Optimal asymptotic bounds for spherical designs

Abstract

In this paper we prove the conjecture of Korevaar and Meyers: for each $N\ge c_dt^d$, there exists a spherical $t$-design in the sphere $S^d$ consisting of $N$ points, where $c_d$ is a constant depending only on $d$.

  • [B] Go to document B. Bajnok, "Construction of spherical $t$-designs," Geom. Dedicata, vol. 43, iss. 2, pp. 167-179, 1992.
    @article {B, MRKEY = {1180648},
      AUTHOR = {Bajnok, Bela},
      TITLE = {Construction of spherical {$t$}-designs},
      JOURNAL = {Geom. Dedicata},
      FJOURNAL = {Geometriae Dedicata},
      VOLUME = {43},
      YEAR = {1992},
      NUMBER = {2},
      PAGES = {167--179},
      ISSN = {0046-5755},
      CODEN = {GEMDAT},
      MRCLASS = {05B30},
      MRNUMBER = {1180648},
      MRREVIEWER = {J. J. Seidel},
      DOI = {10.1007/BF00147866},
      ZBLNUMBER = {0765.05032},
      }
  • [Bannai1] Go to document E. Bannai and R. M. Damerell, "Tight spherical designs. I," J. Math. Soc. Japan, vol. 31, iss. 1, pp. 199-207, 1979.
    @article {Bannai1, MRKEY = {0519045},
      AUTHOR = {Bannai, E. and Damerell, R. M.},
      TITLE = {Tight spherical designs. {I}},
      JOURNAL = {J. Math. Soc. Japan},
      FJOURNAL = {Journal of the Mathematical Society of Japan},
      VOLUME = {31},
      YEAR = {1979},
      NUMBER = {1},
      PAGES = {199--207},
      ISSN = {0025-5645},
      CODEN = {NISUBC},
      MRCLASS = {05B30},
      MRNUMBER = {0519045},
      MRREVIEWER = {J. J. Seidel},
      DOI = {10.2969/jmsj/03110199},
      ZBLNUMBER = {0403.05022},
      }
  • [Bannai2] Go to document E. Bannai and R. M. Damerell, "Tight spherical designs. II," J. London Math. Soc., vol. 21, iss. 1, pp. 13-30, 1980.
    @article {Bannai2, MRKEY = {0576179},
      AUTHOR = {Bannai, E. and Damerell, R. M.},
      TITLE = {Tight spherical designs. {II}},
      JOURNAL = {J. London Math. Soc.},
      FJOURNAL = {The Journal of the London Mathematical Society. Second Series},
      VOLUME = {21},
      YEAR = {1980},
      NUMBER = {1},
      PAGES = {13--30},
      ISSN = {0024-6107},
      CODEN = {JLMSAK},
      MRCLASS = {05B05},
      MRNUMBER = {0576179},
      MRREVIEWER = {J. J. Seidel},
      DOI = {10.1112/jlms/s2-21.1.13},
      ZBLNUMBER = {0436.05018},
      }
  • [BV1] Go to document 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.
    @article {BV1, MRKEY = {2600661},
      AUTHOR = {Bondarenko, Andriy V. and Viazovska, Maryna S.},
      TITLE = {Spherical designs via {B}rouwer fixed point theorem},
      JOURNAL = {SIAM J. Discrete Math.},
      FJOURNAL = {SIAM Journal on Discrete Mathematics},
      VOLUME = {24},
      YEAR = {2010},
      NUMBER = {1},
      PAGES = {207--217},
      ISSN = {0895-4801},
      MRCLASS = {52C35 (41A55 41A63)},
      MRNUMBER = {2600661},
      DOI = {10.1137/080738313},
      ZBLNUMBER = {1229.05057},
      }
  • [BL] Go to document 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.
    @article {BL, MRKEY = {0981745},
      AUTHOR = {Bourgain, J. and Lindenstrauss, J.},
      TITLE = {Distribution of points on spheres and approximation by zonotopes},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {64},
      YEAR = {1988},
      NUMBER = {1},
      PAGES = {25--31},
      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {46B20 (52A25 52A45)},
      MRNUMBER = {0981745},
      MRREVIEWER = {T. Figiel},
      DOI = {10.1007/BF02767366},
      ZBLNUMBER = {0667.52001},
      }
  • [BDN] Go to document 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.
    @article {BDN, MRKEY = {1661307},
      AUTHOR = {Boyvalenkov, P. and Danev, D. and Nikova, S.},
      TITLE = {Nonexistence of certain spherical designs of odd strengths and cardinalities},
      JOURNAL = {Discrete Comput. Geom.},
      FJOURNAL = {Discrete \& Computational Geometry. An International Journal of Mathematics and Computer Science},
      VOLUME = {21},
      YEAR = {1999},
      NUMBER = {1},
      PAGES = {143--156},
      ISSN = {0179-5376},
      CODEN = {DCGEER},
      MRCLASS = {05B05},
      MRNUMBER = {1661307},
      MRREVIEWER = {Giuseppe Pellegrino},
      DOI = {10.1007/PL00009406},
      ZBLNUMBER = {0921.05016},
      }
  • [CCEK] Go to document 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.
    @article {CCEK, MRKEY = {2367321},
      AUTHOR = {Cohn, Henry and Conway, John H. and Elkies, Noam D. and Kumar, Abhinav},
      TITLE = {The {$D\sb 4$} root system is not universally optimal},
      JOURNAL = {Experiment. Math.},
      FJOURNAL = {Experimental Mathematics},
      VOLUME = {16},
      YEAR = {2007},
      NUMBER = {3},
      PAGES = {313--320},
      ISSN = {1058-6458},
      MRCLASS = {52C17 (05B40 52A40)},
      MRNUMBER = {2367321},
      DOI = {10.1080/10586458.2007.10129008},
      ZBLNUMBER = {1137.05020},
      }
  • [CK] Go to document H. Cohn and A. Kumar, "Universally optimal distribution of points on spheres," J. Amer. Math. Soc., vol. 20, iss. 1, pp. 99-148, 2007.
    @article {CK, MRKEY = {2257398},
      AUTHOR = {Cohn, Henry and Kumar, Abhinav},
      TITLE = {Universally optimal distribution of points on spheres},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {20},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {99--148},
      ISSN = {0894-0347},
      MRCLASS = {52A40 (41A63 52C17)},
      MRNUMBER = {2257398},
      MRREVIEWER = {P. McMullen},
      DOI = {10.1090/S0894-0347-06-00546-7},
      ZBLNUMBER = {1198.52009},
      }
  • [CFL] Go to document X. Chen, A. Frommer, and B. Lang, "Computational existence proofs for spherical $t$-designs," Numer. Math., vol. 117, iss. 2, pp. 289-305, 2011.
    @article {CFL, MRKEY = {2754852},
      AUTHOR = {Chen, Xiaojun and Frommer, Andreas and Lang, Bruno},
      TITLE = {Computational existence proofs for spherical {$t$}-designs},
      JOURNAL = {Numer. Math.},
      FJOURNAL = {Numerische Mathematik},
      VOLUME = {117},
      YEAR = {2011},
      NUMBER = {2},
      PAGES = {289--305},
      ISSN = {0029-599X},
      CODEN = {NUMMA7},
      MRCLASS = {65D30 (65C05 65G20)},
      MRNUMBER = {2754852},
      MRREVIEWER = {Michael Gnewuch},
      DOI = {10.1007/s00211-010-0332-5},
      ZBLNUMBER = {1208.65032},
      }
  • [CW] Go to document 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.
    @article {CW, MRKEY = {2272596},
      AUTHOR = {Chen, Xiaojun and Womersley, Robert S.},
      TITLE = {Existence of solutions to systems of underdetermined equations and spherical designs},
      JOURNAL = {SIAM J. Numer. Anal.},
      FJOURNAL = {SIAM Journal on Numerical Analysis},
      VOLUME = {44},
      YEAR = {2006},
      NUMBER = {6},
      PAGES = {2326--2341},
      ISSN = {0036-1429},
      MRCLASS = {65H10 (65G20)},
      MRNUMBER = {2272596},
      MRREVIEWER = {Ljiljana Petkovi{ć}},
      DOI = {10.1137/050626636},
      ZBLNUMBER = {1129.65035},
      }
  • [CS] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Third ed., New York: Springer-Verlag, 1999, vol. 290.
    @book {CS, MRKEY = {1662447},
      AUTHOR = {Conway, J. H. and Sloane, N. J. A.},
      TITLE = {Sphere Packings, Lattices and Groups},
      SERIES = {Grundl. Math. Wissen.},
      VOLUME = {290},
      EDITION = {Third},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1999},
      PAGES = {lxxiv+703},
      ISBN = {0-387-98585-9},
      MRCLASS = {11H31 (05B40 11H06 20D08 52C07 52C17 94B75 94C30)},
      MRNUMBER = {1662447},
      MRREVIEWER = {Renaud Coulangeon},
      ZBLNUMBER = {0915.52003},
      }
  • [DGS] Go to document P. Delsarte, J. M. Goethals, and J. J. Seidel, "Spherical codes and designs," Geometriae Dedicata, vol. 6, iss. 3, pp. 363-388, 1977.
    @article {DGS, MRKEY = {0485471},
      AUTHOR = {Delsarte, P. and Goethals, J. M. and Seidel, J. J.},
      TITLE = {Spherical codes and designs},
      JOURNAL = {Geometriae Dedicata},
      FJOURNAL = {Geometriae Dedicata},
      VOLUME = {6},
      YEAR = {1977},
      NUMBER = {3},
      PAGES = {363--388},
      ISSN = {0046-5755},
      MRCLASS = {05B99},
      MRNUMBER = {0485471},
      MRREVIEWER = {Mikhail Deza},
      ZBLNUMBER = {0376.05015},
      DOI = {10.1007/BF03187604},
     }
  • [HS] Go to document 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.
    @article {HS, MRKEY = {1384885},
      AUTHOR = {Hardin, R. H. and Sloane, N. J. A.},
      TITLE = {Mc{L}aren's improved snub cube and other new spherical designs in three dimensions},
      JOURNAL = {Discrete Comput. Geom.},
      FJOURNAL = {Discrete \& Computational Geometry. An International Journal of Mathematics and Computer Science},
      VOLUME = {15},
      YEAR = {1996},
      NUMBER = {4},
      PAGES = {429--441},
      ISSN = {0179-5376},
      CODEN = {DCGEER},
      MRCLASS = {52B11 (05B30)},
      MRNUMBER = {1384885},
      MRREVIEWER = {J. J. Seidel},
      DOI = {10.1007/BF02711518},
      ZBLNUMBER = {0858.05024},
      }
  • [KM] Go to document 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.
    @article {KM, MRKEY = {1421438},
      AUTHOR = {Korevaar, J. and Meyers, J. L. H.},
      TITLE = {Spherical {F}araday cage for the case of equal point charges and {C}hebyshev-type quadrature on the sphere},
      JOURNAL = {Integral Transform. Spec. Funct.},
      FJOURNAL = {Integral Transforms and Special Functions. An International Journal},
      VOLUME = {1},
      YEAR = {1993},
      NUMBER = {2},
      PAGES = {105--117},
      ISSN = {1065-2469},
      CODEN = {ITSFEQ},
      MRCLASS = {41A55 (78A30)},
      MRNUMBER = {1421438},
      DOI = {10.1080/10652469308819013},
      ZBLNUMBER = {0823.41026},
      }
  • [SK2] Go to document 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.
    @article {SK2, MRKEY = {1458327},
      AUTHOR = {Kuijlaars, A. B. J. and Saff, E. B.},
      TITLE = {Asymptotics for minimal discrete energy on the sphere},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {350},
      YEAR = {1998},
      NUMBER = {2},
      PAGES = {523--538},
      ISSN = {0002-9947},
      CODEN = {TAMTAM},
      MRCLASS = {11K41 (78A30)},
      MRNUMBER = {1458327},
      MRREVIEWER = {M. Lawrence Glasser},
      DOI = {10.1090/S0002-9947-98-02119-9},
      ZBLNUMBER = {0896.52019},
      }
  • [MNW] Go to document 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.
    @article {MNW, MRKEY = {1710640},
      AUTHOR = {Mhaskar, H. N. and Narcowich, F. J. and Ward, J. D.},
      TITLE = {Spherical {M}arcinkiewicz-{Z}ygmund inequalities and positive quadrature},
      JOURNAL = {Math. Comp.},
      FJOURNAL = {Mathematics of Computation},
      VOLUME = {70},
      YEAR = {2001},
      NUMBER = {235},
      PAGES = {1113--1130},
      ISSN = {0025-5718},
      CODEN = {MCMPAF},
      MRCLASS = {41A55 (42C10 65D30)},
      MRNUMBER = {1710640},
      DOI = {10.1090/S0025-5718-00-01240-0},
      ZBLNUMBER = {0980.76070},
      }
  • [M] Go to document O. R. Musin, "The kissing number in four dimensions," Ann. of Math., vol. 168, iss. 1, pp. 1-32, 2008.
    @article {M, MRKEY = {2415397},
      AUTHOR = {Musin, Oleg R.},
      TITLE = {The kissing number in four dimensions},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {168},
      YEAR = {2008},
      NUMBER = {1},
      PAGES = {1--32},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {52C17 (52B40 52C07 52C15 52C45)},
      MRNUMBER = {2415397},
      MRREVIEWER = {Nadia Benakli},
      DOI = {10.4007/annals.2008.168.1},
      ZBLNUMBER = {1169.52008},
      }
  • [OCC] Go to document D. O’Regan, Y. J. Cho, and Y. Chen, Topological Degree Theory and Applications, Chapman & Hall/CRC, Boca Raton, FL, 2006, vol. 10.
    @book {OCC, MRKEY = {2223854},
      AUTHOR = {O'Regan, Donal and Cho, Yeol Je and Chen, Yu-Qing},
      TITLE = {Topological Degree Theory and Applications},
      SERIES = {Ser. Math. Anal. Appl.},
      VOLUME = {10},
      PUBLISHER = {Chapman \& Hall/CRC, Boca Raton, FL},
      YEAR = {2006},
      PAGES = {iv+221},
      ISBN = {1-58488-648-X; 978-1-58488-648-8},
      MRCLASS = {47H11 (47H10 47H14 47N20 58C30)},
      MRNUMBER = {2223854},
      MRREVIEWER = {Maria Patrizia Pera},
      DOI = {10.1201/9781420011487},
      ZBLNUMBER = {1095.47001},
      }
  • [SK] Go to document E. B. Saff and A. B. J. Kuijlaars, "Distributing many points on a sphere," Math. Intelligencer, vol. 19, iss. 1, pp. 5-11, 1997.
    @article {SK, MRKEY = {1439152},
      AUTHOR = {Saff, E. B. and Kuijlaars, A. B. J.},
      TITLE = {Distributing many points on a sphere},
      JOURNAL = {Math. Intelligencer},
      FJOURNAL = {The Mathematical Intelligencer},
      VOLUME = {19},
      YEAR = {1997},
      NUMBER = {1},
      PAGES = {5--11},
      ISSN = {0343-6993},
      CODEN = {MAINDC},
      MRCLASS = {70F10 (52C99 82B05)},
      MRNUMBER = {1439152},
      MRREVIEWER = {T. Erber},
      DOI = {10.1007/BF03024331},
      ZBLNUMBER = {0901.11028},
      }
  • [SZ] Go to document 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.
    @article {SZ, MRKEY = {0744857},
      AUTHOR = {Seymour, P. D. and Zaslavsky, Thomas},
      TITLE = {Averaging sets: a generalization of mean values and spherical designs},
      JOURNAL = {Adv. in Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {52},
      YEAR = {1984},
      NUMBER = {3},
      PAGES = {213--240},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {05B30 (26B15)},
      MRNUMBER = {0744857},
      MRREVIEWER = {J. J. Seidel},
      DOI = {10.1016/0001-8708(84)90022-7},
      ZBLNUMBER = {0596.05012},
      }
  • [Wag] Go to document G. Wagner, "On averaging sets," Monatsh. Math., vol. 111, iss. 1, pp. 69-78, 1991.
    @article {Wag, MRKEY = {1089385},
      AUTHOR = {Wagner, Gerold},
      TITLE = {On averaging sets},
      JOURNAL = {Monatsh. Math.},
      FJOURNAL = {Monatshefte für Mathematik},
      VOLUME = {111},
      YEAR = {1991},
      NUMBER = {1},
      PAGES = {69--78},
      ISSN = {0026-9255},
      CODEN = {MNMTA2},
      MRCLASS = {65D32 (41A55)},
      MRNUMBER = {1089385},
      MRREVIEWER = {John Todd},
      DOI = {10.1007/BF01299278},
      ZBLNUMBER = {0721.65011},
     }
  • [Yu] Go to document V. A. Yudin, "Lower bounds for spherical designs," Izv. Ross. Akad. Nauk Ser. Mat., vol. 61, iss. 3, pp. 213-223, 1997.
    @article {Yu, MRKEY = {1478566},
      AUTHOR = {Yudin, V. A.},
      TITLE = {Lower bounds for spherical designs},
      JOURNAL = {Izv. Ross. Akad. Nauk Ser. Mat.},
      FJOURNAL = {Rossiĭskaya Akademiya Nauk. Izvestiya. Seriya Matematicheskaya},
      VOLUME = {61},
      YEAR = {1997},
      NUMBER = {3},
      PAGES = {213--223},
      ISSN = {0373-2436},
      MRCLASS = {05B30},
      MRNUMBER = {1478566},
      MRREVIEWER = {Vladimir D. Tonchev},
      DOI = {10.1070/im1997v061n03ABEH000132},
      ZBLNUMBER = {0890.05015},
      }

Authors

Andriy Bondarenko

Centre de Recerca Matemàtica, Bellaterra (Barcelona), Spain,
National Taras Shevchenko University, Kyiv, Ukraine,
and
Norwegian University of Science and Technology, Trondheim, Norway

Danylo Radchenko

Max Planck Institute for Mathematics, Bonn, Germany
and
National Taras Shevchenko, University, Kyiv, Ukraine

Maryna Viazovska

University of Cologne, Cologne, Germany
and
Max Planck Institute for Mathematics, Bonn, Germany