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Highly irregular separated nets

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Abstract

In 1998 Burago and Kleiner and (independently) McMullen gave examples of separated nets in Euclidean space which are bilipschitz non-equivalent to the integer lattice. We study weaker notions than bilipschitz equivalence and demonstrate that such notions also distinguish between separated nets. Put differently, we find occurrences of particularly strong divergence of separated nets from the integer lattice. Our approach generalises that of Burago and Kleiner and McMullen which takes place largely in a continuous setting. Existence of irregular separated nets is verified via the existence of non-realisable density functions ρ: [0,1]d → (0, ∞). In the present work we obtain stronger types of non-realisable densities.

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Acknowledgements

We would like to thank the anonymous referees for their very careful reading of the paper and many comments and suggestions that greatly improved the paper.

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Author notes

  1. Michael Dymond

    Present address: School of Mathematics, University of Birmingham, Birmingham, B15 2TT, UK

  2. Vojtěch Kaluža

    Present address: Institute of Science and Technology Austria, Am Campus 1, 3400, Klosterneuburg, Austria

Authors and Affiliations

  1. Institut für Mathematik, Universität Innsbruck, Technikerstraße 13, 6020, Innsbruck, Austria

    Michael Dymond & Vojtěch Kaluža

Authors
  1. Michael Dymond
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  2. Vojtěch Kaluža
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Corresponding author

Correspondence to Michael Dymond.

Additional information

This work was done while both authors were employed at the University of Innsbruck and enjoyed the full support of Austrian Science Fund (FWF): P 30902-N35.

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Dymond, M., Kaluža, V. Highly irregular separated nets. Isr. J. Math. 253, 501–554 (2023). https://doi.org/10.1007/s11856-022-2448-6

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  • DOI: https://doi.org/10.1007/s11856-022-2448-6

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