Questions in Arithmetic Combinatorics

该项目重点关注数学研究中两个最容易获得的领域：组合学和数论。一般来说，组合数学侧重于计算难以直接计算的对象，而数论则研究整数的丰富属性。拟议的研究旨在进一步增加组合论证在研究数论问题中的使用。该项目还包括教育部分，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 Title I 高中数学小组的活动。算术和几何组合的相互交织的领域是该提案的重点。将调查三组相关的开放问题。第一个涉及其 Minkowski 和包含在以给定素数为模的非零二次余数组中的集合的大小。第二个研究有限域大子集的距离和和积问题。第三个是获得在重复组添加下表现出接近最大增长的组的特征。该奖项反映了 NSF 的法定使命，并且通过使用基金会的智力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.