Equiangular lines with a fixed angle
Name
1907.12466.pdf
Description
Accepted version
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194.18 KB
Format
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Checksum (MD5)
25618bb6c6727072643fe241ad1449a6
Author(s)
Jiang, Zilin
Tidor, Jonathan
Yao, Yuan
Zhang, Shengtong
Zhao, Yufei
Date Issued
2021
Journal
Annals of Mathematics
Publisher
Annals of Mathematics
Citation
Jiang, Zilin, Tidor, Jonathan, Yao, Yuan, Zhang, Shengtong and Zhao, Yufei. 2021. "Equiangular lines with a fixed angle." Annals of Mathematics, 194 (3).
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.4007/ANNALS.2021.194.3.3
