Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-23T09:07:16.141Z Has data issue: false hasContentIssue false
  • English
  • Français

POINT DISTRIBUTIONS IN COMPACT METRIC SPACES

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

Published online by Cambridge University Press:  29 November 2017

M. M. Skriganov
M. M. Skriganov*
Affiliation:
St. Petersburg Department, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia email [email protected]
Get access

Abstract

We consider finite point subsets (distributions) in compact metric spaces. In the case of general rectifiable metric spaces, non-trivial bounds for sums of distances between points of distributions and for discrepancies of distributions in metric balls are given (Theorem 1.1). We generalize Stolarsky’s invariance principle to distance-invariant spaces (Theorem 2.1). For arbitrary metric spaces, we prove a probabilistic invariance principle (Theorem 3.1). Furthermore, we construct equal-measure partitions of general rectifiable compact metric spaces into parts of small average diameter (Theorem 4.1).

MSC classification

Primary: 11K38: Irregularities of distribution, discrepancy 52C99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 63 , Issue 3 , 2017 , pp. 1152 - 1171
Copyright
Copyright © University College London 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alexander, J. R., On the sum of distances between n points on a sphere. Acta Math. Hungar. 23 1972, 443–448.CrossRefGoogle Scholar
Alexander, J. R., Beck, J. and Chen, W. W. L., Geometric discrepancy theory and uniform distribution. In Handbook of Discrete and Computational Geometry, 3rd edn. (eds Toth, C. D., Goodman, J. E. and O’Rourke, J.), Taylor and Francis (Boca Raton, FL, 2017), 331–357.Google Scholar
Barg, A. and Skriganov, M. M., Association schemes on general measure spaces and zero-dimensional Abelian groups. Adv. Math. 281 2015, 142–247.Google Scholar
Beck, J., Sums of distances between points on a sphere—an application of the theory of irregularities of distribution to distance geometry. Mathematika 31 1984, 33–41.Google Scholar
Beck, J. and Chen, W. W. L., Irregularities of Distribution (Cambridge Tracts in Mathematics 89 ), Cambridge University Press (Cambridge, 1987).CrossRefGoogle Scholar
Bilyk, D., Dai, F. and Matzke, R., Stolarsky principle and energy optimization on the sphere. Preprint, 2016, arXiv:1611.04420 [math.CA].Google Scholar
Brauchart, J. S. and Dick, J., A simple proof of Stolarsky’s invariance principle. Proc. Amer. Math. Soc. 141 2013, 2085–2096.Google Scholar
Helgason, S., Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press (New York, 1978).Google Scholar
Levenshtein, V. I., Universal bounds for codes and designs. In Handbook of Coding Theory (eds Pless, V. S. and Huffman, W. C.), Elsevier (Amsterdam, 1998), 499–648.Google Scholar
Mattila, P., Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability, Cambridge University Press (Cambridge, 1995).Google Scholar
Rakhmanov, E. A., Saff, E. B. and Zhou, Y. M., Minimal discrete energy on the sphere. Math. Res. Lett. 1 1994, 647–662.CrossRefGoogle Scholar
Skriganov, M. M., Point distributions in compact metric spaces, III. Two-point homogeneous spaces. Preprint POMI 7/2016, 2016, http://www.pdmi.ras.ru/preprint/2016/16-07.html, in Russian; English version available at arXiv:1701.04545 [math.MG].Google Scholar
Stolarsky, K. B., Sums of distances between points on a sphere, II. Proc. Amer. Math. Soc. 41 1973, 575–582.Google Scholar