Highly irregular separated nets

Michael Dymond; Vojtěch Kaluža

doi:10.48550/arXiv.1903.05923

Highly irregular separated nets

Michael Dymond
, Vojtěch Kaluža

Mathematics

Research output: Working paper/Preprint › Preprint

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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.

Original language	English
Publisher	arXiv
Number of pages	52
DOIs	https://doi.org/10.48550/arXiv.1903.05923
Publication status	Published - 14 Mar 2019

Bibliographical note

52 pages. A part of this work is an extensive refinement of a part of arXiv:1704.01940. v4: Changes according to referee's comments, correction of minor typos and errors, especially in the volume bound in Lemma 4.10. To appear in the Israel Journal of Mathematics

Keywords

math.MG
cs.DM
math.FA
51F99, 51M05, 52C99, 26B35, 26B10

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10.48550/arXiv.1903.05923Licence: Other (please provide link to licence statement

1 Article

Highly irregular separated nets
Dymond, M. & Kaluža, V., Mar 2023, In: Israel Journal of Mathematics. 253, p. 501-554 54 p.
Research output: Contribution to journal › Article › peer-review

Cite this

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title = "Highly irregular separated nets",

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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N2 - 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.

AB - 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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U2 - 10.48550/arXiv.1903.05923

DO - 10.48550/arXiv.1903.05923

M3 - Preprint

BT - Highly irregular separated nets

PB - arXiv

ER -

Highly irregular separated nets

Abstract

Bibliographical note

Keywords

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Highly irregular separated nets

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