The Supremum Norm of the Discrepancy Function: Recent Results and Connections

Dmitriy Bilyk; Michael Lacey

doi:10.1007/978-3-642-41095-6_2

The Supremum Norm of the Discrepancy Function: Recent Results and Connections

Dmitriy Bilyk, Michael Lacey

Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

7 Scopus citations

Abstract

A great challenge in the analysis of the discrepancy function D_N is to obtain universal lower bounds on the L^∞ norm of DN in dimensions d≥3. It follows from the L² bound of Klaus Roth that {pipe}DN{pipe}_∞≥ {pipe}DN{pipe}₂ > (logN)^(d-1)=2 It is conjectured that the L^∞ bound is significantly larger, but the only definitive result is that ofWolfgang 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.

Original language	English (US)
Title of host publication	Monte Carlo and Quasi-Monte Carlo Methods 2012
Pages	23-38
Number of pages	16
DOIs	https://doi.org/10.1007/978-3-642-41095-6_2
State	Published - 2013
Event	10th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, MCQMC 2012 - Sydney, NSW, Australia Duration: Feb 13 2012 → Feb 17 2012

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	65
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Other

Other	10th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, MCQMC 2012
Country/Territory	Australia
City	Sydney, NSW
Period	2/13/12 → 2/17/12

Bibliographical note

Funding Information:
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 Monte Carlo and Quasi-Monte Carlo Methods 2012. Springer Proceedings in Mathematics and Statistics, vol. 65, pp. 23-38, 10th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, MCQMC 2012, Sydney, NSW, Australia, 2/13/12. https://doi.org/10.1007/978-3-642-41095-6_2

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abstract = "A great challenge in the analysis of the discrepancy function DN is to obtain universal lower bounds on the L∞ norm of DN in dimensions d≥3. It follows from the L2 bound of Klaus Roth that {pipe}DN{pipe}∞≥ {pipe}DN{pipe}2 > (logN)(d-1)=2 It is conjectured that the L∞ bound is significantly larger, but the only definitive result is that ofWolfgang 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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N2 - A great challenge in the analysis of the discrepancy function DN is to obtain universal lower bounds on the L∞ norm of DN in dimensions d≥3. It follows from the L2 bound of Klaus Roth that {pipe}DN{pipe}∞≥ {pipe}DN{pipe}2 > (logN)(d-1)=2 It is conjectured that the L∞ bound is significantly larger, but the only definitive result is that ofWolfgang 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.

AB - A great challenge in the analysis of the discrepancy function DN is to obtain universal lower bounds on the L∞ norm of DN in dimensions d≥3. It follows from the L2 bound of Klaus Roth that {pipe}DN{pipe}∞≥ {pipe}DN{pipe}2 > (logN)(d-1)=2 It is conjectured that the L∞ bound is significantly larger, but the only definitive result is that ofWolfgang 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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The Supremum Norm of the Discrepancy Function: Recent Results and Connections

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