Additive Combinatorics in Groups and Geometric Combinatorics on Spheres

Abstract

We embark on a tour that takes us through four closely related topics: the dual concepts of independence and spanning in finite abelian groups and the analogous dual concepts of designs and distance sets on spheres. We review some of the main known results in each area, mention several open questions, and discuss some connections among these four interesting topics.

This paper is based on the first of two plenary talks given by the author at the 53rd Southeastern International Conference on Combinatorics, Graph Theory \& Computing, held at Florida Atlantic University, March 7--11, 2022