Abstract
A great challenge in the analysis of the discrepancy function D N is to obtain universal lower bounds on the L ∞ norm of D N in dimensions d ≥ 3. It follows from the L 2 bound of Klaus Roth that \(\Vert D_{N}\Vert _{\infty }\geq \Vert D_{N}\Vert _{2} \gtrsim {(\log N)}^{(d-1)/2}\). It is conjectured that the L ∞ bound is significantly larger, but the only definitive result is that of Wolfgang Schmidt in dimension d = 2. Partial improvements of the Roth exponent (d − 1)∕2 in higher dimensions have been established by the authors and Armen Vagharshakyan. We survey these results, the underlying methods, and some of their connections to other subjects in probability, approximation theory, and analysis.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Beck, J.: A two-dimensional van Aardenne-Ehrenfest theorem in irregularities of distribution. Compos. Math. 72, 269–339 (1989)
Beck, J., Chen, W.W.L.: Irregularities of Distribution. Cambridge University Press, Cambridge (1987)
Bilyk, D.: On Roth’s orthogonal function method in discrepancy theory. Unif. Distrib. Theory 6, 143–184 (2011)
Bilyk, D.: Roth’s orthogonal function method in discrepancy theory and some new connections. In: Chen, W., Srivastav, A., Travaglini, G. (eds.) Panorama of Discrepancy Theory. Springer (2013–14, to appear)
Bilyk, D., Lacey, M.T.: On the small ball inequality in three dimensions. Duke Math. J. 143, 81–115 (2008)
Bilyk, D., Lacey, M.T., Parissis, I., Vagharshakyan, A.: Exponential squared integrability of the discrepancy function in two dimensions. Mathematika 55, 1–27 (2009)
Bilyk, D., Lacey, M.T., Parissis, I., Vagharshakyan, A.: A three-dimensional signed small ball inequality. In: Dependence in Probability, Analysis and Number Theory, pp. 73–87. Kendrick, Heber City (2010)
Bilyk, D., Lacey, M.T., Vagharshakyan, A.: On the small ball inequality in all dimensions. J. Funct. Anal. 254, 2470–2502 (2008)
Bilyk, D., Lacey, M.T., Vagharshakyan, A.: On the signed small ball inequality. Online J. Anal. Comb. 3 (2008)
Chang, S.-Y.A., Wilson, J.M., Wolff, T.H.: Some weighted norm inequalities concerning the Schrödinger operators. Comment. Math. Helv. 60, 217–246 (1985)
Dick, J., Pillichshammer, F.: Digital Nets and Sequences. Cambridge University Press, Cambridge (2010)
Drmota, M., Tichy, R.: Sequences, Discrepancies and Applications. Springer, Berlin (1997)
Dunker, T., Kühn, T., Lifshits, M., Linde, W.: Metric entropy of the integration operator and small ball probabilities for the brownian sheet. C. R. Acad. Sci. Paris Sér. I Math. 326, 347–352 (1998)
Fefferman, R., Pipher, J.: Multiparameter operators and sharp weighted inequalities. Amer. J. Math. 119, 337–369 (1997)
Halász, G.: On Roth’s method in the theory of irregularities of point distributions. In: Recent Progress in Analytic Number Theory, vol. 2, pp. 79–94. Academic, London (1981)
Kuelbs, J., Li, W.V.: Metric entropy and the small ball problem for Gaussian measures. C. R. Acad. Sci. Paris Sér. I Math. 315, 845–850 (1992)
Kuelbs, J., Li, W.V.: Metric entropy and the small ball problem for Gaussian measures. J. Funct. Anal. 116, 133–157 (1993)
Lacey, M.: On the discrepancy function in arbitrary dimension, close to L 1. Anal. Math. 34, 119–136 (2008)
Matoušek, J.: Geometric Discrepancy. Springer, Berlin (2010)
Riesz, F.: Über die Fourierkoeffizienten einer stetigen Funktion von beschränkter Schwankung. Math. Z. 2, 312–315 (1918)
Roth, K.F.: On irregularities of distribution. Mathematika 1, 73–79 (1954)
Schmidt, W.M.: Irregularities of distribution, VII. Acta Arith. 21, 45–50 (1972)
Schmidt, W.M.: Irregularities of distribution, X. In: Zassenhaus, H. (ed.) Number Theory and Algebra, pp. 311–329. Academic, New York (1977)
Sidon, S.: Verallgemeinerung eines Satzes über die absolute Konvergenz von Fourierreihen mit Lücken. Math. Ann. 97, 675–676 (1927)
Sidon, S.: Ein Satz über trigonometrische Polynome mit Lücken und seine Anwendung in der Theorie der Fourier-Reihen. J. Reine Angew. Math. 163, 251–252 (1930)
Talagrand, M.: The small ball problem for the Brownian sheet. Ann. Probab. 22, 1331–1354 (1994)
Temlyakov, V.N.: An inequality for trigonometric polynomials and its application for estimating the entropy numbers. J. Complexity 11, 293–307 (1995)
Zygmund, A.: Trigonometric Series, vols. I, II. Cambridge University Press, Cambridge (2002)
Acknowledgements
This research is supported in part by NSF grants DMS 1101519, 1260516 (Dmitriy Bilyk), DMS 0968499, and a grant from the Simons Foundation #229596 (Michael Lacey).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bilyk, D., Lacey, M. (2013). The Supremum Norm of the Discrepancy Function: Recent Results and Connections. In: Dick, J., Kuo, F., Peters, G., Sloan, I. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2012. Springer Proceedings in Mathematics & Statistics, vol 65. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41095-6_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-41095-6_2
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-41094-9
Online ISBN: 978-3-642-41095-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)