Spherical two-distance sets and eigenvalues of signed graphs

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 $-1\le \beta <0\le \alpha <1$, we propose a conjecture for the limit of $N_{\alpha ,\beta }(d)\mathrm{/}d$ as $d\to \mathrm{\infty }$ in terms of eigenvalue multiplicities of signed graphs. We determine this limit when $\alpha +2\beta <0$ or $(1-\alpha )\mathrm{/}(\alpha -\beta )\in \{1,\sqrt{2},\sqrt{3}\}$. 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 ${\lim }_{d\to \mathrm{\infty }}{N}_{\alpha ,\beta }(d)\mathrm{/}d$ for any nontrivial fixed values of $\alpha$ and $\beta$ outside of the equiangular lines setting.