Direct and Inverse Problems for Cardinality Questions in Additive Combinatorics

组合数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，假设一个人将二十面体的边缘和对角线涂成红色或蓝色。你总能找到边都是红色的三角形还是边都是蓝色的三角形吗？虽然组合数学传统上在计算机科学、信息论和数学模型中得到应用，但这个项目的目标是进一步研究它在一定是数学中最古老的部分：数论中的应用。PI将让高中生、本科生和研究生参与该项目。正在研究的问题适合于研究培训，因为它们很容易获得，并为掌握组合数学中使用的一些核心技术提供了一个很好的环境。与该项目的研究活动平行，并得到该项目的支持，国际数学联合会的目标是写一本关于组合学在数论中的应用的短书，目标是面向数学本科生。近年来，逆定理促进了加性组合数学的发展。然而，一些非常基本的反问题在很大程度上仍未被触及。这个项目专注于组合方法可能有用的开放问题的研究。三个说明性的例子很容易表达，因此将影响到广泛的数学受众：对于具有接近最大数目的h重和的交换群中的有限集，可以得到什么结构信息；关于其指数和具有几乎最小范数的有限整数集，可以说什么；当不同和的数目已知时，关于有限集的元素对形成的不同差的数目可以推导出什么？希望这一领域的最新进展和新奇想法的结合将导致进步。由于这些问题代表了人们在加法组合学中必须克服的更广泛的基数问题中的挑战。人们希望，任何发现都将应用于其他背景。该项目的方法具有很强的跨学科成分，可以挖掘组合学和调和分析之间的新联系。

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.