Skip to main content Accessibility help



The discrepancy function measures the deviation of the empirical distribution of a point set in $[0,1]^{d}$ from the uniform distribution. In this paper, we study the classical discrepancy function with respect to the bounded mean oscillation and exponential Orlicz norms, as well as Sobolev, Besov and Triebel–Lizorkin norms with dominating mixed smoothness. We give sharp bounds for the discrepancy function under such norms with respect to infinite sequences.



Hide All
1. Beck, J. and Chen, W. W. L., Irregularities of Distribution, Cambridge University Press (Cambridge, 1987).
2. Bilyk, D., Lacey, M. T., Parissis, I. and Vagharshakyan, A., Exponential squared integrability of the discrepancy function in two dimensions. Mathematika 55 2009, 24702502.
3. Bilyk, D., Lacey, M. T. and Vagharshakyan, A., On the small ball inequality in all dimensions. J. Funct. Anal. 254 2008, 24702502.
4. Bilyk, D. and Markhasin, L., BMO and exponential Orlicz space estimates of the discrepancy function in arbitrary dimension. J. Anal. Math. (to appear).
5. Chang, S.-Y. A. and Fefferman, R., A continuous version of duality of H 1 with BMO on the bidisc. Ann. of Math. (2) 112 1980, 179201.
6. Chang, S.-Y. A., Wilson, J. M. and Wolff, T. H., Some weighted norm inequalities concerning the Schrödinger operators. Comment. Math. Helv. 60 1985, 217246.
7. Chen, W. W. L., On irregularities of distribution. Mathematika 27 1981, 153170.
8. Chen, W. W. L. and Skriganov, M. M., Explicit constructions in the classical mean squares problem in irregularities of point distribution. J. Reine Angew. Math. 545 2002, 6795.
9. Dick, J., Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high-dimensional periodic functions. SIAM J. Numer. Anal. 45 2007, 21412176.
10. Dick, J., Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order. SIAM J. Numer. Anal. 46 2008, 15191553.
11. Dick, J., Discrepancy bounds for infinite-dimensional order two digital sequences over F2 . J. Number Theory 136 2014, 204232.
12. Dick, J. and Baldeaux, J., Equidistribution properties of generalized nets and sequences. In Monte Carlo and Quasi-Monte Carlo Methods 2008 (eds L’Ecuyer, P. and Owen, A.), Springer (Berlin–Heidelberg, 2009), 305323.
13. Dick, J., Hinrichs, A., Markhasin, L. and Pillichshammer, F., Optimal L p -discrepancy bounds for second order digital sequences. Israel J. Math. 221 2017, 489510.
14. Dick, J. and Pillichshammer, F., Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration, Cambridge University Press (Cambridge, 2010).
15. Dick, J. and Pillichshammer, F., Optimal L2 discrepancy bounds for higher order digital sequences over the finite field F2 . Acta Arith. 162 2014, 6599.
16. Dick, J. and Pillichshammer, F., Explicit constructions of point sets and sequences with low discrepancy. In Uniform Distribution and Quasi-Monte Carlo Methods – Discrepancy, Integration and Applications (Radon Series of Computational and Applied Mathematics) (eds Kritzer, P., Niederreiter, H., Pillichshammer, F. and Winterhof, A.), De Gruyter (Berlin/Boston, 2014), 6386.
17. Drmota, M. and Tichy, R. F., Sequences, Discrepancies and Applications (Lecture Notes in Mathematics 1651 ), Springer (Berlin, 1997).
18. Hinrichs, A., Discrepancy of Hammersley points in Besov spaces of dominating mixed smoothness. Math. Nachr. 283 2010, 478488.
19. Kritzinger, R., L p - and S p, q r B-discrepancy of the symmetrized van der Corput sequence and modified Hammersley point sets in arbitrary bases. J. Complexity 33 2016, 145168.
20. Kuipers, L. and Niederreiter, H., Uniform Distribution of Sequences, John Wiley (New York, NY, 1974) , reprinted by Dover (Mineola, NY, 2006).
21. Leobacher, G. and Pillichshammer, F., Introduction to Quasi-Monte Carlo Integration and Applications (Compact Textbooks in Mathematics), Birkhäuser/Springer (Cham, 2014).
22. Lindenstrauss, J. and Tzafriri, L., Classical Banach Spaces, I, Springer (Berlin, 1977).
23. Markhasin, L., Discrepancy of generalized Hammersley type point sets in Besov spaces with dominating mixed smoothness. Unif. Distrib. Theory 8 2013, 135164.
24. Markhasin, L., Discrepancy and integration in function spaces with dominating mixed smoothness. Dissertationes Math. (Rozprawy Mat.) 494 2013, 181.
25. Markhasin, L., Quasi-Monte Carlo methods for integration of functions with dominating mixed smoothness in arbitrary dimension. J. Complexity 29 2013, 370388.
26. Markhasin, L., L p - and S p, q r B-discrepancy of (order 2) digital nets. Acta Arith. 168 2015, 139159.
27. Matoušek, J., Geometric Discrepancy. An Illustrated Guide (Algorithms and Combinatorics 18 ), Springer (Berlin, 1999).
28. Niederreiter, H., Point sets and sequences with small discrepancy. Monatsh. Math. 104 1987, 273337.
29. Niederreiter, H., Random Number Generation and Quasi-Monte Carlo Methods (CBMS-NSF Regional Conference Series in Applied Mathematics 63 ), SIAM (Philadelphia, 1992).
30. Niederreiter, H. and Xing, C. P., Low-discrepancy sequences and global function fields with many rational places. Finite Fields Appl. 2 1996, 241273.
31. Proinov, P. D., On irregularities of distribution. C. R. Acad. Bulgare Sci. 39 1986, 3134.
32. Roth, K. F., On irregularities of distribution. Mathematika 1 1954, 7379.
33. Roth, K. F., On irregularities of distribution. IV. Acta Arith. 37 1980, 6775.
34. Schmidt, W. M., Irregularities of distribution. VII. Acta Arith. 21 1972, 4550.
35. Schmidt, W. M., Irregularities of distribution X. In Number Theory and Algebra, Academic Press (New York, NY, 1977), 311329.
36. Skriganov, M. M., Harmonic analysis on totally disconnected groups and irregularities of point distributions. J. Reine Angew. Math. 600 2006, 2549.
37. Sobol’, I. M., The distribution of points in a cube and the approximate evaluation of integrals. Zh. Vychisl. Mat. Mat. Fiz. 7 1967, 784802.
38. Tao, T., A type diagram for function spaces. What’s new – a blog by T. Tao, 11 March 2010,
39. Tezuka, S., Polynomial arithmetic analogue of Halton sequences. ACM Trans. Model. Comput. Simul. 3 1993, 99107.
40. Triebel, H., Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration, European Mathematical Society (Zürich, 2010).
41. Triebel, H., Numerical integration and discrepancy. A new approach. Math. Nachr. 283 2010, 139159.
MathJax is a JavaScript display engine for mathematics. For more information see

MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed