Cubature for the sphere and the discrete spherical harmonic transform
SIAM Journal on Numerical Analysis
Maximum planar sets that determine k distances
Discrete Mathematics
Journal of Combinatorial Theory Series A
Distinct distances in finite planar sets
Discrete Mathematics
Classification of three-distance sets in two dimensional Euclidean space
European Journal of Combinatorics
On antipodal Euclidean tight (2e + 1)-designs
Journal of Algebraic Combinatorics: An International Journal
Note: Spherical two-distance sets
Journal of Combinatorial Theory Series A
Bounds on three- and higher-distance sets
European Journal of Combinatorics
Bounds on s-Distance Sets with Strength t
SIAM Journal on Discrete Mathematics
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A subset X in the d-dimensional Euclidean space is called a k-distance set if there are exactly k distinct distances between two distinct points in X and a subset X is called a locally k-distance set if for any point x in X, there are at most k distinct distances between x and other points in X. Delsarte, Goethals, and Seidel gave the Fisher type upper bound for the cardinalities of k-distance sets on a sphere in 1977. In the same way, we are able to give the same bound for locally k-distance sets on a sphere. In the first part of this paper, we prove that if X is a locally k-distance set attaining the Fisher type upper bound, then determining a weight function w, (X,w) is a tight weighted spherical 2k-design. This result implies that locally k-distance sets attaining the Fisher type upper bound are k-distance sets. In the second part, we give a new absolute bound for the cardinalities of k-distance sets on a sphere. This upper bound is useful for k-distance sets for which the linear programming bound is not applicable. In the third part, we discuss about locally two-distance sets in Euclidean spaces. We give an upper bound for the cardinalities of locally two-distance sets in Euclidean spaces. Moreover, we prove that the existence of a spherical two-distance set in (d-1)-space which attains the Fisher type upper bound is equivalent to the existence of a locally two-distance set but not a two-distance set in d-space with more than d(d+1)/2 points. We also classify optimal (largest possible) locally two-distance sets for dimensions less than eight. In addition, we determine the maximum cardinalities of locally two-distance sets on a sphere for dimensions less than forty.