Point Sets on the Sphere S 2 with Small Spherical Cap Discrepancy

Discrete and Computational Geometry

Volumn 48, Issue 4, 2012, Pages 990-1024

  Aistleitner, C ^a   Brauchart, J S ^b   Dick, J ^b  

^a GRAZ UNIVERSITY OF TECHNOLOGY   (Austria)

^b UNIVERSITY OF NEW SOUTH WALES   (Australia)

Author keywords

Discrepancy; Isotropic discrepancy; Lambert map; Level curve; Level set; Numerical integration; Quasi Monte Carlo; Spherical cap discrepancy

Indexed keywords

EID: 84870330820     PISSN: 01795376     EISSN: 14320444     Source Type: Journal    
DOI: 10.1007/s00454-012-9451-3     Document Type: Article

References (54)

1. Aistleitner, C.: Covering numbers, dyadic chaining and discrepancy. J. Complex. 27(6), 531-540 (2011).
2. Alexander, K. S.: The central limit theorem for empirical processes on Vapnik-Červonenkis classes. Ann. Probab. 15(1), 178-203 (1987).
3. Alexander, K. S., Talagrand, M.: The law of the iterated logarithm for empirical processes on Vapnik-Červonenkis classes. J. Multivar. Anal. 30(1), 155-166 (1989).
4. Andrievskii, V. V., Blatt, H.-P., Götz, M.: Discrepancy estimates on the sphere. Monatshefte Math. 128(3), 179-188 (1999).
5. Barvinok, A.: A Course in Convexity. Graduate Studies in Mathematics, vol. 54. American Mathematical Society, Providence (2002).
6. Beck, J.: Sums of distances between points on a sphere-an application of the theory of irregularities of distribution to discrete geometry. Mathematika 31(1), 33-41 (1984).
7. Beck, J.: On the discrepancy of convex plane sets. Monatshefte Math. 105(2), 91-106 (1988).
8. Beck, J., Chen, W. W. L.: Irregularities of Distribution. Cambridge Tracts in Mathematics, vol. 89. Cambridge University Press, Cambridge (2008). Reprint of the 1987 original. [MR0903025].
9. Bendito, E., Carmona, A., Encinas, A. M., Gesto, J. M., Gómez, A., Mouriño, C., Sánchez, M. T.: Computational cost of the Fekete problem. I. The forces method on the 2-sphere. J. Comput. Phys. 228(9), 3288-3306 (2009).
10. Bondarenko, A., Radchenko, D., Viazovska, M.: Optimal asymptotic bounds for spherical designs (2011). arXiv: 1009. 4407v3 [math. MG].
11. Brauchart, J. S.: Optimal logarithmic energy points on the unit sphere. Math. Comput. 77(263), 1599-1613 (2008).
12. 2. Numer. Math. 121(3), 473-502 (2012).
13. Brauchart, J. S., Saff, E. B., Sloan, I. H., Womersley, R. S.: Sequences of approximate spherical designs. Manuscript, 62 pages.
14. 2-discrepancy and sum of distances. Manuscript, 26 pages.
15. Cassels, J. W. S.: An Introduction to the Geometry of Numbers. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, vol. 99. Springer, Berlin (1959).
16. Chen, X., Frommer, A., Lang, B.: Computational existence proofs for spherical t-designs. Numer. Math. 117(2), 289-305 (2011).
17. Damelin, S. B., Grabner, P. J.: Energy functionals, numerical integration and asymptotic equidistribution on the sphere. J. Complex. 19(3), 231-246 (2003). Numerical integration and its complexity (Oberwolfach, 2001).
18. Damelin, S. B., Grabner, P. J.: Corrigendum to: "Energy functionals, numerical integration and asymptotic equidistribution on the sphere" [J. Complexity 19 (2003), no. 3, 231-246; mr1984111]. J. Complex. 20(6), 883-884 (2004).
19. Delsarte, P., Goethals, J. M., Seidel, J. J.: Spherical codes and designs. Geom. Dedic. 6(3), 363-388 (1977).
20. Dick, J., Pillichshammer, F.: Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge (2010).
21. Dudley, R. M.: Central limit theorems for empirical measures. Ann. Probab. 6(6), 899-929 (1979). 1978.
22. Durst, M., Dudley, R. M.: Empirical processes, Vapnik-Chervonenkis classes and Poisson processes. Probab. Math. Stat. 1(2), 109-115 (1981). 1980.
23. Faure, H.: Discrépance de suites associées à un système de numération (en dimension s). Acta Arith. 41(4), 337-351 (1982).
24. d, d≥3. Potential Anal. 13(4), 345-359 (2000).
25. Grabner, P. J.: Erdo{double acute}s-Turán type discrepancy bounds. Monatshefte Math. 111(2), 127-135 (1991).
26. Grabner, P. J., Tichy, R. F.: Spherical designs, discrepancy and numerical integration. Math. Comput. 60(201), 327-336 (1993).
27. Hannay, J. H., Nye, J. F.: Fibonacci numerical integration on a sphere. J. Phys. A 37(48), 11591-11601 (2004).
28. Hardin, D. P., Saff, E. B.: Discretizing manifolds via minimum energy points. Not. Am. Math. Soc. 51(10), 1186-1194 (2004).
29. Hardin, R. H., Sloane, N. J. A.: McLaren's improved snub cube and other new spherical designs in three dimensions. Discrete Comput. Geom. 15(4), 429-441 (1996).
30. Haussler, D.: Sphere packing numbers for subsets of the Boolean n-cube with bounded Vapnik-Chervonenkis dimension. J. Comb. Theory, Ser. A 69(2), 217-232 (1995).
31. Heinrich, S., Novak, E., Wasilkowski, G. W., Woźniakowski, H.: The inverse of the star-discrepancy depends linearly on the dimension. Acta Arith. 96(3), 279-302 (2001).
32. Korevaar, J.: Fekete extreme points and related problems. In: Approximation Theory and Function Series, Budapest, 1995. Bolyai Soc. Math. Stud., vol. 5, pp. 35-62. János Bolyai Math. Soc., Budapest (1996).
33. Korevaar, J., Meyers, J. L. H.: Spherical Faraday cage for the case of equal point charges and Chebyshev-type quadrature on the sphere. Integral Transforms Spec. Funct. 1(2), 105-117 (1993).
34. Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Pure and Applied Mathematics. Wiley-Interscience, New York (1974).
35. Larcher, G.: Optimale Koeffizienten bezüglich zusammengesetzter Zahlen. Monatshefte Math. 100(2), 127-135 (1985).
36. Larcher, G.: On the distribution of s-dimensional Kronecker-sequences. Acta Arith. 51(4), 335-347 (1988).
37. Larcher, G.: Corrigendum: "On the distribution of s-dimensional Kronecker-sequences" [Acta Arith. 51 (1988), no. 4, 335-347; MR0971085 (90f: 11065)]. Acta Arith. 60(1), 93-95 (1991).
38. Li, X.-J., Vaaler, J. D.: Some trigonometric extremal functions and the Erdo{double acute}s-Turán type inequalities. Indiana Univ. Math. J. 48(1), 183-236 (1999).
39. Lubotzky, A., Phillips, R., Sarnak, P.: Hecke operators and distributing points on the sphere. I. Commun. Pure Appl. Math. 39(S, suppl.), S149-S186 (1986). Frontiers of the Mathematical Sciences: 1985 (New York, 1985).
40. 2. II. Commun. Pure Appl. Math. 40(4), 401-420 (1987).
41. 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), 1256-1278 (2010).
42. Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63. SIAM, Philadelphia (1992).
43. Niederreiter, H., Sloan, I. H.: Integration of nonperiodic functions of two variables by Fibonacci lattice rules. J. Comput. Appl. Math. 51(1), 57-70 (1994).
44. Philipp, W.: Empirical distribution functions and uniform distribution mod 1. In: Proc. Conf. Diophantine Approximation and Its Application, Washington, DC, 1972, pp. 211-234. Academic Press, New York (1973).
45. Saff, E. B., Kuijlaars, A. B. J.: Distributing many points on a sphere. Math. Intell. 19(1), 5-11 (1997).
46. Sjögren, P.: Estimates of mass distributions from their potentials and energies. Ark. Mat. 10, 59-77 (1972).
47. Sloan, I. H., Womersley, R. S.: Extremal systems of points and numerical integration on the sphere. Adv. Comput. Math. 21(1-2), 107-125 (2004).
48. Sloan, I. H., Womersley, R. S.: A variational characterisation of spherical designs. J. Approx. Theory 159(2), 308-318 (2009).
49. Sobol', I. M.: Distribution of points in a cube and approximate evaluation of integrals. Ž. Vyčisl. Mat. i Mat. Fiz. 7, 784-802 (1967).
50. Stolarsky, K. B.: Sums of distances between points on a sphere. II. Proc. Am. Math. Soc. 41, 575-582 (1973).
51. Talagrand, M.: Sharper bounds for Gaussian and empirical processes. Ann. Probab. 22(1), 28-76 (1994).
52. Talagrand, M.: Vapnik-Chervonenkis type conditions and uniform Donsker classes of functions. Ann. Probab. 31(3), 1565-1582 (2003).
53. Vapnik, V. N., Červonenkis, A. J.: The uniform convergence of frequencies of the appearance of events to their probabilities. Teor. Verojatnost. i Primenen. 16, 264-279 (1971).
54. Wagner, G.: Erdo{double acute}s-Turán inequalities for distance functions on spheres. Mich. Math. J. 39(1), 17-34 (1992).