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 yielding bounds which bridge the gap between the aforementioned regimes and are best possible either exactly or up to a small multiplicative constant. Our approach relies on orthogonal projection of matrices with respect to the Frobenius inner product and as a byproduct, it 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$\mathbb{R}^r$中夹角为$\arccos(\alpha)$的等角线最大数量的问题，并在$r \leq 1/\alpha^2 - 2$的区域给出了部分答案。在另一个极端，即$r$至少为$1/\alpha$的指数函数时，近期突破使得该问题几乎完全解决。本文提出了一种新的上界获取方法，该方法统一并改进了现有方法，从而得到的上界能够弥合上述区域之间的差距，并且要么精确最优，要么在常数因子内最优。我们的方法依赖于基于Frobenius内积的矩阵正交投影，作为副产品，它首次将Alon-Boppana定理推广到稠密图，且对于对应$\mathbb{R}^r$中$\binom{r+1}{2}$条等角线的强正则图，该定理取等。此外，还将讨论该方法在复数域中的应用。