Questions in Arithmetic Combinatorics

该项目侧重于数学研究中最容易触及的两个领域：组合学和数论。从广义上讲，组合学侧重于对难以直接计数的对象进行计数，而数论研究整数的丰富性质。提出的研究旨在进一步增加组合论证在研究数论问题中的应用。该项目还有一个教育组成部分，即培训下一代数学家，提高科学素养和公众对科学的参与。该项目将支持当地一所第一中学数学小组的活动。算术和几何组合的相互交织的领域是这个建议的重点。我们将研究三组相关的开放问题。第一个问题是闵可夫斯基和包含在以给定素数为模的非零二次残数群中的集合的大小。第二部分研究有限域大子集的距离和和积问题。第三部分是关于在重复集相加下表现出接近最大增长的集的一个性质。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。

The project focuses on two of the most accessible areas of mathematical research: combinatorics and number theory. Generally and broadly speaking, combinatorics focuses on counting objects that are hard to count directly, and number theory studies the rich properties of whole numbers. The proposed research aims to increase even further the use of combinatorial arguments in studying questions in number theory. The project also has an educational component, on training the next generation of mathematicians and advancing scientific literacy and public engagement with science. The project will support the activities of the Math Team of a local Title I High School. The intertwined areas of arithmetic and geometric combinatorics are the focus of this proposal. Three groups of related open problems will be investigated. The first concerns the size of sets whose Minkowski sums is contained in the group of non-zero quadratic residues modulo a given prime. The second investigates distance and sum-product questions for large subset of finite fields. The third is on obtaining a characterization of sets that exhibit near maximum growth under repeated set addition.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.