Questions in Arithmetic Combinatorics

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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.