匀分布时，该单个随机变量在统计上接近于具有熵$(1-δ) \cdot N$。我们需要的种子只有$(1/δ)\cdot \log Λ$ -位长，这大大改善了以前的合并构造。(C)使用我们的新的合并，我们展示了如何构建随机提取器，使用对数长度的种子，同时提取$1 - o(1)$分数的最小熵的源。“方法的多重性”，如在以前的工作中使用的，分析子集的向量空间在有限域上，通过构建有点低程度的插值多项式，消失在每个点的子集{ \em与高的多重性}。这种方法的典型应用包括证明插值多项式在子集外的某些点上也消失，然后使用简单的零点数量界限来完成分析。我们的扩增，这种技术是，我们证明，在适当的条件下，插值多项式消失{ \em与高的多重性}的集合之外。这种新颖性导致了更严格的分析。
英文摘要：
We extend the "method of multiplicities" to get the following results, of interest in combinatorics and randomness extraction. (A) We show that every Kakeya set (a set of points that contains a line in every direction) in $\F_q^n$ must be of size at least $q^n/2^n$. This bound is tight to within a $2 + o(1)$ factor for every $n$ as $q \to \infty$, compared to previous bounds that were off by exponential factors in $n$. (B) We give improved randomness extractors and "randomness mergers". Mergers are seeded functions that take as input $Λ$ (possibly correlated) random variables in $\{0,1\}^N$ and a short random seed and output a single random variable in $\{0,1\}^N$ that is statistically close to having entropy $(1-δ) \cdot N$ when one of the $Λ$ input variables is distributed uniformly. The seed we require is only $(1/δ)\cdot \log Λ$-bits long, which significantly improves upon previous construction of mergers. (C) Using our new mergers, we show how to construct randomness extractors that use logarithmic length seeds while extracting $1 - o(1)$ fraction of the min-entropy of the source. The "method of multiplicities", as used in prior work, analyzed subsets of vector spaces over finite fields by constructing somewhat low degree interpolating polynomials that vanish on every point in the subset {\em with high multiplicity}. The typical use of this method involved showing that the interpolating polynomial also vanished on some points outside the subset, and then used simple bounds on the number of zeroes to complete the analysis. Our augmentation to this technique is that we prove, under appropriate conditions, that the interpolating polynomial vanishes {\em with high multiplicity} outside the set. This novelty leads to significantly tighter analyses.
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