The set of points in a metric space is called an $s$-distance set if pairwise
distances between these points admit only $s$ distinct values. Two-distance
spherical sets with the set of scalar products $\{\alpha, -\alpha\}$,
$\alpha\in[0,1)$, are called equiangular. The problem of determining the
maximum size of $s$-distance sets in various spaces has a long history in
mathematics. We suggest a new method of bounding the size of an $s$-distance
set in compact two-point homogeneous spaces via zonal spherical functions. This
method allows us to prove that the maximum size of a spherical two-distance set
in $\mathbb{R}^n$, $n\geq 7$, is $\frac{n(n+1)}2$ with possible exceptions for
some $n=(2k+1)^2-3$, $k \in \mathbb{N}$. We also prove the universal upper
bound $\sim \frac 2 3 n a^2$ for equiangular sets with $\alpha=\frac 1 a$ and,
employing this bound, prove a new upper bound on the size of equiangular sets
in all dimensions. Finally, we classify all equiangular sets reaching this new
bound.

利用 zonal spherical functions 方法对紧致二点齐次空间中 s-distance set 的大小进行了限制。通过此方法我们证明了在 R^n 中 (当 n≥7 时), 球面二距离集的最大大小为 (n (n+1))/2 (对于一些 n=(2k+1)^2-3，其中 k∈N，可能存在例外)。我们还证明了呈现此新上限的所有 equiangular set，并利用其证明了所有维度中 equiangular set 大小的新上限。