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BMO and exponential Orlicz space estimates of the discrepancy function in arbitrary dimension

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Abstract

In the current paper, we obtain discrepancy estimates in exponential Orlicz and BMO spaces in arbitrary dimension d ≥ 3. In particular, we use dyadic harmonic analysis to prove that the dyadic product BMO and exp(L2/(d−1)) norms of the discrepancy function of so-called digital nets of order two are bounded above by (logN)(d−1)/2. The latter bound has been recently conjectured in several papers and is consistent with the best known low-discrepancy constructions. Such estimates play an important role as an intermediate step between the well-understood Lp bounds and the notorious open problem of finding the precise L∞ asymptotics of the discrepancy function in higher dimensions, which is still elusive.

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References

  1. G. Amirkhanyan, D. Bilyk, and M. Lacey, Estimates of the discrepancy function in exponential Orlicz spaces, (2014), availiable at https://doi.org/arxiv.org/abs/1306.1766.

    MATH  Google Scholar 

  2. J. Beck and W. W. L. Chen, Irregularities of Distribution, Cambridge University Press, Cambridge, 1987.

    Book  MATH  Google Scholar 

  3. A. Bernard, Espaces H1 de martingales à deux indices. Dualité avec des martingales de type “BMO’, Bull. Sci. Math. 103 (1979), 297–303.

    MathSciNet  MATH  Google Scholar 

  4. D. Bilyk, On Roth’s orthogonal function method in discrepancy theory, Unif. Distrib. Theory 6 (2011), 143–184.

    MathSciNet  MATH  Google Scholar 

  5. D. Bilyk, M. T. Lacey, I. Parissis, and A. Vagharshakyan, Exponential squared integrability of the discrepancy function in two dimensions, Mathematika 55 (2009), 2470–2502.

    Article  MathSciNet  MATH  Google Scholar 

  6. D. Bilyk, M. T. Lacey, and A. Vagharshakyan, On the small ball inequality in all dimensions, J. Funct. Anal. 254 (2008), 2470–2502.

    Article  MathSciNet  MATH  Google Scholar 

  7. S.-Y. A. Chang and R. Fefferman, A continuous version of duality of H1 with BMO on the bidisc, Ann. of Math. 112 (1980), 179–201.

    Article  MathSciNet  MATH  Google Scholar 

  8. S.-Y. A. Chang, J. M. Wilson, and T. H. Wolff, Some weighted norm inequalities concerning the Schrdinger operators, Comment. Math. Helv. 60 (1985), 217–246.

    Article  MathSciNet  MATH  Google Scholar 

  9. W. W. L. Chen, On irregularities of distribution, Mathematika 27 (1981), 153–170.

    Article  MATH  Google Scholar 

  10. W.W. L. Chen and M. M. Skriganov, Explicit constructions in the classical mean squares problem in irregularities of point distribution, J. Reine Angew. Math. 545 (2002), 67–95.

    MathSciNet  MATH  Google Scholar 

  11. W.W. L. Chen and M. M. Skriganov, Orthogonality and digit shifts in the classical mean squares problem in irregularities of point distribution, in Diophantine Approximation, Springer, Vienna, 2008, pp. 141–159.

    Chapter  Google Scholar 

  12. H. Davenport, Note on irregularities of distribution, Mathematika 3 (1956), 131–135.

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Dick, Explicit constructions of quasi-Monte Carlo rules for the numerical integration of highdimensional periodic functions, SIAM J. Numer. Anal. 45 (2007), 2141–2176.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Dick, Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order, SIAM J. Numer. Anal. 46 (2008), 1519–1553.

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Dick, Discrepancy bounds for infinite-dimensional order two digital sequences over F2, J. Number Theory 136 (2014), 204–232.

    Article  MathSciNet  MATH  Google Scholar 

  16. J. Dick and F. Pillichshammer, Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration, Cambridge University Press, Cambridge, 2010.

    Book  MATH  Google Scholar 

  17. J. Dick and F. Pillichshammer, OptimalL2 discrepancy bounds for higher order digital sequences over the finite field F2, Acta Arith. 162 (2014), 65–99.

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Dick and F. Pillichshammer, Explicit constructions of point sets and sequences with low discrepancy, in Uniform Distribution and Quasi-Monte Carlo Methods: Discrepancy, Integration and Applications, Walter DeGruyter GmbH, Berlin/Boston, 2014, pp. 63–86.

    Google Scholar 

  19. H. Faure, F. Pillichshammer, G. Pirsic, and W. Ch. Schmid, L2 discrepancy of generalized twodimensional Hammersley point sets scrambled with arbitrary permutations, Acta Arith. 141 (2010), 395–418.

    Article  MathSciNet  MATH  Google Scholar 

  20. G. Halász, On Roth’s method in the theory of irregularities of point distributions, in Recent Progress in Analytic Number Theory, Vol. 2, Academic Press, London-New York, 1981, pp. 79–94.

    Google Scholar 

  21. J. H. Halton, On the efficiency of certain quasi-random sequences of points in evaluating multidimensional integrals, Numer. Math. 2 (1960), 84–90.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Hinrichs, Discrepancy of Hammersley points in Besov spaces of dominating mixed smoothness, Math. Nachr. 283 (2010), 478–488.

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Hinrichs, Discrepancy, integration and tractability, in Monte Carlo and Quasi-Monte Carlo Methods 2012, Springer, Berlin-Heidelberg, 2013, pp. 129–172.

    Chapter  Google Scholar 

  24. A. Hinrichs and L. Markhasin, On lower bounds for the L2-discrepancy, J. Complexity 27 (2011), 127–132.

    Article  MathSciNet  MATH  Google Scholar 

  25. L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, John Wiley & Sons, Ltd., New York, 1974.

    MATH  Google Scholar 

  26. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. I., Springer, Berlin, 1977.

    Book  MATH  Google Scholar 

  27. L. Markhasin, Discrepancy of generalized Hammersley type point sets in Besov spaces with dominating mixed smoothness, Unif. Distrib. Theory 8 (2013), 135–164.

    MathSciNet  MATH  Google Scholar 

  28. L. Markhasin, Quasi-Monte Carlo methods for integration of functions with dominating mixed smoothness in arbitrary dimension, J. Complexity 29 (2013), 370–388.

    Article  MathSciNet  MATH  Google Scholar 

  29. L. Markhasin, Discrepancy and integration in function spaces with dominating mixed smoothness, Dissertationes Math. 494 (2013), 1–81.

    Article  MathSciNet  MATH  Google Scholar 

  30. L. Markhasin, L2- and Srp, qB-discrepancy of (order 2) digital nets, Acta Arith. 168 (2015), 139–160.

    MathSciNet  MATH  Google Scholar 

  31. J. Matoušek, Geometric discrepancy. An Illustrated Guide, Springer-Verlag, Berlin, 1999.

    Book  MATH  Google Scholar 

  32. H. Niederreiter, Point sets and sequences with small discrepancy, Monatsh. Math. 104 (1987), 273–337.

    Article  MathSciNet  MATH  Google Scholar 

  33. E. Novak and H. Woźniakowski, Tractability of Multivariate Problems. Volume II: Standard Information for Functionals European Mathematical Society Publishing House, Zürich, 2010.

    Book  MATH  Google Scholar 

  34. J. Pipher and L. Ward, BMO from dyadic BMO on the bydisc, J. London Math. Soc. 77 (2008), 524–544.

    Article  MathSciNet  MATH  Google Scholar 

  35. K. F. Roth, On irregularities of distribution, Mathematika 1 (1954), 73–79.

    Article  MathSciNet  MATH  Google Scholar 

  36. K. F. Roth, On irregularities of distribution. IV, Acta Arith. 37 (1980), 67–75.

    Article  MathSciNet  MATH  Google Scholar 

  37. W. M. Schmidt, Irregularities of distribution. VII, Acta Arith. 21 (1972), 45–50.

    Article  MathSciNet  MATH  Google Scholar 

  38. W. M. Schmidt, Irregularities of distribution X, in Number Theory and Algebra, Academic Press, New York, 1977, pp. 311–329.

    Google Scholar 

  39. M. M. Skriganov, Harmonic analysis on totally disconnected groups and irregularities of point distributions, J. Reine Angew. Math. 600 (2006), 25–49.

    MathSciNet  MATH  Google Scholar 

  40. M. M. Skriganov, On mean values of the Lq-discrepancies of point distributions, St. Petersburg Math. J. 24 (2013), 991–1012.

    Article  MathSciNet  MATH  Google Scholar 

  41. M. M. Skriganov, Dyadic shift randomizations in classical discrepancy theory, Mathematikca 62 (2016), 183–209.

    Article  MathSciNet  MATH  Google Scholar 

  42. I. M. Sobol, The distribution of points in a cube and the approximate evaluation of integrals, Zh. Vychisl. Mat. i Mat. Fiz. 7 (1967), 784–802.

    MathSciNet  Google Scholar 

  43. H. Triebel, Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration, European Mathematical Society Publishing House, Zürich, 2010.

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, University of Minnesota, 206 Church St. SE, Minneapolis, MN, 55408, USA

    Dmitriy Bilyk

  2. Institut für Stochastik und Anwendungen, Universität Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart, Germany

    Lev Markhasin

Authors
  1. Dmitriy Bilyk
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  2. Lev Markhasin
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Correspondence to Dmitriy Bilyk.

Additional information

The first author was supported by the NSF grant DMS-1260516.

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Bilyk, D., Markhasin, L. BMO and exponential Orlicz space estimates of the discrepancy function in arbitrary dimension. JAMA 135, 249–269 (2018). https://doi.org/10.1007/s11854-018-0035-x

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  • DOI: https://doi.org/10.1007/s11854-018-0035-x

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