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POINT DISTRIBUTIONS IN TWO-POINT HOMOGENEOUS SPACES

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Discrete geometry Noncompact transformation groups

Published online by Cambridge University Press:  26 March 2019

M. M. Skriganov
M. M. Skriganov*
Affiliation:
St. Petersburg Department, Steklov Mathematical Institute, Russian Academy of Sciences, St. Petersburg 191023, Russia email mmskrig@gmail.com
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Abstract

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).

MSC classification

Primary: 11K38: Irregularities of distribution, discrepancy
Secondary: 22F30: Homogeneous spaces 52C99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 65 , Issue 3 , 2019 , pp. 557 - 587
Copyright
Copyright © University College London 2019 

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