AccĂšs libre

Metrical Star Discrepancy Bounds for Lacunary Subsequences of Digital Kronecker-Sequences and Polynomial Tractability

À propos de cet article

Citez

[1] AISTLEITNER, CH.: Covering numbers, dyadic chaining and discrepancy, J. Complexity 27 (2011), 531–540.10.1016/j.jco.2011.03.001Search in Google Scholar

[2] AISTLEITNER, CH.: On the inverse of the discrepancy for infinite dimensional infinite sequences, J. Complexity 29 (2013), 182–194.10.1016/j.jco.2012.06.002Search in Google Scholar

[3] BECK, J.: Probabilistic diophantine approximation, I. Kronecker-sequences, Ann. Math. 140 (1994), 451–502.10.2307/2118607Search in Google Scholar

[4] BERNSTEIN, S. N.: The Theory of Probabilities. Gastehizdat Publishing House, Moscow, 1946.Search in Google Scholar

[5] BILYK, D.—LACEY, M. T. — VAGHARSHAKYAN, A.: On the small ball inequality in all dimensions, J. Funct. Anal. 254 (2008), 2470–2502.10.1016/j.jfa.2007.09.010Search in Google Scholar

[6] BOREL,É.: Les probabilitĂ©s denombrables et leurs applications arithmĂ©tiques, Rend. Circ. Mat. Palermo 27 (1909), 247–271.10.1007/BF03019651Search in Google Scholar

[7] DICK, J.: A note on the existence of sequences with small star discrepancy, J. Complexity 23 (2007), 649–652.10.1016/j.jco.2007.01.004Search in Google Scholar

[8] DICK, J.—KUO, F. Y.—SLOAN, I. H.: High-dimensional integration: the quasi-Monte Carlo way, Acta Numer. 22 (2013), 133–288.10.1017/S0962492913000044Search in Google Scholar

[9] DICK, J.—PILLICHSHAMMER, F.: Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge, 2010.10.1017/CBO9780511761188Search in Google Scholar

[10] DOERR, B.—GNEWUCH, M.—SRIVASTAV, A.: Bounds and constructions for the star-discrepancy via ή-covers, J. Complexity 21 (2005), 691–709.10.1016/j.jco.2005.05.002Search in Google Scholar

[11] DRMOTA, M.—TICHY, R. F.: Sequences, Discrepancies and Applications. In: Lecture Notes in Math. Vol. 1651, Springer-Verlag, Berlin, 1997.Search in Google Scholar

[12] GNEWUCH, M.: Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy, J. Complexity 24 (2008), 154–172.10.1016/j.jco.2007.08.003Search in Google Scholar

[13] GNEWUCH, M.: Gnewuch, M. Entropy, randomization, derandomization, and discrepancy. In: Monte Carlo and Quasi-Monte Carlo Methods 2010. Springer Proc. Math. Stat. Vol. 23, Springer-Verlag, Heidelberg, 2012, pp. 43–78,10.1007/978-3-642-27440-4_3Search in Google Scholar

[14] HEINRICH, S.—NOVAK, E.—WASILKOWSKI, G. W.—WOĆčNIAKOWSKI, H.: The inverse of the star discrepancy depends linearly on the dimension, Acta Arith. 96 (2001), 279–302.10.4064/aa96-3-7Search in Google Scholar

[15] HINRICHS, A.: Covering numbers, Vapnik-Červonenkis classes and bounds on the star-discrepancy, J. Complexity 20 (2004), 477–483.10.1016/j.jco.2004.01.001Search in Google Scholar

[16] KUIPERS, L.—NIEDERREITER, H.: Uniform Distribution of Sequences. John Wiley, New York (1974). Reprint, Dover Publications, Mineola, NY, 2006.Search in Google Scholar

[17] LARCHER, G.: On the distribution of an analog to classical Kronecker-sequences, J. Number Theory 52 (1995), 198–215.10.1006/jnth.1995.1065Search in Google Scholar

[18] LARCHER, G.—NIEDERREITER, H.: Kronecker-type sequences and nonarchimedean diophantine approximation, Acta Arith. 63 (1993), 380–396.Search in Google Scholar

[19] LARCHER, G.—PILLICHSHAMMER, F.: Metrical lower bounds on the discrepancy of digital Kronecker-sequences, J. Number Th. 135 (2014), 262–283.10.1016/j.jnt.2013.08.010Search in Google Scholar

[20] LEOBACHER, G.—PILLICHSHAMMER, F.: Introduction to Quasi-Monte Carlo Integration and Applications. In: Compact Textbooks in Mathematics, BirkhĂ€user, 2014.Search in Google Scholar

[21] LÖBBE, TH.: Probabilistic star discrepancy bounds for lacunary point sets (2014), see arXiv:1408.2220.Search in Google Scholar

[22] NIEDERREITER, H.: Random Number Generation and Quasi-Monte Carlo Methods. In: CBMS-NSF Series in Applied Mathematics Vol. 63, SIAM, Philadelphia, 1992.10.1137/1.9781611970081Search in Google Scholar

[23] NOVAK, E.—WOĆčNIAKOWSKI, H.: Tractability of Multivariate Problems, Volume I: Linear Information. EMS, ZĂŒrich, 2008.10.4171/026Search in Google Scholar

[24] NOVAK, E.—WOĆčNIAKOWSKI, H.: Tractability of Multivariate Problems, Volume II: Standard Information for Functionals. EMS, ZĂŒrich, 2010.10.4171/084Search in Google Scholar