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Bondarenko, Andriy; Radchenko, Danylo; Viazovska, Maryna

Optimal asymptotic bounds for spherical designs. (English) Zbl 1270.05026

Ann. Math. (2) 178, No. 2, 443-452 (2013).
Summary: In this paper we prove the conjecture of J. Korevaar and J. L. H. Meyers [Integral Transforms Spec. Funct. 1, No. 2, 105–117 (1993; Zbl 0823.41026)]: for each \(N\geq c_dt^d\), there exists a spherical \(t\)-design in the sphere \(S^d\) consisting of \(N\) points, where \(c_d\) is a constant depending only on \(d\).

Cited in 3 Reviews
Cited in 80 Documents

MSC:

05B30 Other designs, configurations

Keywords:

area-regular partitions; Brouwer degree; Marcinkiewicz-Zygmund inequalities; spherical designs

Citations:

Zbl 0823.41026
Cite Review PDF
Full Text: DOI arXiv

References:

[1] B. Bajnok, ”Construction of spherical \(t\)-designs,” Geom. Dedicata, vol. 43, iss. 2, pp. 167-179, 1992. · Zbl 0765.05032 · doi:10.1007/BF00147866
[2] E. Bannai and R. M. Damerell, ”Tight spherical designs. I,” J. Math. Soc. Japan, vol. 31, iss. 1, pp. 199-207, 1979. · Zbl 0403.05022 · doi:10.2969/jmsj/03110199
[3] E. Bannai and R. M. Damerell, ”Tight spherical designs. II,” J. London Math. Soc., vol. 21, iss. 1, pp. 13-30, 1980. · Zbl 0436.05018 · doi:10.1112/jlms/s2-21.1.13
[4] A. V. Bondarenko and M. S. Viazovska, ”Spherical designs via Brouwer fixed point theorem,” SIAM J. Discrete Math., vol. 24, iss. 1, pp. 207-217, 2010. · Zbl 1229.05057 · doi:10.1137/080738313
[5] J. Bourgain and J. Lindenstrauss, ”Distribution of points on spheres and approximation by zonotopes,” Israel J. Math., vol. 64, iss. 1, pp. 25-31, 1988. · Zbl 0667.52001 · doi:10.1007/BF02767366
[6] P. Boyvalenkov, D. Danev, and S. Nikova, ”Nonexistence of certain spherical designs of odd strengths and cardinalities,” Discrete Comput. Geom., vol. 21, iss. 1, pp. 143-156, 1999. · Zbl 0921.05016 · doi:10.1007/PL00009406
[7] H. Cohn, J. H. Conway, N. D. Elkies, and A. Kumar, ”The \(D_4\) root system is not universally optimal,” Experiment. Math., vol. 16, iss. 3, pp. 313-320, 2007. · Zbl 1137.05020 · doi:10.1080/10586458.2007.10129008
[8] H. Cohn and A. Kumar, ”Universally optimal distribution of points on spheres,” J. Amer. Math. Soc., vol. 20, iss. 1, pp. 99-148, 2007. · Zbl 1198.52009 · doi:10.1090/S0894-0347-06-00546-7
[9] X. Chen, A. Frommer, and B. Lang, ”Computational existence proofs for spherical \(t\)-designs,” Numer. Math., vol. 117, iss. 2, pp. 289-305, 2011. · Zbl 1208.65032 · doi:10.1007/s00211-010-0332-5
[10] X. Chen and R. S. Womersley, ”Existence of solutions to systems of underdetermined equations and spherical designs,” SIAM J. Numer. Anal., vol. 44, iss. 6, pp. 2326-2341, 2006. · Zbl 1129.65035 · doi:10.1137/050626636
[11] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Third ed., New York: Springer-Verlag, 1999, vol. 290. · Zbl 0915.52003
[12] P. Delsarte, J. M. Goethals, and J. J. Seidel, ”Spherical codes and designs,” Geometriae Dedicata, vol. 6, iss. 3, pp. 363-388, 1977. · Zbl 0376.05015 · doi:10.1007/BF03187604
[13] R. H. Hardin and N. J. A. Sloane, ”McLaren’s improved snub cube and other new spherical designs in three dimensions,” Discrete Comput. Geom., vol. 15, iss. 4, pp. 429-441, 1996. · Zbl 0858.05024 · doi:10.1007/BF02711518
[14] J. Korevaar and J. L. H. Meyers, ”Spherical Faraday cage for the case of equal point charges and Chebyshev-type quadrature on the sphere,” Integral Transform. Spec. Funct., vol. 1, iss. 2, pp. 105-117, 1993. · Zbl 0823.41026 · doi:10.1080/10652469308819013
[15] A. B. J. Kuijlaars and E. B. Saff, ”Asymptotics for minimal discrete energy on the sphere,” Trans. Amer. Math. Soc., vol. 350, iss. 2, pp. 523-538, 1998. · Zbl 0896.52019 · doi:10.1090/S0002-9947-98-02119-9
[16] H. N. Mhaskar, F. J. Narcowich, and J. D. Ward, ”Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature,” Math. Comp., vol. 70, iss. 235, pp. 1113-1130, 2001. · Zbl 0980.76070 · doi:10.1090/S0025-5718-00-01240-0
[17] O. R. Musin, ”The kissing number in four dimensions,” Ann. of Math., vol. 168, iss. 1, pp. 1-32, 2008. · Zbl 1169.52008 · doi:10.4007/annals.2008.168.1
[18] D. O’Regan, Y. J. Cho, and Y. Chen, Topological Degree Theory and Applications, Chapman & Hall/CRC, Boca Raton, FL, 2006, vol. 10. · Zbl 1095.47001 · doi:10.1201/9781420011487
[19] E. B. Saff and A. B. J. Kuijlaars, ”Distributing many points on a sphere,” Math. Intelligencer, vol. 19, iss. 1, pp. 5-11, 1997. · Zbl 0901.11028 · doi:10.1007/BF03024331
[20] P. D. Seymour and T. Zaslavsky, ”Averaging sets: a generalization of mean values and spherical designs,” Adv. in Math., vol. 52, iss. 3, pp. 213-240, 1984. · Zbl 0596.05012 · doi:10.1016/0001-8708(84)90022-7
[21] G. Wagner, ”On averaging sets,” Monatsh. Math., vol. 111, iss. 1, pp. 69-78, 1991. · Zbl 0721.65011 · doi:10.1007/BF01299278
[22] V. A. Yudin, ”Lower bounds for spherical designs,” Izv. Ross. Akad. Nauk Ser. Mat., vol. 61, iss. 3, pp. 213-223, 1997. · Zbl 0890.05015 · doi:10.1070/im1997v061n03ABEH000132
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