Equiangular lines with a fixed angle

Abstract: A configuration of N lines through the origin in d-dimensional Euclidean space is called equiangular if the lines are pairwise separated by the same angle. A natural and long-standing problem in discrete geometry is to determine the maximum size of a configuration of equiangular lines in a given dimension.

We determine, for each fixed angle and in all sufficiently large dimensions, the maximum number of equiangular lines separated by this given angle. Surprisingly, this maximum depends on spectral graph theoretic properties of the fixed angle.

Our proof involves the following novel result that seems to be of independent interest: A bounded degree connected graph has sublinear second eigenvalue multiplicity (that is, the multiplicity of the second-largest eigenvalue of the adjacency matrix of the graph is sublinear in the number of vertices).

Joint work with Zilin Jiang, Yuan Yao, Shengtong Zhang, and Yufei Zhao.

combinatoricsprobability

Audience: researchers in the topic

Extremal and probabilistic combinatorics webinar

Series comments: We've added a password: concatenate the 6 first prime numbers (hence obtaining an 8-digit password).