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On Pattern Entropy of Weak Model Sets

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Abstract

We study point sets arising from cut-and-project constructions. An important class is that of weak model sets, which include squarefree numbers and visible lattice points. For such model sets, we give a non-trivial upper bound on their pattern entropy in terms of the volume of the window boundary in internal space. This proves a conjecture by R.V. Moody.

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Acknowledgments

This work was initiated during a visit of one of the authors to Bielefeld University in spring 2014. The support by the German Research Council (DFG) within the CRC 701 “Spectral Structures and Topological Methods in Mathematics” is gratefully acknowledged. Results of this paper have been presented [30] during the Mini-Workshop “Dynamical versus Diffraction Spectra in the Theory of Quasicrystals” at the MFO in Oberwolfach in December 2014. We thank the participants for discussions and Michael Baake for helpful comments on the manuscript. We also thank the referee for careful reading, comments and a correction.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501, Bielefeld, Germany

    Christian Huck

  2. Department für Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstraße 11, 91058, Erlangen, Germany

    Christoph Richard

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  1. Christian Huck
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  2. Christoph Richard
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Correspondence to Christian Huck.

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Editor in Charge: Günter M. Ziegler.

To the memory of Peter A. B. Pleasants.

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Huck, C., Richard, C. On Pattern Entropy of Weak Model Sets. Discrete Comput Geom 54, 741–757 (2015). https://doi.org/10.1007/s00454-015-9718-6

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  • DOI: https://doi.org/10.1007/s00454-015-9718-6

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