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Topics in Mathematical Analysis and Applications pp 583–599Cite as

Integer Points in Large Bodies

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Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 94))

Abstract

For a compact body \(\mathcal{B}\) in three-dimensional Euclidean space with sufficiently smooth boundary, the number \(N(\mathcal{B};t)\) of points with integer coordinates in a linearly enlarged copy \(t\mathcal{B}\) is approximated in first order by the volume \(\mathrm{vol}(\mathcal{B})t^{3}\). This article provides a survey on the state of art of research on the lattice discrepancy \(D(\mathcal{B};t) = N(\mathcal{B};t) -\mathrm{vol}(\mathcal{B})t^{3}\), starting from the classic theory and emphasizing recent developments and advances.

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Notes

  1. 1.

    For the definitions of the order symbols O, Ω, ≪ , \(\asymp \), etc., see, e.g., Krätzel’s book [22].

  2. 2.

    A weaker version of the first asymptotics, with error term \(O(t^{3/2-1/286+\varepsilon })\), has been established by Popov [45].

  3. 3.

    This subsection describes quite recent research by the author which is in course of publication elsewhere [43].

  4. 4.

    Or not, if one has learned the right lesson from the example of the torus.

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Acknowledgements

This survey article is based on the author’s plenary lecture at the International Conference on Elementary and Analytic Number Theory 2012 (ELAZ’12), held at Schloß Schney near Würzburg, Germany, in August 2012. The author is glad to use this opportunity to thank the organizers of that conference, Mr. and Mrs. Jörn and Rasa Steuding, along with their amazing team, for the wonderful organization and all their kindness and hospitality.

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Authors and Affiliations

  1. Department of Integrative Biology, Institute of Mathematics, BOKU Wien (University of Natural Resources and Life Sciences, Vienna), Vienna, Austria

    Werner Georg Nowak

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  1. Werner Georg Nowak
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Correspondence to Werner Georg Nowak .

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Editors and Affiliations

  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

  2. Department of Mathematics, University of Pécs, Pécs, Hungary

    László Tóth

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Nowak, W.G. (2014). Integer Points in Large Bodies. In: Rassias, T., Tóth, L. (eds) Topics in Mathematical Analysis and Applications. Springer Optimization and Its Applications, vol 94. Springer, Cham. https://doi.org/10.1007/978-3-319-06554-0_26

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