Spherical two-distance sets and eigenvalues of signed graphs

Abstract

We study the problem of determining the maximum size of a spherical two-distance set with two fixed angles (one acute and one obtuse) in high dimensions. Let $N_{\alpha,\beta}(d)$ denote the maximum number of unit vectors in $\mathbb R^d$ where all pairwise inner products lie in $\{\alpha,\beta\}$. For fixed $-1\leq\beta<0\leq\alpha<1$, we propose a conjecture for the limit of $N_{\alpha,\beta}(d)/d$ as $d \to \infty$ in terms of eigenvalue multiplicities of signed graphs. We determine this limit when $\alpha+2\beta<0$ or $(1-\alpha)/(\alpha-\beta) \in \{1, \sqrt{2}, \sqrt{3}\}$. Our work builds on our recent resolution of the problem in the case of $\alpha = -\beta$ (corresponding to equiangular lines). It is the first determination of $\lim_{d \to \infty} N_{\alpha,\beta}(d)/d$ for any nontrivial fixed values of $\alpha$ and $\beta$ outside of the equiangular lines setting.