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Irregularities of Distribution and Average Decay of Fourier Transforms

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A Panorama of Discrepancy Theory

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

Abstract

In Geometric Discrepancy we usually test a distribution of N points against a suitable family of sets. If this family consists of dilated, translated and rotated copies of a given d-dimensional convex body \(D \subset \left [0,1\right )^{d}\), then a result proved by W. Schmidt, J. Beck and H. Montgomery shows that the corresponding L 2 discrepancy cannot be smaller than \(c_{d}N^{\left (d-1\right )/2d}\). Moreover, this estimate is sharp, thanks to results of D. Kendall, J. Beck and W. Chen. Both lower and upper bounds are consequences of estimates of the decay of \(\left \Vert \hat{\chi }_{D}\left (\rho \cdot \right )\right \Vert _{L^{2}\left (\varSigma _{d-1}\right )}\) for large ρ, where \(\hat{\chi }_{D}\) is the Fourier transform (expressed in polar coordinates) of the characteristic function of the convex body D, while Σ d−1 is the unit sphere in \(\mathbb{R}^{d}\). In this chapter we provide the Fourier analytic background and we carefully investigate the relation between the L 2 discrepancy and the estimates of \(\left \Vert \hat{\chi }_{D}\left (\rho \cdot \right )\right \Vert _{L^{2}\left (\varSigma _{d-1}\right )}\).

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Notes

  1. 1.

    Actually the convexity hypothesis allows us to integrate by parts at least once without using any regularity assumption on ∂ B. In this way we get the bound ρ −1 (uniformly in σ), which is enough to prove the theorem in the dimensions d = 2 and d = 3.

  2. 2.

    D. Kendall seems to have been the first one to realize that certain lattice points problems can be handled using multi-dimensional Fourier analysis.

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Authors and Affiliations

  1. Dipartimento di Ingegneria, Università di Bergamo, Viale Marconi 5, 24044, Dalmine, Bergamo, Italia

    Luca Brandolini & Giacomo Gigante

  2. Dipartimento di Statistica e Metodi Quantitativi, Edificio U7, Università di Milano-Bicocca, Via Bicocca degli Arcimboldi 8, 20126, Milano, Italia

    Giancarlo Travaglini

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  1. Luca Brandolini
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  2. Giacomo Gigante
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  3. Giancarlo Travaglini
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Correspondence to Luca Brandolini or Giancarlo Travaglini .

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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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Brandolini, L., Gigante, G., Travaglini, G. (2014). Irregularities of Distribution and Average Decay of Fourier Transforms. 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_3

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