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 all 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与Seidel提出了确定$\mathbb{R}^r$空间中夹角为$\arccos(\alpha)$的等角直线最大数目的问题，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$r$至少与$1/\alpha$呈指数关系）下，近期突破性进展已使该问题近乎完全解决。本文提出了一种新的上界获取方法，该方法统一并改进了所有现有途径，从而得到能够衔接上述两个极端区间的上界，且这些上界要么精确最优，要么在较小乘法常数范围内达到最优。我们的方法依赖于矩阵关于Frobenius内积的正交投影，作为副产品，该方法首次将Alon-Boppana定理推广至稠密图，并在对应于$\mathbb{R}^r$空间中$\binom{r+1}{2}$条等角直线的强正则图上达到等号。本文还将讨论该方法在复数域中的应用。