Equiangular Lines and Few-Angle Spherical Codes

Definitions and mathematical statement

For an integer $n\geq 1$, let \[ S^{n-1}=\{v\in{\mathbb{R}}^n:\langle v,v\rangle=1\}. \] For $0<\alpha<1$, define $N_\alpha(n)$ to be the maximum cardinality of a family $\mathcal L$ of distinct one-dimensional linear subspaces of ${\mathbb{R}}^n$ such that \[ |\langle u,v\rangle|=\alpha \] whenever $u$ and $v$ are unit representatives of distinct members of $\mathcal L$. The absolute value makes this condition independent of the choices of representatives.

The first source question asks whether, for each fixed admissible parameter $r\geq2$, \[ N_{1/(2r-1)}(n)=\frac{r(n-1)}{r-1}+O_r(1) \qquad(n\to\infty). \] Explicitly, this means asking whether there exist constants $C_r\geq0$ and $n_r\in{\mathbb{N}}$ such that, for every integer $n\geq n_r$, \[ \left|N_{1/(2r-1)}(n)-\frac{r(n-1)}{r-1}\right|\leq C_r. \] The source does not specify the admissible domain of $r$, particularly whether $r$ must be an integer. This qualification remains unresolved.

A spherical code in ${\mathbb{R}}^n$ means a finite subset $X\subseteq S^{n-1}$. For a fixed admissible real parameter $\alpha$, the second source question asks whether there exist constants $C_\alpha>0$ and $n_\alpha\in{\mathbb{N}}$ such that, for every integer $n\geq n_\alpha$ and every spherical code $X\subseteq S^{n-1}$ satisfying \[ \langle x,y\rangle\in[-1,0)\cup\{\alpha\} \qquad\text{for all distinct }x,y\in X, \] one has \[ |X|\leq C_\alpha n^2. \] The constants may depend on $\alpha$, but not on $n$ or $X$. The intended range of $\alpha$ is not supplied. The endpoint $-1$ is allowed; zero is allowed only if $\alpha=0$.

The supplied dated curation labels the record partially solved and reports that the fixed-angle conjecture for $\alpha=1/(2k-1)$ is proved for every fixed $k$ in sufficiently large dimension. It also reports that a broader multi-angle construction and related problems remain unresolved, but does not state those additional problems. These are supplied literature-status descriptions, not independently verified mathematical conclusions.

Short English statement

The first question asks whether the maximum number of equiangular lines with absolute unit-vector inner product $1/(2r-1)$ differs from $r(n-1)/(r-1)$ by a bounded amount for fixed $r$. The second asks whether spherical codes with pairwise inner products in $[-1,0)\cup\{\alpha\}$ have size $O_\alpha(n^2)$. The intended parameter domains need clarification. Supplied dated curation reports the first question solved in sufficiently large dimensions and labels the overall record partially solved; this assessment has not been independently verified.

Sources

• Supplied local registry record, permanent UnsolvedMath ID $30003329$, title: Equiangular Lines and Few-Angle Spherical Codes. Exact source problem code: catalogue entry. Collection: Oberwolfach Reports Open Problems. The record gives proposed year 2017 and supplies no named proposer.
• Supplied source citation: Combinatorics (2018), Oberwolfach Reports, DOI 10.4171/owr/2017/1, pp. 5--81. Original OWR report: https://doi.org/10.4171/owr/2017/1. The supplied background describes the reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Jiang--Tidor--Yao--Zhang--Zhao, fixed-angle lines: https://annals.math.princeton.edu/2021/194-3/p03.
• Supplied literature reference: Balla--Draxler--Keevash--Sudakov: https://arxiv.org/abs/1606.06620.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the record partially solved and reports that the fixed-angle conjecture for $\alpha=1/(2k-1)$ is proved for every fixed $k$ in sufficiently large dimension, while a broader multi-angle construction and related problems in the extracted passage remain unresolved. The supplied background calls this a dated literature triage and requests independent verification of time-sensitive status and cited claims.