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Equality Case in van der Corput’s Inequality and Collisions in Multiple Lattice Tilings

  • Gennadiy AverkovEmail author
Article
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Abstract

Van der Corput’s provides the sharp bound \(\mathop {\mathrm {vol}}\nolimits (C) \le m 2^d\) on the volume of a d-dimensional origin-symmetric convex body C that has \(2m-1\) points of the integer lattice in its interior. For \(m=1\), a characterization of the equality case \(\mathop {\mathrm {vol}}\nolimits (C)= m 2^d\) is equivalent to the well-known problem of characterizing tilings by translations of a convex body. It is rather surprising that so far, for \(m \ge 2\), no characterization of the equality case has been available, though a hint to the respective characterization problem can be found in the 1987 monograph of Gruber and Lekkerkerker. We give an explicit characterization of the equality case for all \(m \ge 2\). Our result reveals that, the equality case for \(m \ge 2\) is more restrictive than for \(m=1\). We also present consequences of our characterization in the context of multiple lattice tilings.

Keywords

Lattice Multiple tiling Tiling Van der Corput’s inequality 

Mathematics Subject Classification

05B45 11H06 52C22 

Notes

Acknowledgements

Research supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—314838170, GRK 2297 MathCoRe.

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of MathematicsOtto-von-Guericke-Universität MagdeburgMagdeburgGermany

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