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Upper Bounds in Classical Discrepancy Theory

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A Panorama of Discrepancy Theory

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

Abstract

We discuss some of the ideas behind the study of upper bound questions in classical discrepancy theory. The many ideas involved come from diverse areas of mathematics and include diophantine approximation, probability theory, number theory and various forms of Fourier analysis. We illustrate these ideas by largely restricting our discussion to two dimensions.

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Notes

  1. 1.

    It was put to the first author by a rather preposterous engineering colleague many years ago that this could be achieved easily by a square lattice in the obvious way. Not quite the case, as an obvious way would be far from so to this colleague.

  2. 2.

    For those readers not familiar with the theory of diophantine approximation, just take any quadratic irrational like \(\sqrt{2}\) or \(\sqrt{3}\).

  3. 3.

    In their paper, Halton and Zaremba have an exact expression for the integral under study.

  4. 4.

    This is not the case if we wish to study Theorem 10 for k > 2.

  5. 5.

    Note that the set \(\mathcal{Q}_{h}\) here is different from that in the last section. However, since we are working with rectangles inside [0, 1) × [0, 2h), our statements here concerning \(\mathcal{Q}_{h}\) remain valid for the set \(\mathcal{Q}_{h}\) defined in the last section.

  6. 6.

    Simply imagine that we use Fourier analysis but with the Walsh functions replacing the exponential functions.

  7. 7.

    Here we somewhat abuse notation, as t clearly has more coordinates than p. In the sequel, W l (t) is really W l (0 ⊕t), notation abused again.

  8. 8.

    The assumption that p ≥ k − 1 cannot be relaxed, as noted by Chen [9].

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Acknowledgements

The research of the second author has been supported by RFFI Project No. 08-01-00182.

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Authors and Affiliations

  1. Department of Mathematics, Macquarie University, Sydney, NSW, 2109, Australia

    William Chen

  2. Steklov Mathematical Institute, Fontanka 27, St Petersburg, 191011, Russia

    Maxim Skriganov

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  1. William Chen
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Correspondence to William Chen .

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Editors and Affiliations

  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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Chen, W., Skriganov, M. (2014). Upper Bounds in Classical Discrepancy Theory. 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_1

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