Direct and Inverse Problems for Cardinality Questions in Additive Combinatorics

组合学可以说是纯数学中最容易理解的分支。它的核心是一些基本的问题，这些问题可能会让高中生和专家一样兴奋。例如，假设有人将二十面体的边缘和对角线涂成红色或蓝色。我们总能找到一个边都是红色的三角形或者一个边都是蓝色的三角形吗？虽然组合学传统上在计算机科学、信息论和数学模型中有应用，但这个项目旨在进一步研究它在数学中最古老的部分：数论中的应用。该项目将包括高中生、本科生和研究生。所研究的问题适合用于研究训练，因为它们易于理解，并为掌握组合学中使用的一些核心技术提供了良好的环境。与该项目的研究活动并行并得到该项目的支持，PI的目标是编写一本针对数学本科生的关于组合学在数论中的应用的小书。近年来，逆定理促进了加性组合学的发展。然而，一些非常基本的逆向问题在很大程度上仍未触及。这个项目的重点是研究可能使用组合方法的开放性问题。三个指示性的例子，很容易表述，因此将达到广泛的数学受众，是：什么结构信息可以导出的有限集合在交换群接近最大数量的h-fold和；对于指数和具有接近最小范数的有限整数集，有什么结论？当一个有限集合中不同的和的数量已知时，对于由元素对形成的不同的差的数量，我们能推导出什么呢？希望该领域的最新进展和新思想的结合将导致进步。因为这些问题代表了在加性组合学中必须克服的各种各样的基数问题的挑战。希望任何发现也能应用于其他情况。该项目的方法论具有很强的跨学科成分，可以发现组合学和谐波分析之间的新联系。

Combinatorics is arguably the most accessible branch of pure mathematics. At its core lie elementary questions that might excite a high school student as much as an expert. For example, imagine that one colors the edges and diagonals of an icosahedron red or blue. Can one always find a triangle whose edges are all red or a triangle whose edges are all blue? While combinatorics has traditionally found applications in computer science, information theory, and mathematical models, this project aims to further investigate its applications to what must be the oldest part of mathematics: number theory. The PI will involve high school, undergraduate, and graduate students in the project. The problems under study are suitable for training in research as they are easily accessible and offer an excellent setting for grasping some of the core techniques used in combinatorics. Parallel to and supported by the research activities of the project, the PI's goal is to write a short book on the application of combinatorics to number theory aimed at undergraduates in mathematics. Inverse theorems have catalyzed the development of additive combinatorics in recent years. However, some very basic inverse questions remain largely untouched. This project focuses on the study of open questions where combinatorial methods are likely to be of use. Three indicative examples, which are easy to formulate and hence will reach a wide mathematical audience, are: what structural information can be derived for finite sets in a commutative group that have near maximum number of h-fold sums; what can be said about finite sets of integers whose exponential sum has nearly minimum norm; and what can be deduced about the number of distinct differences that are formed from pairs of elements of a finite set, when the number of distinct sums is known? It is hoped that a combination of recent advances in the field and novel ideas will lead to progress. As these questions are representative of the challenges one must overcome in a wider variety of cardinality questions in additive combinatorics. It is hoped that any discoveries will be applied to other contexts as well. The project's methodology has a strong interdisciplinary component, which could unearth new connections between combinatorics and harmonic analysis.