The Kakeya problem and additive combinatorics

Kakeya 问题和加性组合数学

• 批准号: 1001607
• 负责人: Nets Katz
• 金额: $ 15.24万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001607&HistoricalAwards=false
• 关键词: Kakeya; problem; additive; combinatorics

This project is concerned with various problems in additive combinatorics and in particular with their applications to the Kakeya problem. The Kakeya problem asks, loosely speaking, how small a set in Euclidean space can be if it contains a unit line segment in every direction. More precisely, one would like lower bounds on the Hausdorff dimension, and it is conjectured that the lower bound should be the dimension of the Euclidean space. The conjecture has long been known true in the plane, but is much more difficult in higher dimensions and has inspired a great deal of work using techniques from fields ranging from harmonic analysis to commutative algebra to logic. In a paper with Laba and Tao, the author helped develop the connection of the Kakeya problem to sum-product theory. This is the theory of lower bounds on the sumset and product set of some finite subset of a field (or ring.) Sum product theory has been developing quickly over the past decade, and we now hope we know enough to come close to resolution in dimension 3, using some recent techniques of Bourgain, Konyagin, and others. In particular, we hope to show that the Assouad dimension of a Kakeya set in three dimensions is 3.From the earliest part of our mathematical educations, we learn that addition and multiplication are two of the most important mathematical processes, and that they are widely applicable to a number of real world problems. Surprisingly, not everything about the way in which addition and multiplication are connected is completely understood. We have recently learned that in a certain sense, addition and multiplication do not go well together - if a set does not expand much under addition, it must expand under multiplication, and vice versa. This principle is expressed in a number of "sum product" theorems. Sum Product theory has led to a lot of excitement in the last few years. It has applications in the combinatorics and geometric measure theory, the parts of mathematics for which it was developed, as well as in theoretical computer science, where expanding sets are used to generate pseudo-randomness. This project deals both with the basic theory of sums and products as well as with their deep mathematical applications. Continued work in this area is certain to lead to further applications.

本项目涉及加性组合学中的各种问题，特别是它们在Kakeya问题中的应用。粗略地说，Kakeya问题问的是，如果欧几里得空间中的一个集合在每个方向上都包含一个单位线段，它可以有多小。更准确地说，我们想要的是豪斯多夫维的下界，我们推测这个下界应该是欧几里德空间的维数。这个猜想在平面上早就被认为是正确的，但在高维上就困难得多，并且激发了大量的工作，这些工作使用了从调和分析到交换代数再到逻辑等领域的技术。在与Laba和Tao合著的一篇论文中，作者帮助建立了Kakeya问题与和积理论的联系。这是关于域（或环）的有限子集的sumset和product set下界的理论。和积理论在过去的十年中发展迅速，我们现在希望我们有足够的知识来接近三维的分辨率，使用布尔甘、科尼亚金和其他人的一些最新技术。特别地，我们希望证明三维空间中Kakeya集合的Assouad维数是3。从我们数学教育的最初部分开始，我们就知道加法和乘法是两个最重要的数学过程，它们被广泛应用于许多现实世界的问题。令人惊讶的是，并不是所有关于加法和乘法的联系方式都被完全理解了。我们最近了解到，在某种意义上，加法和乘法不能很好地结合在一起——如果一个集合在加法下不能扩展很多，那么它必须在乘法下扩展，反之亦然。这个原理用一些“和积”定理来表示。和积理论在过去几年中引起了许多兴奋。它在组合学和几何测量理论中有应用，这是它发展的数学部分，在理论计算机科学中也有应用，其中扩展集用于生成伪随机性。这个项目既涉及和和乘积的基本理论，也涉及它们的深层数学应用。在这一领域的持续工作肯定会导致进一步的应用。