The Kakeya problem and additive combinatorics

这个项目涉及加性组合数学中的各种问题，特别是它们在Kakeya问题中的应用。粗略地说，Kakeya问题问的是，如果欧几里德空间中的一个集合在每个方向上都包含一条单位线段，那么它可以有多小。更准确地说，人们希望得到Hausdorff维度的下界，并且推测该下界应该是欧几里德空间的维度。长期以来，人们一直知道这个猜想在平面上是正确的，但在更高的维度上要困难得多，并利用从调和分析到交换代数再到逻辑等领域的技术，激发了大量工作。在与拉巴和陶的一篇论文中，作者帮助发展了Kakeya问题与和积理论的联系。这是关于域(或环)的某个有限子集的和集和积集的下界理论。和乘积理论在过去的十年里发展迅速，我们现在希望我们知道足够多的知识来接近3维空间的解析，使用一些最新的技术，布尔加，科尼亚金，和其他。特别地，我们希望证明三维Kakeya集的Asourad维数为3。从我们的数学教育的最早部分开始，我们就知道加法和乘法是最重要的两个数学过程，它们广泛适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连接方式都被完全理解。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很好地扩展，那么它必须在乘法下扩展，反之亦然。这一原理在许多“和积”定理中得到了表达。和积理论在过去的几年里引起了很多人的兴奋。它在组合学和几何测度论中有应用，它是为其开发的数学部分，以及在理论计算机科学中，扩展集被用于产生伪随机性。这个项目既涉及和和乘积的基本理论，也涉及它们的深入数学应用。在这一领域的持续工作肯定会导致更多的应用。

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.