Highly irregular separated nets

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Journal Article | Epub ahead of print | English

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Author
Dymond, Michael; Kaluza, VojtechISTA
Department
Abstract
In 1998 Burago and Kleiner and (independently) McMullen gave examples of separated nets in Euclidean space which are non-bilipschitz equivalent to the integer lattice. We study weaker notions of equivalence of separated nets and demonstrate that such notions also give rise to distinct equivalence classes. 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.
Publishing Year
Date Published
2022-12-27
Journal Title
Israel Journal of Mathematics
Acknowledgement
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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IST-REx-ID
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2021-07-14
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arXiv 1903.05923

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