# Equiangular lines with a fixed angle

### Abstract

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 through the origin in $\mathbb {R}^d$ with pairwise common angle arccos$\alpha$. Let $k$ denote the minimum number (if it exists) of vertices in 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 $k\ge 2$ and all sufficiently large $d$. A key ingredient is a new result in spectral graph theory: the adjacency matrix of a connected bounded degree graph has sublinear second eigenvalue multiplicity.

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## Authors

Zilin Jiang

Arizona State University, Tempe, AZ

Jonathan Tidor

Massachusetts Institute of Technology, Cambridge, MA

Yuan Yao

Massachusetts Institute of Technology, Cambridge, MA

Shengtong Zhang

Massachusetts Institute of Technology, Cambridge, MA

Yufei Zhao

Massachusetts Institute of Technology, Cambridge, MA