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

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U2 - 10.4007/annals.2021.194.3.3

DO - 10.4007/annals.2021.194.3.3

M3 - Article

AN - SCOPUS:85119945033

VL - 194

SP - 729

EP - 743

JO - Annals of Mathematics

JF - Annals of Mathematics

SN - 0003-486X

IS - 3

ER -