# MathSciDoc: An Archive for Mathematician ∫

#### CombinatoricsMetric Geometrymathscidoc:2203.06002

Annals of Mathematics, 194, (3), 729-743, 2021.1
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<α<1. Let N_α(d) denote the maximum number of lines through the origin in R^d with pairwise common angle arccos α. Let k denote the minimum number (if it exists) of vertices in a graph whose adjacency matrix has spectral radius exactly (1−α)/(2α). If k<∞, then N_α(d)= ⌊ k(d−1)/(k−1) ⌋ for all sufficiently large d, and otherwise N_α(d)=d+o(d). In particular, N_{1/(2k−1)}(d)= ⌊ k(d−1)/(k−1) ⌋ for every integer k≥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.
```@inproceedings{zilin2021equiangular,
title={Equiangular lines with a fixed angle},
author={Zilin Jiang, Jonathan Tidor, Yuan Yao, Shengtong Zhang, and Yufei Zhao},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220318131157455833008},
booktitle={Annals of Mathematics},
volume={194},
number={3},
pages={729-743},
year={2021},
}
```
Zilin Jiang, Jonathan Tidor, Yuan Yao, Shengtong Zhang, and Yufei Zhao. Equiangular lines with a fixed angle. 2021. Vol. 194. In Annals of Mathematics. pp.729-743. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220318131157455833008.