Additive Combinatorics in Groups and Geometric Combinatorics on Spheres

This chapter delves into the fascinating intersection of additive combinatorics in finite abelian groups and geometric combinatorics on spheres. It introduces and explores the concepts of independence and spanning in groups, and their counterparts in Euclidean space: designs and distance sets. The author establishes strong parallels between these seemingly disparate topics, demonstrating how results from one area can be profitably applied to the other. The chapter also presents intriguing problems and conjectures, such as finding the size of the largest t-independent set in an abelian group and classifying spherical t-designs and s-distance sets. Through detailed definitions and examples, the text offers a rich and engaging exploration of these advanced mathematical topics, making it a valuable resource for specialists in combinatorics and related fields.