Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdos-Falconer distance conjecture
Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdos-Falconer distance conjecture
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超平面上的平均值、有限域上向量空间中的和积理论以及 Erdos-Falconer 距离猜想
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发表时间：
2007
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通讯作者：
M. Rudnev
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作者：
D. Hart;A. Iosevich;Doowon Koh;M. Rudnev
证明了有限域上向量空间中点与超平面的关联数的逐点平均界。虽然我们的估计是，在一般情况下，尖锐的，我们观察到一个改进的产品集和集包含在一个领域。我们利用这些关联界得到了用A · A + ··· + A · A覆盖有限域Fq的算术问题的显著改进，其中A是一个足够大的子集Fq.我们还使用的发病机制和发展的算术结构，研究有限域上的向量空间中的埃尔多斯-法尔科纳距离猜想。我们证明，自然模拟的欧氏埃尔多-法尔科纳距离猜想不持有在此设置。在积极的一面，我们得到了良好的指数的埃尔多斯-法尔科纳距离问题的子集的单位球F d q，并讨论其锋利度。这导致在一个合理的完整描述的埃尔多斯-法尔科纳距离问题在高维向量空间一般有限域。
We prove a pointwise and average bound for the number of incidences between points and hyperplanes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering F q , the finite field with q elements, by A · A + ··· + A · A, where A is a subset F q of sufficiently large size. We also use the incidence machinery and develop arithmetic constructions to study the Erdos-Falconer distance conjecture in vector spaces over finite fields. We prove that the natural analog of the Euclidean Erdos-Falconer distance conjecture does not hold in this setting. On the positive side, we obtain good exponents for the Erdos-Falconer distance problem for subsets of the unit sphere in F d q and discuss their sharpness. This results in a reasonably complete description of the Erdos-Falconer distance problem in higher-dimensional vector spaces over general finite fields.