Equiangular lines with a fixed angle. (English) Zbl 1478.52015

Summary: 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 \(\mathbb{R}^d\) with pairwise common angle \(\operatorname{arccos}\alpha\). 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 \(k<\infty\), 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 \(k\ge 2\) 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.


52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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