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Roth’s Orthogonal Function Method in Discrepancy Theory and Some New Connections

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A Panorama of Discrepancy Theory

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2107))

Abstract

In this survey we give a comprehensive, but gentle introduction to the circle of questions surrounding the classical problems of discrepancy theory, unified by the same approach originated in the work of Klaus Roth (Mathematika 1:73–79, 1954) and based on multiparameter Haar (or other orthogonal) function expansions. Traditionally, the most important estimates of the discrepancy function were obtained using variations of this method. However, despite a large amount of work in this direction, the most important questions in the subject remain wide open, even at the level of conjectures. The area, as well as the method, has enjoyed an outburst of activity due to the recent breakthrough improvement of the higher-dimensional discrepancy bounds and the revealed important connections between this subject and harmonic analysis, probability (small deviation of the Brownian motion), and approximation theory (metric entropy of spaces with mixed smoothness). Without assuming any prior knowledge of the subject, we present the history and different manifestations of the method, its applications to related problems in various fields, and a detailed and intuitive outline of the latest higher-dimensional discrepancy estimate.

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  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55408, USA

    Dmitriy Bilyk

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  1. Department of Mathematics, Macquarie University, Sydney, New South Wales, Australia

    William Chen

  2. Department of Computer Science, Kiel University, Kiel, Germany

    Anand Srivastav

  3. Dipartimento di Statistica e Metodi Quantitativi, Università di Milano-Bicocca, Milano, Italy

    Giancarlo Travaglini

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Bilyk, D. (2014). Roth’s Orthogonal Function Method in Discrepancy Theory and Some New Connections. In: Chen, W., Srivastav, A., Travaglini, G. (eds) A Panorama of Discrepancy Theory. Lecture Notes in Mathematics, vol 2107. Springer, Cham. https://doi.org/10.1007/978-3-319-04696-9_2

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