TY - JOUR
T1 - Equiangular lines with a fixed angle
AU - Jiang, Zilin
AU - Tidor, Jonathan
AU - Yao, Yuan
AU - Zhang, Shengtong
AU - Zhao, Yufei
N1 - Funding Information:
Keywords: equiangular lines, spectral graph theory, eigenvalue multiplicity AMS Classification: Primary: 52C35; Secondary: 05C50, 05D10. Jiang was supported by an AMS Simons Travel Grant and NSF Award DMS-1953946. Tidor was supported by the NSF Graduate Research Fellowship Program DGE-1745302. Zhao was supported by NSF Award DMS-1764176, the MIT Solomon Buchsbaum Fund, and a Sloan Research Fellowship. © 2021 Department of Mathematics, Princeton University.
Funding Information:
Jiang was supported by an AMS Simons Travel Grant and NSF Award DMS-1953946. Tidor was supported by the NSF Graduate Research Fellowship Program DGE-1745302. Zhao was supported by NSF Award DMS-1764176, the MIT Solomon Buchsbaum Fund, and a Sloan Research Fellowship
Publisher Copyright:
© 2021. Department of Mathematics, Princeton University
PY - 2021/11
Y1 - 2021/11
N2 - 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 Nα(d) denote the maximum number of lines through the origin in Rd 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 Nα(d) = bk(d - 1)/(k - 1)c for all sufficiently large d, and otherwise Nα(d)) = d+o(d). In particular, (Formula Presented) for every integer k ≥ 2 and all sufficiently large d.
AB - 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 Nα(d) denote the maximum number of lines through the origin in Rd 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 Nα(d) = bk(d - 1)/(k - 1)c for all sufficiently large d, and otherwise Nα(d)) = d+o(d). In particular, (Formula Presented) for every integer k ≥ 2 and all sufficiently large d.
KW - Eigenvalue multiplicity
KW - Equiangular lines
KW - Spectral graph theory
UR - http://www.scopus.com/inward/record.url?scp=85119945033&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85119945033&partnerID=8YFLogxK
U2 - 10.4007/annals.2021.194.3.3
DO - 10.4007/annals.2021.194.3.3
M3 - Article
AN - SCOPUS:85119945033
SN - 0003-486X
VL - 194
SP - 729
EP - 743
JO - Annals of Mathematics
JF - Annals of Mathematics
IS - 3
ER -