Direct and Inverse Problems for Cardinality Questions in Additive Combinatorics

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Combinatorics is arguably the most accessible branch of pure mathematics. At its core lie elementary questions that might excite a high school student as much as an expert. For example, imagine that one colors the edges and diagonals of an icosahedron red or blue. Can one always find a triangle whose edges are all red or a triangle whose edges are all blue? While combinatorics has traditionally found applications in computer science, information theory, and mathematical models, this project aims to further investigate its applications to what must be the oldest part of mathematics: number theory. The PI will involve high school, undergraduate, and graduate students in the project. The problems under study are suitable for training in research as they are easily accessible and offer an excellent setting for grasping some of the core techniques used in combinatorics. Parallel to and supported by the research activities of the project, the PI's goal is to write a short book on the application of combinatorics to number theory aimed at undergraduates in mathematics. Inverse theorems have catalyzed the development of additive combinatorics in recent years. However, some very basic inverse questions remain largely untouched. This project focuses on the study of open questions where combinatorial methods are likely to be of use. Three indicative examples, which are easy to formulate and hence will reach a wide mathematical audience, are: what structural information can be derived for finite sets in a commutative group that have near maximum number of h-fold sums; what can be said about finite sets of integers whose exponential sum has nearly minimum norm; and what can be deduced about the number of distinct differences that are formed from pairs of elements of a finite set, when the number of distinct sums is known? It is hoped that a combination of recent advances in the field and novel ideas will lead to progress. As these questions are representative of the challenges one must overcome in a wider variety of cardinality questions in additive combinatorics. It is hoped that any discoveries will be applied to other contexts as well. The project's methodology has a strong interdisciplinary component, which could unearth new connections between combinatorics and harmonic analysis.