Upper bounds for s-distance sets and equiangular lines

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 {α,−α}, α∈[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 Rn, n≥7, is n(n+1)2 with possible exceptions for some n=(2k+1)2−3, k∈N. We also prove the universal upper bound ∼23na2 for equiangular sets with α=1a 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.