Equiangular lines with a fixed angle

Zilin Jiang, Jonathan Tidor, Yuan Yao, Shengtong Zhang, Yufei Zhao

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

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 < 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.

Original languageEnglish (US)
Pages (from-to)729-743
Number of pages15
JournalAnnals of Mathematics
Volume194
Issue number3
DOIs
StatePublished - Nov 2021
Externally publishedYes

Keywords

  • Eigenvalue multiplicity
  • Equiangular lines
  • Spectral graph theory

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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