In 1973, Lemmens and Seidel posed the problem of determining the maximum number of equiangular lines in $\mathbb{R}^r$ with angle $\arccos(\alpha)$ and gave a partial answer in the regime $r \leq 1/\alpha^2 - 2$. At the other extreme where $r$ is at least exponential in $1/\alpha$, recent breakthroughs have led to an almost complete resolution of this problem. In this paper, we introduce a new method for obtaining upper bounds which unifies and improves upon previous approaches, thereby bridging the gap between the aforementioned regimes, as well as significantly extending or improving all previously known bounds when $r \geq 1/\alpha^2 - 2$. Our method is based on orthogonal projection of matrices with respect to the Frobenius inner product and it also yields the first extension of the Alon-Boppana theorem to dense graphs, with equality for strongly regular graphs corresponding to $\binom{r+1}{2}$ equiangular lines in $\mathbb{R}^r$. Applications of our method in the complex setting will be discussed as well.

翻译：1973年，Lemmens和Seidel提出了确定$\mathbb{R}^r$空间中夹角为$\arccos(\alpha)$的等角线最大数目问题，并在$r \leq 1/\alpha^2 - 2$范围内给出了部分解答。在另一个极端情形（即$r$至少为$1/\alpha$的指数级）下，近年来的突破性进展已近乎完整解决了该问题。本文提出一种新的上界获取方法，该方法统一并改进了既有研究途径，从而弥合了上述两种情形之间的理论鸿沟，同时显著扩展或改进了当$r \geq 1/\alpha^2 - 2$时所有已知上界。我们的方法基于Frobenius内积下的矩阵正交投影，并首次将Alon-Boppana定理推广至稠密图情形——当强正则图对应于$\mathbb{R}^r$空间中$\binom{r+1}{2}$条等角线时，该推广呈现严格相等性。此外，本文还将讨论该方法在复空间中的若干应用。