Equiangular lines with a fixed angle
Equiangular lines with a fixed angle
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DOI：
10.4007/annals.2021.194.3.3
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发表时间：
2019-07
影响因子：
4.9
通讯作者：
Zilin Jiang;Jonathan Tidor;Yuan Yao;Shengtong Zhang;Yufei Zhao
中科院分区：
文献类型：
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作者：
Zilin Jiang;Jonathan Tidor;Yuan Yao;Shengtong Zhang;Yufei Zhao
我们解决了一个长期存在的关于等角线的问题，对于每个给定的固定角度和所有足够大的维度，我们确定了被给定角度成对分开的线的最大数量。修复$0 < \alpha < 1$。设$N_\alpha(d)$表示$\mathbb{R}^d$中具有对公角$\arccos \alpha$的最大行数。设$k$表示图的最小顶点数（如果存在），其邻接矩阵的谱半径恰好为$(1-\alpha)/(2\alpha)$。如果$k < \infty$，则$N_\alpha(d) = \lfloor k(d-1)/(k-1) \rfloor$对于所有足够大的$d$，否则$N_\alpha(d) = d + o(d)$。特别地，$N_{1/(2k-1)}(d) = \lfloor k(d-1)/(k-1) \rfloor$对于所有整数$k\geq 2$和所有足够大的$d$。谱图理论的一个新成果是一个关键的组成部分：连通有界度图的邻接矩阵具有次线性二阶特征值多重性。
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 in $\mathbb{R}^d$ with pairwise common angle $\arccos \alpha$. Let $k$ denote the minimum number (if it exists) of vertices of 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 $k\geq 2$ 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.