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Slice discrepancy and irregularities of distribution on spheres

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  26 February 2010

Martin Blümlinger
Martin Blümlinger
Affiliation:
Dr. M. Blümlinger, Institut für Analysis, Technische Mathematik und Versicherungsmathematik, Technische Universität Wien, Austria.
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Abstract

We improve W. Schmidt's lower bound for the slice (intersection of two halfspheres) discrepancy of point distributions on spheres and show that this estimate is up to a logarithmic factor best possible. It is shown that the slice and spherical cap discrepancies are equivalent for the definition of uniformly distributed sequences on spheres.

MSC classification

Secondary: 11K38: Irregularities of distribution, discrepancy
Type
Research Article
Information
Mathematika , Volume 38 , Issue 1 , June 1991 , pp. 105 - 116
Copyright
Copyright © University College London 1991

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References

Al. Alexander, R.. Principles of a new method in the study of irregularities of distribution, To appear in Invent. Math.Google Scholar
BC. Beck, J. and Chen, W.. Irregularities of Distribution (Cambridge University Press, Cambridge, 1987).Google Scholar
Be. Beck, J.. Sums of distances between points on a spherean application of the theory of irregularities of distribution to discrete geometry. Mathematika, 31 (1984), 33–41.Google Scholar
Bl. Blümlinger, M.. Asymptotic distribution and weak convergence on compact Riemannian manifolds. Monatshefte Math., 101 (1990), 177–188.Google Scholar
DR. Dunkl, C. and Ramirez, D.. Topics in Harmonic Analysis (Meredith, New York, 1971).Google Scholar
Er. Erdös, P.. Problems and results on Diophantine approximation. Compositio Math., 16 (1964), 52–66.Google Scholar
KN. Kuipers, L. and Niederreiter, H.. Uniform Distribution of Sequences (Wiley, New York, 1974).Google Scholar
LPS. Lubotzky, A., Phillips, R. and Samak, P.. Hecke Operators and Distributing Points on the Sphere I, Comm. Pure a. Appl. Math., 39 (1986), S139–S186.Google Scholar
Mü. Müller, C.. Spherical Harmonics (Springer, Lecture Notes, Berlin, 1966).Google Scholar
S1. Schmidt, W.. Irregularities of distribution III, Pacific J. of Math., 29 (1969), 225–234.Google Scholar
S2. Schmidt, W.. Irregularities of distribution IV, Invent. Math., 7 (1969), 55–82.Google Scholar
Vi. Vilenkin, N. J.. Special Functions and the Theory of Group Representations (A.M.S., Providence, 1968).Google Scholar