Expanders, Fourier analysis and point-curve incidence theory

本课题的主要目的是研究环境场对集膨胀器性能的影响。这是通过研究最近在有限域上的向量空间中研究的两个问题的欧几里得类似来探讨的。第一个问题是Falconer距离问题对点构型研究的自然推广。第二个问题是从一大类被称为和积问题的问题中得出的。第一个问题的连续性质和第二个问题的离散性质允许研究人员将来自不同领域的各种方法结合起来，以便更好地说明所涉及的一般原则。这些包括从算术和几何组合、数论和经典调和分析中得出的方法。将欧几里得情况下的方法与有限场几何中的方法并置，可以更完整地描述集合展开现象。该项目涉及使用解决问题的方法研究几个数学领域之间的相互作用，其中考虑两个问题，其类似问题在一个称为有限域几何的数学子领域中有已知的解决方案。这些问题是在两个额外的相关子领域，即离散和连续欧几里得几何中进行研究的。该项目的目标不仅是在这些交替的环境中找到这两个问题的解决方案，而且要探索各种已知方法和技术之间的相互作用，以便对所涉及的一般原则的范围获得更完整的理解。这些问题的主要数学领域包括加法和几何组合、数论和经典调和分析。这些领域在加密、数据挖掘、信号处理和生物信息学方面有各种各样的应用。

The main goal of the proposed project is to study the effect that the ambient field has on the behavior of set expanders. This is explored by studying the Euclidean analogs of two problems which were recently investigated in the context of vector spaces over finite fields. The first problem is a natural generalization of the Falconer distance problem to the study of point configurations. The second problem is drawn from a large class of problems known as the sum-product problems. The continuous nature of the first problem and the discrete nature of the second allows the researcher to incorporate a variety of methods from various fields in order to better illustrate the general principles involved. These include methods drawn from arithmetic and geometric combinatorics, number theory, and classical harmonic analysis. The juxtaposition of the methods from the Euclidean cases with the methods in the context of the finite field geometry could provide a more complete picture of the set expansion phenomenon.This project involves studying the interaction between several areas of mathematics using a problem-solving approach, in which two problems are considered whose analogs have known solutions in one subfield of mathematics known as finite field geometry. The problems are examined in two additional related subfields known as discrete and continuous Euclidean geometry. The goal of the project is to not only find solutions to the two problems in these alternate contexts, but to explore the interplay between a variety of known methods and techniques in order to obtain a more complete understanding of the scope of the general principles involved. The major areas of mathematics these problems are taken from include additive and geometric combinatorics, number theory, and classical harmonic analysis. These areas have a variety of applications in encryption, data mining, signal processing, and bioinformatics.