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.

MSC:

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