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POINT DISTRIBUTIONS IN TWO‐POINT HOMOGENEOUS SPACES
Author(s) -
Skriganov M.M.
Publication year - 2019
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579319000019
Subject(s) - mathematics , pure mathematics , point (geometry) , euclidean geometry , rank (graph theory) , corollary , mathematical analysis , combinatorics , geometry
We consider point distributions in compact connected two‐point homogeneous spaces (Riemannian symmetric spaces of rank one). All such spaces are known: they are the spheres in the Euclidean spaces, the real, complex and quaternionic projective spaces and the octonionic projective plane. For all such spaces the best possible bounds for the quadratic discrepancies and sums of pairwise distances are obtained in the paper (Theorems 2.1 and 2.2). Distributions of points of t ‐designs on such spaces are also considered (Theorem 2.3). In particular, it is shown that the optimal t ‐designs meet the best possible bounds for quadratic discrepancies and sums of pairwise distances (Corollary 2.1). Our approach is based on the Fourier analysis on two‐point homogeneous spaces and explicit spherical function expansions for discrepancies and sums of distances (Theorems 4.1 and 4.2).