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.
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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).
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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
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