Equiangular lines with a fixed angle

Zilin Jiang Arizona State University, Tempe AZ 85287 Jonathan Tidor Massachusetts Institute of Technology, Cambridge MA 02141 Yuan Yao Massachusetts Institute of Technology, Cambridge MA 02141 Shengtong Zhang Massachusetts Institute of Technology, Cambridge MA 02141 Yufei Zhao Massachusetts Institute of Technology, Cambridge MA 02141

Combinatorics Metric Geometry mathscidoc: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.
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@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.
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