Spherical two-distance sets and eigenvalues of signed graphs

TL;DR

The paper addresses the asymptotic size of spherical $\{\alpha,\beta\}$-codes in high dimensions by linking the problem to spectral theory of signed graphs. It introduces a conjecture that the limit $\lim_{d\to\infty} N_{\alpha,\beta}(d)/d$ equals $k_p(\lambda)/(k_p(\lambda)-1)$ with $\lambda=(1-\alpha)/(\alpha-\beta)$ and $p=\left\lfloor-\alpha/\beta\right\rfloor+1$, and proves this in several regimes, notably when $p\le2$ or $\lambda\in\{1,\sqrt{2},\sqrt{3}\}$. The authors develop a structural theorem showing the associated graphs are near-complete $p$-partite graphs and employ a suite of techniques—bounded-degree eigenvalue-multiplicity bounds, a forbidden-subgraph framework, a third-moment method, and an algebraic-degree argument—to bound $N_{\alpha,\beta}(d)$. They also construct signed-graph examples with large eigenvalue multiplicities, highlighting intrinsic obstacles to a full resolution of the conjecture within this framework. Overall, the work extends the equiangular-lines results to broader fixed-angle regimes and advances the understanding of how spectral properties control two-distance spherical codes.

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 $N_{α,β}(d)$ denote the maximum number of unit vectors in $\mathbb R^d$ where all pairwise inner products lie in $\{α,β\}$. For fixed $-1\leqβ<0\leqα<1$, we propose a conjecture for the limit of $N_{α,β}(d)/d$ as $d \to \infty$ in terms of eigenvalue multiplicities of signed graphs. We determine this limit when $α+2β<0$ or $(1-α)/(α-β) \in \{1, \sqrt{2}, \sqrt{3}\}$. Our work builds on our recent resolution of the problem in the case of $α= -β$ (corresponding to equiangular lines). It is the first determination of $\lim_{d \to \infty} N_{α,β}(d)/d$ for any nontrivial fixed values of $α$ and $β$ outside of the equiangular lines setting.

Figures (9)

• Figure 1: A valid $3$-coloring of a signed graph. Throughout this paper, the positive edges are represented by solid segments and the negative edges are represented by dashed segments.
• Figure 2: The Paley graph of order $9$.
• Figure 3: The Shrikhande graph.
• Figure 4: $H_3^\pm$
• Figure 5: $\hat{H}_3^\pm$

Theorems & Definitions (73)

• Definition 1.2
• Theorem 1.3: Equiangular lines with a fixed angle JTYZZ21
• Theorem 1.4: BDKS18 and JTYZZ21
• Theorem 1.5: JTYZZ21
• Definition 1.6
• Theorem 1.7
• Remark
• Definition 1.8
• Definition 1.9
• Definition 1.10