Equiangular lines in $\mathbb{R}^n$ are lines through origin
pairwise separated by the same angle

What's the maximum number $E(n)$ of
equiangular lines in $\mathbb{R}^n$?

What's the maximum number $E(n)$ of equiangular lines in $\mathbb{R}^n$?

Angles $\to$ 90° as $n\to\infty$

What happens if the angles are held fixed?

What's the maximum number $E_\alpha(n)$ of
equiangular lines in $\mathbb{R}^n$ with a fixed angle $\arccos\alpha$?

What's the maximum number $E_{1/5}(n)$ of
equiangular lines in $\mathbb{R}^n$ with a fixed angle $\arccos(1/5)$?

Known results on $E(d)$ and $E_{1/5}(n)$ agree for $n \in \{14, \dots , 41\}$

Lemmens–Seidel 1973 for $d \le 48$
$E(n) \le \max(E_{1/3}(d), N_{1/5}(n), \lfloor \frac{48d}{49-d}\rfloor)$

What's the maximum number $E_\alpha(n)$ of equiangular lines
in $\mathbb{R}^n$ with a fixed angle $\arccos\alpha$ in high dimensions?

What's the maximum number $E_\alpha(n)$ of equiangular lines
in $\mathbb{R}^n$ with a fixed angle $\arccos\alpha$ in high dimensions?

Spectral radius order $k(\lambda) := $ smallest $k$ such that
$\exists$ $k$-vertex graph $G$ whose adjacency matrix has spectral radius $\lambda$

What's the maximum number $E_\alpha(n)$ of equiangular lines
in $\mathbb{R}^n$ with a fixed angle $\arccos\alpha$ in high dimensions?

me

Jonathan Tidor

Yufei Zhao

Yuan Yao

Shengtong Zhang

Spectral radius order $k(\lambda) := $ smallest $k$ such that
$\exists$ $k$-vertex graph $G$ whose adjacency matrix has spectral radius $\lambda$

1. When $k < \infty$, proof needs $n \ge 2^{2^{C\lambda k}}$. Balla showed $n \ge 2^{k^{Ck}}$; Balla and Bucić showed $n \ge 2^{k^C}$ for $\alpha = 1/(2k-1)$.
2. When $k = \infty$, $o(n) = O_{\alpha}(n/\log\log n)$.
   Schildkraut constructed $\Omega_{\alpha}(\log\log n)$.

Connection to Spectral Graph Theory

Equiangular lines in $\mathbb{R}^n$ with angle $\arccos\alpha$
$\Leftrightarrow$ Unit vectors in $\mathbb{R}^n$ with inner product $\pm \alpha$
$\Leftarrow$$\Rightarrow$ Graph $G$, where $u \sim v$ in $G$ iff $\langle u, v\rangle = -\alpha$
s.t. $(1-\alpha) I - 2\alpha A_G + \alpha J \succeq 0$ and rank $\le n$

Equivalently, given $\alpha$ and $n$, find $G$ with max order s.t.
$(1-\alpha) I - 2\alpha A_G + \alpha J \succeq 0$ and rank $\le n$

Example when $\alpha = 1/5$, take $\lfloor (n-1)/2 \rfloor$ copies of

Upper bound on $E_\alpha(n)$

Given $\alpha$ and $n$, find $G$ with max order s.t.
$M := (1-\alpha) I - 2\alpha A_G + \alpha J \succeq 0$ and rank $\le n$

Rank-nullity
$E_\alpha(n) \le n + \mathrm{null}(M)$$\le \mathrm{null}((1-\alpha) I - 2\alpha A_G) + n + 1$

Since $M \succeq 0$, $\lambda := \frac{1-\alpha}{2\alpha}$ could be

1. largest eigenvalue of $G$ (equality case)
2. second largest eigenvalue of connected $G$ (need to rule out)

$\lambda := \frac{1-\alpha}{2\alpha}$ could be second largest eigenvalue of connected $G$

Switching $v$ to $-v$ in $\mathbb{R}^n$

Sublinear Second Eigenvalue Multiplicity

Proof Sketch

Multiplicity of second largest eigenvalue $\lambda$ of an $n$-vertex connected graph $G$ of bounded degree is at most $Cn/\log\log n$.

$r = c\log\log n$ and $s = c \log n$

• $H$ is $G$ with small $r$-net removed
• Radius $s$ balls in $H$ have spectral radius $\le (\lambda^{2r} - 1)^{1/2r}$
  Vertex removal lowers spectral radius
• Bound $\mathrm{mult}(\lambda, H)$ via moments
• $\mathrm{mult}(\lambda, G) \le \mathrm{mult}(\lambda, H) + \lvert r\text{-net} \rvert = o(n)$
  by Cauchy interlacing

What is the maximum second eigenvalue multiplicity of
an $n$-vertex connected graph of bounded degree?

McKenzie, Rasmussen, Srivastava 2021
For regular graphs, $O(n/\log^c n)$

What is the maximum second eigenvalue multiplicity of
an $n$-vertex Cayley graph of bounded degree?

Gromov, Colding–Minicozzi, Kleiner
Lee–Makarychev: Constant for abelian groups

What's the maximum number $E_\alpha(n)$ of
equiangular lines in $\mathbb{R}^n$ with a fixed angle $\arccos\alpha$?

Gossett, J., Teets, Wellner 2026 for $\alpha^* = \tfrac{1}{1+2\sqrt2}$

$E_{\alpha^*}(n) = \max(24, \lfloor 3(n-1)/2 \rfloor)$ for $n \ge 23$

Strongest possible sense

Classify equiangular line systems with fixed angle
$\arccos(1/(1+2\sqrt2))$ in $\mathbb{R}^n$ for all $n$,
up to orthogonal transformations

Classification: when $n \ge 29$, only generic equiangular line systems

For which $\alpha$, are there only finitely many non-generic equiangular line systems?

Spherical Two-distance Sets

$E_{\alpha, \beta}(n) = $ maximum number of unit vectors in $\mathbb{R}^n$
such that pairwise inner products are $\beta < 0 < \alpha$

Bukh 2005: $E_{\alpha, \beta}(n) = O_{\alpha,\beta}(n)$

Problem Determine $\lim_{n\to\infty} E_{\alpha, \beta}(n) /n$

Conjecture Depend on $k_p(\lambda)$ in terms of signed graphs
where $p = \lfloor -\alpha/\beta\rfloor+1$ and $\lambda = (1-\alpha)/(\alpha - \beta)$

Solved in special cases

• J., Tidor, Yao, Zhang, Zhao 2023: $p \le 2$ or $\lambda = 1,\sqrt{2},\sqrt{3}$
• J., Polyanskii 2025: $\lambda < \lambda^* \approx 2.0198$
  $\lambda^* := \sqrt{\rho} + 1/\sqrt{\rho}$, where $\rho$ is root of $x^3 = x + 1$

Solution Framework

1. Local forbidden subgraphs: leverage Gram matrix $\succeq 0$
2. Global graph structure: Ramsey theory
   • Equiangular lines: bounded degree graph
   • Spherical two-distance sets:
     complete $p$-partite XOR bounded degree

Need new insights in spectral graph theory:

• Sublinear eigenvalue multiplicity is false for signed graphs
• J., Polyanskii 2025: local forbidden subgraphs alone can never break $\lambda^*$ barrier
• J., Wang 2026: local forbidden subgraphs + global graph structure can never break $\lambda^*$ barrier as soon as $p \ge 3$

Other "Equiangular Lines"

Fixed angle: determine $\lim_{n\to\infty} E_\alpha^\mathbb{C}(n)/n$

Equiangular subspaces: configurations of $k$-dimensional subspaces in $\mathbb{R}^d$ with given pairwise angles