Abstract

The set of points in a metric space is called an -distance set if pairwise distances between these points admit only distinct values. Two-distance spherical sets with the set of scalar products , , are called equiangular. The problem of determining the maximal size of -distance sets in various spaces has a long history in mathematics. We determine a new method of bounding the size of an -distance set in 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 is with possible exceptions for some , . We also prove the universal upper bound for equiangular sets with and, employing this bound, prove a new upper bound on the size of equiangular sets in an arbitrary dimension.