Expanders, Fourier analysis and point-curve incidence theory

建议项目的主要目标是研究环境电场对SET膨胀器行为的影响。这是通过研究两个问题的欧几里得类比来探索的，这两个问题是最近在有限域上的向量空间中被研究的。第一个问题是将Falconer距离问题自然推广到点构形的研究。第二个问题来自一大类被称为和积问题的问题。第一个问题的连续性和第二个问题的离散性使研究人员可以从不同的领域结合各种方法，以便更好地说明所涉及的一般原则。这些方法来自算术和几何组合学、数论和经典调和分析。将欧几里得情形中的方法与有限域几何中的方法并列在一起，可以提供集合扩展现象的更完整的图景。这个项目涉及使用问题解决方法来研究数学的几个领域之间的相互作用，其中考虑了两个问题，它们的相似之处在数学的一个子领域称为有限域几何中具有已知解。这些问题在另外两个相关的子域中被研究，即离散和连续的欧几里德几何。该项目的目标不仅是在这些交替的情况下找到解决这两个问题的办法，而且探索各种已知方法和技术之间的相互作用，以便更全面地了解所涉及的一般原则的范围。这些问题的主要数学领域包括加法和几何组合学、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中有着广泛的应用。

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.