# Equiangular lines with a fixed angle

## TL;DR

This paper solves a longstanding problem by determining the maximum number of equiangular lines with a fixed angle in high dimensions, using spectral graph theory to establish precise asymptotic bounds.

## Contribution

It provides exact formulas for the maximum number of equiangular lines with a given angle in large dimensions, introducing a new spectral graph theory result about eigenvalue multiplicities.

## Key findings

- For fixed angles, maximum lines grow linearly with dimension.
- Explicit formulas for maximum lines when angle is of the form arccos(1/(2k-1)).
- New spectral graph theory result on eigenvalue multiplicity.

## 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 $0 < \alpha < 1$. Let $N_\alpha(d)$ denote the maximum number of lines through the origin in $\mathbb{R}^d$ 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 $k < \infty$, 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.