Brown University, ICERM
New bounds for spherical two-distance sets and equiangular lines.
ABSTRACT: 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 a and -a, are called equiangular. The problem of determining
the maximal size of s-distance sets in various spaces has a long history in mathematics.
We determine a new method of bounding the size of an s-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 n dimension Euclidean space is n(n+1)/2 with possible exceptions
for some n=(2k+1)^2-3, where k is an positive integer. We also prove the universal upper bound 2/3 n a^2
for equiangular sets with angle 1/a and, employing this bound, prove a new upper bound on the size of
equiangular sets in an arbitrary dimension.
Friday, February 2, 2018
Stratton Hall 203