Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan pp 59-74 | Cite as

# Irregularities of Distributions and Extremal Sets in Combinatorial Complexity Theory

## Abstract

In 2004 the second author of the present paper proved that a point set in [0, 1]^{d} which has star-discrepancy at most *ε* must necessarily consist of at least *c*_{abs}*dε*^{−1} points. Equivalently, every set of *n* points in [0, 1]^{d} must have star-discrepancy at least *c*_{abs}*dn*^{−1}. The original proof of this result uses methods from Vapnik–Chervonenkis theory and from metric entropy theory. In the present paper we give an elementary combinatorial proof for the same result, which is based on identifying a sub-box of [0, 1]^{d} which has approximately *d* elements of the point set on its boundary. Furthermore, we show that a point set for which no such box exists is rather irregular, and must necessarily have a large star-discrepancy.

## Notes

### Acknowledgements

The first author is supported by the Austrian Science Fund (FWF), projects F5507-N26 and I1751-N26, and by the FWF START project Y-901-N35.

## References

- 1.Bilyk, D.: Roth’s orthogonal function method in discrepancy theory and some new connections. In: A Panorama of Discrepancy Theory. Lecture Notes in Mathematics, vol. 2107, pp. 71–158. Springer, Cham (2014)zbMATHGoogle Scholar
- 2.Bilyk, D., Lacey, M.T., Vagharshakyan, A.: On the small ball inequality in all dimensions. J. Funct. Anal.
**254**(9), 2470–2502 (2008)MathSciNetCrossRefGoogle Scholar - 3.Dick, J., Pillichshammer, F.: Digital Nets and Sequences. Cambridge University Press, Cambridge (2010).CrossRefGoogle Scholar
- 4.Drmota, M., Tichy, R.F.: Sequences, Discrepancies and Applications. Lecture Notes in Mathematics, vol. 1651. Springer, Berlin (1997)Google Scholar
- 5.Heinrich, S.: Some open problems concerning the star-discrepancy. J. Complex.
**19**(3), 416–419 (2003)MathSciNetCrossRefGoogle Scholar - 6.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)MathSciNetCrossRefGoogle Scholar - 7.Hinrichs, A.: Covering numbers, Vapnik-Červonenkis classes and bounds for the star-discrepancy. J. Complex.
**20**(4), 477–483 (2004)zbMATHGoogle Scholar - 8.Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Wiley-Interscience [Wiley], New York, London, Sydney (1974)Google Scholar
- 9.Mohri, M., Rostamizadeh, A., Talwalkar, A.: Foundations of Machine Learning. Adaptive Computation and Machine Learning. MIT Press, Cambridge, MA (2012)zbMATHGoogle Scholar
- 10.Novak, E., Woźniakowski, H.: Tractability of Multivariate Problems. Vol. 1: Linear Information. EMS Tracts in Mathematics, vol. 6. European Mathematical Society, Zürich (2008)Google Scholar
- 11.Novak, E., Woźniakowski, H.: Tractability of Multivariate Problems. Vol. 2: Standard Information for Functionals. EMS Tracts in Mathematics, vol. 12. European Mathematical Society, Zürich (2010)Google Scholar
- 12.Roth, K.F.: On irregularities of distribution. Mathematika
**1**, 73–79 (1954)MathSciNetCrossRefGoogle Scholar