Additive Combinatorics in Groups and Geometric Combinatorics on Spheres

TL;DR

This paper explores the interplay between additive combinatorics in finite abelian groups and geometric combinatorics on spheres, highlighting known results, open questions, and interconnections among these areas.

Contribution

It provides a comprehensive review of dual concepts in additive and geometric combinatorics, emphasizing their relationships and recent developments.

Findings

Reviewed key results in additive combinatorics and geometric combinatorics

Identified open questions in the study of designs and distance sets

Discussed connections between group theory and sphere geometry

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.