Equiangular lines with a fixed angle

Zilin Jiang, Jonathan Tidor, Yuan Yao, Shengtong Zhang, Yufei Zhao

Published 2019-07-29Version 1

Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle. Fix $0 < \alpha < 1$. Let $N_\alpha(d)$ denote the maximum number of lines in $\mathbb{R}^d$ with pairwise common angle $\arccos \alpha$. Let $k$ denote the minimum number (if it exists) of vertices of a graph whose adjacency matrix has spectral radius exactly $(1-\alpha)/(2\alpha)$. If $k < \infty$, then $N_\alpha(d) = \lfloor k(d-1)/(k-1) \rfloor$ for all sufficiently large $d$, and otherwise $N_\alpha(d) = d + o(d)$. In particular, $N_{1/(2k-1)}(d) = \lfloor k(d-1)/(k-1) \rfloor$ for every integer $k\geq 2$ and all sufficiently large $d$.