Equiangular lines with a fixed angle

• 成果类型: Article
• 署名作者: Jiang, Zilin; Tidor, Jonathan; Yao, Yuan; Zhang, Shengtong; Zhao, Yufei
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2021.194.3.3
• 发表日期: 2021
• 页码: 729-743
• 关键词: Bounds; sets

摘要

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 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 < infinity, then N-alpha(d) = left perpediculark(d - 1)/(k - 1)]right perpendicualr for all sufficiently large d, and otherwise N-alpha(d) = d+o(d). In particular, N1/(2k-1)(d) = laft perpendiculark(d-1)/(k-1)right perpendicular 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.