{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,22]],"date-time":"2025-10-22T17:52:19Z","timestamp":1761155539534},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2012,1,6]],"date-time":"2012-01-06T00:00:00Z","timestamp":1325808000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2012,7]]},"DOI":"10.1007\/s00211-011-0444-6","type":"journal-article","created":{"date-parts":[[2012,1,5]],"date-time":"2012-01-05T22:20:40Z","timestamp":1325802040000},"page":"473-502","source":"Crossref","is-referenced-by-count":17,"title":["Quasi\u2013Monte Carlo rules for numerical integration over the unit sphere $${\\mathbb{S}^2}$$"],"prefix":"10.1007","volume":"121","author":[{"given":"Johann S.","family":"Brauchart","sequence":"first","affiliation":[]},{"given":"Josef","family":"Dick","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2012,1,6]]},"reference":[{"issue":"3","key":"444_CR1","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1007\/s006050050056","volume":"128","author":"V.V. Andrievskii","year":"1999","unstructured":"Andrievskii V.V., Blatt H.-P., G\u00f6tz M.: Discrepancy estimates on the sphere. Monatsh. Math. 128(3), 179\u2013188 (1999)","journal-title":"Monatsh. Math."},{"key":"444_CR2","doi-asserted-by":"crossref","unstructured":"Antonov, I.A., Saleev, V.M.: An effective method for the computation of \u03bb P\u03c4-sequences. Zh. Vychisl. Mat. i Mat. Fiz. 19(1), 243\u2013245, 271 (1979)","DOI":"10.1016\/0041-5553(79)90085-5"},{"issue":"3","key":"444_CR3","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1080\/10586458.2009.10129052","volume":"18","author":"B. Ballinger","year":"2009","unstructured":"Ballinger B., Blekherman G., Cohn H., Giansiracusa N., Kelly E., Sch\u00fcrmann A.: Experimental study of energy-minimizing point configurations on spheres. Exp. Math. 18(3), 257\u2013283 (2009)","journal-title":"Exp. Math."},{"issue":"1","key":"444_CR4","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1112\/S0025579300010639","volume":"31","author":"J. Beck","year":"1984","unstructured":"Beck J.: Sums of distances between points on a sphere\u2014an application of the theory of irregularities of distribution to discrete geometry. Mathematika 31(1), 33\u201341 (1984)","journal-title":"Mathematika"},{"issue":"9","key":"444_CR5","doi-asserted-by":"crossref","first-page":"3288","DOI":"10.1016\/j.jcp.2009.01.021","volume":"228","author":"E. Bendito","year":"2009","unstructured":"Bendito E., Carmona A., Encinas A.M., Gesto J.M., G\u00f3mez A., Mouri\u00f1o C., S\u00e1nchez M.T.: Computational cost of the Fekete problem. I. The forces method on the 2-sphere. J. Comput. Phys. 228(9), 3288\u20133306 (2009)","journal-title":"J. Comput. Phys."},{"issue":"5\u20136","key":"444_CR6","doi-asserted-by":"crossref","first-page":"1301","DOI":"10.1023\/B:JOSS.0000041741.27244.ac","volume":"116","author":"M.K. Berkenbusch","year":"2004","unstructured":"Berkenbusch M.K., Claus I., Dunn C., Kadanoff L.P., Nicewicz M., Venkataramani S.C.: Discrete charges on a two dimensional conductor. J. Stat. Phys. 116(5\u20136), 1301\u20131358 (2004)","journal-title":"J. Stat. Phys."},{"issue":"3","key":"444_CR7","doi-asserted-by":"crossref","first-page":"1559","DOI":"10.1090\/S0002-9947-07-04416-9","volume":"360","author":"S.V. Borodachov","year":"2008","unstructured":"Borodachov S.V., Hardin D.P., Saff E.B.: Asymptotics for discrete weighted minimal Riesz energy problems on rectifiable sets. Trans. Am. Math. Soc. 360(3), 1559\u20131580 (2008) (electronic)","journal-title":"Trans. Am. Math. Soc."},{"key":"444_CR8","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1145\/42288.214372","volume":"14","author":"P. Bratley","year":"1988","unstructured":"Bratley P., Fox B.L.: Algorithm 659: Implementing sobol\u2019s quasirandom sequence generator. ACM Trans. Math. Softw. 14, 88\u2013100 (1988)","journal-title":"ACM Trans. Math. Softw."},{"issue":"263","key":"444_CR9","doi-asserted-by":"crossref","first-page":"1599","DOI":"10.1090\/S0025-5718-08-02085-1","volume":"77","author":"J.S. Brauchart","year":"2008","unstructured":"Brauchart J.S.: Optimal logarithmic energy points on the unit sphere. Math. Comp. 77(263), 1599\u20131613 (2008)","journal-title":"Math. Comp."},{"key":"444_CR10","unstructured":"Brauchart, J.S., Dick, J.: A simple Proof of Stolarsky\u2019s Invariance Principle. arXiv:1101.4448v1 [math.NA] (submitted)"},{"key":"444_CR11","unstructured":"Brauchart, J.S., Womersley, R.S.: Numerical integration over the unit sphere, $${\\mathcal{L}_2}$$ -discrepancy and sum of distances"},{"issue":"1","key":"444_CR12","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1090\/S0894-0347-06-00546-7","volume":"20","author":"H. Cohn","year":"2007","unstructured":"Cohn H., Kumar A.: Universally optimal distribution of points on spheres. J. Am. Math. Soc. 20(1), 99\u2013148 (2007) (electronic)","journal-title":"J. Am. Math. Soc."},{"issue":"2","key":"444_CR13","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1137\/S1064827595281344","volume":"18","author":"J. Cui","year":"1997","unstructured":"Cui J., Freeden W.: Equidistribution on the sphere. SIAM J. Sci. Comput. 18(2), 595\u2013609 (1997)","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"444_CR14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1515\/156939606776886643","volume":"12","author":"J. Dick","year":"2006","unstructured":"Dick J., Kritzer P.: A best possible upper bound on the star discrepancy of (t, m, 2)-nets. Monte Carlo Methods Appl. 12(1), 1\u201317 (2006)","journal-title":"Monte Carlo Methods Appl."},{"key":"444_CR15","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511761188","volume-title":"Digital Nets and Sequences. Discrepancy Theory and Quasi\u2013Monte Carlo Integration","author":"J. Dick","year":"2010","unstructured":"Dick J., Pillichshammer F.: Digital Nets and Sequences. Discrepancy Theory and Quasi\u2013Monte Carlo Integration. Cambridge University Press, Cambridge (2010)"},{"key":"444_CR16","unstructured":"Erd\u00f6s, P., Tur\u00e1n, P.: On a problem in the theory of uniform distribution. I. Nederl. Akad. Wetensch., Proc. 51, 1146\u20131154 (Indagationes Math. 10, 370\u2013378, 1948) (1948)"},{"key":"444_CR17","unstructured":"Erd\u00f6s, P., Tur\u00e1n, P.: On a problem in the theory of uniform distribution. II. Nederl. Akad. Wetensch., Proc. 51, 1262\u20131269 (Indagationes Math. 10, 406\u2013413, 1948) (1948)"},{"issue":"4","key":"444_CR18","doi-asserted-by":"crossref","first-page":"337","DOI":"10.4064\/aa-41-4-337-351","volume":"41","author":"H. Faure","year":"1982","unstructured":"Faure H.: Discr\u00e9pance de suites associ\u00e9es \u00e0 \u00e0 un syst\u00e8me de num\u00e9ration (en dimension s). Acta Arith. 41(4), 337\u2013351 (1982)","journal-title":"Acta Arith."},{"issue":"2","key":"444_CR19","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1007\/BF01332351","volume":"111","author":"P.J. Grabner","year":"1991","unstructured":"Grabner P.J.: Erd\u0151s-Tur\u00e1n type discrepancy bounds. Monatsh. Math. 111(2), 127\u2013135 (1991)","journal-title":"Monatsh. Math."},{"issue":"10","key":"444_CR20","first-page":"1186","volume":"51","author":"D.P. Hardin","year":"2004","unstructured":"Hardin D.P., Saff E.B.: Discretizing manifolds via minimum energy points. Not. Am. Math. Soc. 51(10), 1186\u20131194 (2004)","journal-title":"Not. Am. Math. Soc."},{"issue":"1","key":"444_CR21","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1016\/j.aim.2004.05.006","volume":"193","author":"D.P. Hardin","year":"2005","unstructured":"Hardin D.P., Saff E.B.: Minimal Riesz energy point configurations for rectifiable d-dimensional manifolds. Adv. Math. 193(1), 174\u2013204 (2005)","journal-title":"Adv. Math."},{"issue":"6","key":"444_CR22","doi-asserted-by":"crossref","first-page":"790","DOI":"10.1016\/j.jco.2005.07.004","volume":"21","author":"K. Hesse","year":"2005","unstructured":"Hesse K., Sloan I.H.: Optimal lower bounds for cubature error on the sphere S 2. J. Complex. 21(6), 790\u2013803 (2005)","journal-title":"J. Complex."},{"key":"444_CR23","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1145\/641876.641879","volume":"29","author":"S. Joe","year":"2003","unstructured":"Joe S., Kuo F.Y.: Remark on algorithm 659: Implementing sobol\u2032s quasirandom sequence generator. ACM Trans. Math. Softw. 29, 49\u201357 (2003)","journal-title":"ACM Trans. Math. Softw."},{"key":"444_CR24","doi-asserted-by":"crossref","unstructured":"LeVeque, W.J.: An inequality connected with Weyl\u2019s criterion for uniform distribution. In: Proc. Sympos. Pure Math., vol. VIII, pp. 22\u201330. Amer. Math. Soc., Providence, R.I. (1965)","DOI":"10.1090\/pspum\/008\/0179150"},{"issue":"1","key":"444_CR25","first-page":"183","volume":"48","author":"X.-J. Li","year":"1999","unstructured":"Li X.-J., Vaaler J.D.: Some trigonometric extremal functions and the Erd\u0151s-Tur\u00e1n type inequalities. Indiana Univ. Math. J. 48(1), 183\u2013236 (1999)","journal-title":"Indiana Univ. Math. J."},{"key":"444_CR26","unstructured":"Lubotzky, A., Phillips, R., Sarnak, P.: Hecke operators and distributing points on the sphere. I. Commun. Pure Appl. Math., 39(S, suppl.):S149\u2013S186, 1986 [Frontiers of the mathematical sciences: 1985 (New York, 1985)]"},{"issue":"4","key":"444_CR27","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1006\/jcom.1998.0489","volume":"14","author":"J. Matou\u0161ek","year":"1998","unstructured":"Matou\u0161ek J.: On the L 2-discrepancy for anchored boxes. J. Complex. 14(4), 527\u2013556 (1998)","journal-title":"J. Complex."},{"key":"444_CR28","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0094775","volume-title":"Spherical harmonics. Lecture Notes in Mathematics, vol 17","author":"C. M\u00fcller","year":"1966","unstructured":"M\u00fcller C.: Spherical harmonics. Lecture Notes in Mathematics, vol 17. Springer-Verlag, Berlin (1966)"},{"issue":"6","key":"444_CR29","doi-asserted-by":"crossref","first-page":"1256","DOI":"10.1016\/j.jat.2010.01.003","volume":"162","author":"F.J. Narcowich","year":"2010","unstructured":"Narcowich F.J., Sun X., Ward J.D., Wu Z.: Leveque type inequalities and discrepancy estimates for minimal energy configurations on spheres. J. Approx. Theory 162(6), 1256\u20131278 (2010)","journal-title":"J. Approx. Theory"},{"issue":"1","key":"444_CR30","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/0022-314X(88)90025-X","volume":"30","author":"H. Niederreiter","year":"1988","unstructured":"Niederreiter H.: Low-discrepancy and low-dispersion sequences. J. Number Theory 30(1), 51\u201370 (1988)","journal-title":"J. Number Theory"},{"key":"444_CR31","unstructured":"Niederreiter, H.: Random number generation and quasi\u2013Monte Carlo methods. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol 63. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1992)"},{"issue":"3","key":"444_CR32","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1006\/ffta.1996.0016","volume":"2","author":"H. Niederreiter","year":"1996","unstructured":"Niederreiter H., Xing C.: Low-discrepancy sequences and global function fields with many rational places. Finite Fields Appl. 2(3), 241\u2013273 (1996)","journal-title":"Finite Fields Appl."},{"issue":"3","key":"444_CR33","doi-asserted-by":"crossref","first-page":"281","DOI":"10.4064\/aa-72-3-281-298","volume":"72","author":"H. Niederreiter","year":"1995","unstructured":"Niederreiter H., Xing C.P.: Low-discrepancy sequences obtained from algebraic function fields over finite fields. Acta Arith. 72(3), 281\u2013298 (1995)","journal-title":"Acta Arith."},{"key":"444_CR34","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1145\/945511.945518","volume":"12","author":"A.B. Owen","year":"2003","unstructured":"Owen A.B.: Variance with alternative scramblings of digital nets. ACM Trans. Model. Comput. Simul. 12, 363\u2013378 (2003)","journal-title":"ACM Trans. Model. Comput. Simul."},{"key":"444_CR35","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1112\/S0025579300000541","volume":"1","author":"K.F. Roth","year":"1954","unstructured":"Roth K.F.: On irregularities of distribution. Mathematika 1, 73\u201379 (1954)","journal-title":"Mathematika"},{"key":"444_CR36","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/BF01418774","volume":"7","author":"W.M. Schmidt","year":"1969","unstructured":"Schmidt W.M.: Irregularities of distribution. IV. Invent. Math. 7, 55\u201382 (1969)","journal-title":"Invent. Math."},{"key":"444_CR37","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1007\/BF02384802","volume":"10","author":"P. Sj\u00f6gren","year":"1972","unstructured":"Sj\u00f6gren P.: Estimates of mass distributions from their potentials and energies. Ark. Mat. 10, 59\u201377 (1972)","journal-title":"Ark. Mat."},{"issue":"1\u20132","key":"444_CR38","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1023\/B:ACOM.0000016428.25905.da","volume":"21","author":"I.H. Sloan","year":"2004","unstructured":"Sloan I.H., Womersley R.S.: Extremal systems of points and numerical integration on the sphere. Adv. Comput. Math. 21(1\u20132), 107\u2013125 (2004)","journal-title":"Adv. Comput. Math."},{"key":"444_CR39","first-page":"784","volume":"7","author":"I.M. Sobol\u00b4","year":"1967","unstructured":"Sobol\u00b4 I.M.: Distribution of points in a cube and approximate evaluation of integrals. Z\u0306 Vy\u010disl. Mat. i Mat. Fiz. 7, 784\u2013802 (1967)","journal-title":"Z\u0306Vy\u010disl. Mat. i Mat. Fiz."},{"key":"444_CR40","doi-asserted-by":"crossref","first-page":"575","DOI":"10.1090\/S0002-9939-1973-0333995-9","volume":"41","author":"K.B. Stolarsky","year":"1973","unstructured":"Stolarsky K.B.: Sums of distances between points on a sphere. II. Proc. Am. Math. Soc. 41, 575\u2013582 (1973)","journal-title":"Proc. Am. Math. Soc."},{"issue":"2","key":"444_CR41","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1016\/j.jat.2007.09.009","volume":"151","author":"X. Sun","year":"2008","unstructured":"Sun X., Chen Z.: Spherical basis functions and uniform distribution of points on spheres. J. Approx. Theory 151(2), 186\u2013207 (2008)","journal-title":"J. Approx. Theory"},{"issue":"2","key":"444_CR42","doi-asserted-by":"crossref","first-page":"389","DOI":"10.2140\/pjm.1990.144.389","volume":"144","author":"G. Wagner","year":"1990","unstructured":"Wagner G.: On means of distances on the surface of a sphere (lower bounds). Pac. J. Math. 144(2), 389\u2013398 (1990)","journal-title":"Pac. J. Math."},{"issue":"1","key":"444_CR43","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1307\/mmj\/1029004450","volume":"39","author":"G. Wagner","year":"1992","unstructured":"Wagner G.: Erd\u0151s-Tur\u00e1n inequalities for distance functions on spheres. Michigan Math. J. 39(1), 17\u201334 (1992)","journal-title":"Michigan Math. J."},{"issue":"2","key":"444_CR44","doi-asserted-by":"crossref","first-page":"381","DOI":"10.2140\/pjm.1992.154.381","volume":"154","author":"G. Wagner","year":"1992","unstructured":"Wagner G.: On means of distances on the surface of a sphere. II. Upper bounds. Pac. J. Math. 154(2), 381\u2013396 (1992)","journal-title":"Pac. J. Math."},{"issue":"1","key":"444_CR45","doi-asserted-by":"crossref","first-page":"87","DOI":"10.4064\/aa-73-1-87-102","volume":"73","author":"C.P. Xing","year":"1995","unstructured":"Xing C.P., Niederreiter H.: A construction of low-discrepancy sequences using global function fields. Acta Arith. 73(1), 87\u2013102 (1995)","journal-title":"Acta Arith."}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-011-0444-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-011-0444-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-011-0444-6","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,12,24]],"date-time":"2021-12-24T22:50:59Z","timestamp":1640386259000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-011-0444-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,1,6]]},"references-count":45,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2012,7]]}},"alternative-id":["444"],"URL":"https:\/\/doi.org\/10.1007\/s00211-011-0444-6","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,1,6]]}}}