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POINT DISTRIBUTIONS IN TWO-POINT HOMOGENEOUS SPACES

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

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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References

Alexander, J. R., On the sum of distances between n points on a sphere. Acta Math. Hungar. 23(3–4) 1972, 443–448.Google Scholar
Alexander, J. R., Beck, J. and Chen, W. W. L., Geometric discrepancy theory and uniform distributions. 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), 279–304.Google Scholar
Banai, E., On extremal finite sets in the sphere and other metric spaces. In Algebraic, Extremal and Metric Combinatorics (eds Deza, M. M., Frankl, P. and Rosenberg, I. G.), Cambridge University Press (Cambridge, 1988).Google Scholar
Beck, J., Sums of distances between points on a sphere: an application of the theory of irregularities of distributions 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).Google Scholar
Besse, A. L., Manifolds All of Whose Geodesics are Closed (A Series of Modern Surveys in Mathematics 93 ), Springer (Berlin, 1978).Google Scholar
Bilyk, D. and Dai, F., Geodesic distance Riesz energy on the sphere. Trans. Amer. Math. Soc. 2018, published on line, https://dx.doi.org/10.1090/tran/7711.Google Scholar
Bilyk, D., Dai, F. and Matzke, R., Stolarsky principle and energy optimization on the sphere. Constr. Approx. 48(1) 2018, 31–61.Google Scholar
Bondarenko, A., Radchenko, D. and Viazovska, M., Optimal asymptotic bounds for spherical designs. Ann. of Math. 178(2) 2013, 443–452.Google Scholar
Bondarenko, A., Radchenko, D. and Viazovska, M., Well-separated spherical designs. Constr. Approx. 41(1) 2015, 93–112.Google Scholar
Brandolini, L., Chen, W. W. L., Colzani, L., Gigante, G. and Travaglini, G., Discrepancy and numerical integration on metric measure spaces. J. Geom. Anal. 29(1) 2019, 328–369.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
Brauchart, J. S. and Grabner, P. J., Distributing many points on spheres. J. Complexity 31(3) 2015, 293–326.Google Scholar
Cohn, H. and Kumar, A., Universally optimal distribution of points on spheres. J. Amer. Math. Soc. 20(1) 2006, 99–147.Google Scholar
Cohn, H., Kumar, A. and Minton, G., Optimal simplices and codes in projective spaces. Geom. Topol. 20 2016, 1289–1357.Google Scholar
Conway, J., Hardin, R. and Sloane, N. J. A., Packing lines, planes, etc.: packing in Grassmannian spaces. Exp. Math. 5 1996, 139–159.Google Scholar
Deza, M. M. and Laurent, M., Geometry of Cuts and Metrics, Springer (Berlin, 1997).Google Scholar
Etayo, U., Marzo, J. and Ortega-Cerdà, J., Asymptotically optimal designs on compact algebraic manifolds. Monatsh. Math. 186(2) 2018, 235–248.Google Scholar
Gangolli, R., Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy’s Brownian motion of several parameters. Ann. Inst. H. Poincaré III(2) 1967, 121–325.Google Scholar
Gigante, W. and Leopardi, P., Diameter bounded equal measure partitions of Alfors regular metric measure spaces. Discrete Comput. Geom. 57 2017, 419–430.Google Scholar
Helgason, S., Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press (New York, 1978).Google Scholar
Helgason, S., Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions, Academic Press (London, 1984).Google Scholar
Koornwinder, T. H., Jacobi functions and analysis on noncompact semisimple Lie groups. In Special Functions: Group Theoretical Aspects and Application (eds Askey, R. A., Koornwinder, T. H. and Schempp, W.), D. Reidel (Dordrecht, 1984), 1–85.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
Malyarenko, A., Invariant Random Fields on Spaces with a Group Action, Springer (Berlin, 2013).Google Scholar
Saff, E. B. and Kwijlaars, A. B. J., Distributing many points on a sphere. Math. Intelligencer 19(1) 1997, 5–11.Google Scholar
Skriganov, M. M., Point distributions in compact metric spaces. Mathematika 63(3) 2017, 1152–1171.Google Scholar
Skriganov, M. M., Bounds for $L_{p}$ -discrepancies of point distributions in compact metric measure spaces. Preprint, 2018, arXiv:1802.01577.Google Scholar
Skriganov, M. M., Stolarsky’s invariance principle for projective spaces. Preprint, 2018,arXiv:1805.03541.Google Scholar
Stolarsky, K. B., Sums of distances between points on a sphere, II. Proc. Amer. Math. Soc. 41 1973, 575–582.Google Scholar
Szegö, G., Orthogonal Polynomials, American Mathematical Society (Providence, RI, 1950).Google Scholar
Vilenkin, N. J. and Klimyk, A. U., Representation of Lie Groups and Special Functions, vols. 1–3, Kluwer Academic (Dordrecht, 1991–1992).Google Scholar
Wolf, J. A., Spaces of Constant Curvature, University of California (Berkley, 1972).Google Scholar
Wolf, J. A., Harmonic Analysis on Commutative Spaces (Mathematical Surveys and Monographs 142 ), American Mathematical Society (Providence, RI, 2007).Google Scholar