Questions in Arithmetic Combinatorics

该项目侧重于数学研究中最容易进入的两个领域：组合数学和数论。一般来说，组合数学专注于计算难以直接计数的对象，数论研究整数的丰富性质。拟议的研究旨在进一步增加组合参数在数论研究问题中的使用。该项目还有一个教育部分，即培训下一代数学家，提高科学素养和公众参与科学。该项目将支持当地一所一级高中数学队的活动。算术和几何组合学的相互交织的领域是这个建议的重点。三组相关的开放问题将被调查。第一个问题涉及的Minkowski和包含在非零二次剩余模一个给定的素数组的集的大小。第二部分研究了有限域大集的距离与和积问题。第三个奖项是获得在重复添加集的情况下表现出接近最大增长的集的表征。该奖项反映了NSF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.