Additive and Combinatorial Problems in Groups

该项目由组合学项目、刺激竞争研究的既定项目（EPSCoR）以及代数和数论项目共同资助。在这个项目中，PI的目的是分组研究某些问题。群是一种数学结构，我们可以在其中加减两个元素，就像整数一样。群是数学中的基本对象，在现实生活中无处不在；例如，12小时钟上的数字由12个元素组成。所处理的问题是组合性质的；例如，如果我们取一个组的任何子集，那么只要这个子集在某种意义上是“大”的（对其结构没有任何假设），我们就可以预测会出现一些模式。研究生将作为该提案的一部分接受培训。PI还将参与K-12的外展活动。本项目研究的群是：(1)整数及其多项式环类似物，适用于数论技术；(2)有限群，包括有限域上的向量空间，适用于代数技术；(3)更一般的可适应群，适用于傅里叶分析和遍历理论技术。本文研究的问题包括：如果群G的子集a在某种意义上很大（例如a具有正密度，或者来自G的有限划分），那么能否在a - a和a + a - a等集合中找到结构？这个问题与遍历理论和拓扑动力学中的各种递归概念密切相关。如果A是一个稀疏集，例如素数的一个子集，那么这个问题将导致素数中迄今未探索的模式。PI还将使用各种技术研究其他加法问题，如加法基数、基本成分和指数和。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project is jointly funded by the Combinatorics program, the Established Program to Stimulate Competitive Research (EPSCoR), and the Algebra and Number Theory program. In this project, the PI aims to study certain problems in groups. A group is a mathematical structure in which we can add and subtract two elements, just like in the integers. Groups are fundamental objects in mathematics and ubiquitous in real life; for example, the numbers on a 12-hour clock form a group of 12 elements. The addressed problems are of combinatorial nature; for example, if we take any subset of a group, then as long as the subset is ''large'' in some sense (without any assumption on its structure), we predict that some pattern will emerge. Graduate students will be trained as part of this proposal. The PI will also be involved in K-12 outreach.The groups studied in this project are: (1) The integers and their polynomial ring analogs, for which techniques from number theory apply, (2) finite groups, including vector spaces over finite fields, for which techniques from algebra apply, and (3) more general amenable groups, for which techniques from Fourier analysis and ergodic theory apply. Problems studied in this proposal include the following: If a subset A of a group G is large in some sense (e.g. A has positive density, or comes from a finite partition of G), then can one find structures in sumsets such as A-A and A+A-A? This question is intimately related to various notions of recurrence in ergodic theory and topological dynamics. If A is a sparse set, e.g. a subset of the primes, then this question will lead to patterns hitherto unexplored in the primes. The PI will also study other additive problems such as additive bases, essential components, and exponential sums using various techniques.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.