---
OA_place: publisher
OA_type: hybrid
_id: '21766'
abstract:
- lang: eng
  text: We provide a new characterisation of the decades old open problem of extending
    bilipschitz mappings given on a Euclidean separated net. In particular, this allows
    for the complete positive solution of the open problem in dimension two. Along
    the way, we develop a set of tools for bilipschitz extensions of mappings between
    subsets of Euclidean spaces.
acknowledgement: "The present work developed from a research visit of M.D. to V.K.
  at IST Austria, funded by\r\na London Mathematical Society Research in Pairs grant.
  This work was done while V.K. was fully funded by the Austria Science Fund (FWF)
  [M 3100-N]."
article_processing_charge: Yes (in subscription journal)
article_type: original
arxiv: 1
author:
- first_name: Michael
  full_name: Dymond, Michael
  last_name: Dymond
- first_name: Vojtech
  full_name: Kaluza, Vojtech
  id: 21AE5134-9EAC-11EA-BEA2-D7BD3DDC885E
  last_name: Kaluza
  orcid: 0000-0002-2512-8698
citation:
  ama: Dymond M, Kaluza V. Extending bilipschitz mappings between separated nets.
    <i>Annales Fennici Mathematici</i>. 2026;51(1):237-260. doi:<a href="https://doi.org/10.54330/afm.181562">10.54330/afm.181562</a>
  apa: Dymond, M., &#38; Kaluza, V. (2026). Extending bilipschitz mappings between
    separated nets. <i>Annales Fennici Mathematici</i>. Finnish Mathematical Society.
    <a href="https://doi.org/10.54330/afm.181562">https://doi.org/10.54330/afm.181562</a>
  chicago: Dymond, Michael, and Vojtech Kaluza. “Extending Bilipschitz Mappings between
    Separated Nets.” <i>Annales Fennici Mathematici</i>. Finnish Mathematical Society,
    2026. <a href="https://doi.org/10.54330/afm.181562">https://doi.org/10.54330/afm.181562</a>.
  ieee: M. Dymond and V. Kaluza, “Extending bilipschitz mappings between separated
    nets,” <i>Annales Fennici Mathematici</i>, vol. 51, no. 1. Finnish Mathematical
    Society, pp. 237–260, 2026.
  ista: Dymond M, Kaluza V. 2026. Extending bilipschitz mappings between separated
    nets. Annales Fennici Mathematici. 51(1), 237–260.
  mla: Dymond, Michael, and Vojtech Kaluza. “Extending Bilipschitz Mappings between
    Separated Nets.” <i>Annales Fennici Mathematici</i>, vol. 51, no. 1, Finnish Mathematical
    Society, 2026, pp. 237–60, doi:<a href="https://doi.org/10.54330/afm.181562">10.54330/afm.181562</a>.
  short: M. Dymond, V. Kaluza, Annales Fennici Mathematici 51 (2026) 237–260.
corr_author: '1'
date_created: 2026-04-26T22:01:47Z
date_published: 2026-04-17T00:00:00Z
date_updated: 2026-04-28T12:06:00Z
day: '17'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.54330/afm.181562
external_id:
  arxiv:
  - '2507.22007'
file:
- access_level: open_access
  checksum: 442023926a3803d5d6ca8db8dbc4af1c
  content_type: application/pdf
  creator: dernst
  date_created: 2026-04-28T12:03:13Z
  date_updated: 2026-04-28T12:03:13Z
  file_id: '21772'
  file_name: 2026_AnnalesFenniciMath_Dymond.pdf
  file_size: 342082
  relation: main_file
  success: 1
file_date_updated: 2026-04-28T12:03:13Z
has_accepted_license: '1'
intvolume: '        51'
issue: '1'
keyword:
- Lipschitz
- bilipschitz
- extension
- separated net.
language:
- iso: eng
month: '04'
oa: 1
oa_version: Published Version
page: 237-260
project:
- _id: fc35eaa2-9c52-11eb-aca3-88501ab155e9
  grant_number: M03100
  name: Spectra and topology of graphs and of simplicial complexes
publication: Annales Fennici Mathematici
publication_identifier:
  eissn:
  - 2737-114X
  issn:
  - 2737-0690
publication_status: published
publisher: Finnish Mathematical Society
quality_controlled: '1'
scopus_import: '1'
status: public
title: Extending bilipschitz mappings between separated nets
tmp:
  image: /images/cc_by_nc.png
  legal_code_url: https://creativecommons.org/licenses/by-nc/4.0/legalcode
  name: Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0)
  short: CC BY-NC (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 51
year: '2026'
...
---
_id: '9652'
abstract:
- lang: eng
  text: 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.
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.'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Michael
  full_name: Dymond, Michael
  last_name: Dymond
- first_name: Vojtech
  full_name: Kaluza, Vojtech
  id: 21AE5134-9EAC-11EA-BEA2-D7BD3DDC885E
  last_name: Kaluza
  orcid: 0000-0002-2512-8698
citation:
  ama: Dymond M, Kaluza V. Highly irregular separated nets. <i>Israel Journal of Mathematics</i>.
    2023;253:501-554. doi:<a href="https://doi.org/10.1007/s11856-022-2448-6">10.1007/s11856-022-2448-6</a>
  apa: Dymond, M., &#38; Kaluza, V. (2023). Highly irregular separated nets. <i>Israel
    Journal of Mathematics</i>. Springer Nature. <a href="https://doi.org/10.1007/s11856-022-2448-6">https://doi.org/10.1007/s11856-022-2448-6</a>
  chicago: Dymond, Michael, and Vojtech Kaluza. “Highly Irregular Separated Nets.”
    <i>Israel Journal of Mathematics</i>. Springer Nature, 2023. <a href="https://doi.org/10.1007/s11856-022-2448-6">https://doi.org/10.1007/s11856-022-2448-6</a>.
  ieee: M. Dymond and V. Kaluza, “Highly irregular separated nets,” <i>Israel Journal
    of Mathematics</i>, vol. 253. Springer Nature, pp. 501–554, 2023.
  ista: Dymond M, Kaluza V. 2023. Highly irregular separated nets. Israel Journal
    of Mathematics. 253, 501–554.
  mla: Dymond, Michael, and Vojtech Kaluza. “Highly Irregular Separated Nets.” <i>Israel
    Journal of Mathematics</i>, vol. 253, Springer Nature, 2023, pp. 501–54, doi:<a
    href="https://doi.org/10.1007/s11856-022-2448-6">10.1007/s11856-022-2448-6</a>.
  short: M. Dymond, V. Kaluza, Israel Journal of Mathematics 253 (2023) 501–554.
date_created: 2021-07-14T07:01:28Z
date_published: 2023-03-01T00:00:00Z
date_updated: 2023-08-14T11:26:34Z
day: '01'
ddc:
- '515'
- '516'
department:
- _id: UlWa
doi: 10.1007/s11856-022-2448-6
external_id:
  arxiv:
  - '1903.05923'
  isi:
  - '000904950300003'
file:
- access_level: open_access
  checksum: 6fa0a3207dd1d6467c309fd1bcc867d1
  content_type: application/pdf
  creator: vkaluza
  date_created: 2021-07-14T07:41:50Z
  date_updated: 2021-07-14T07:41:50Z
  file_id: '9653'
  file_name: separated_nets.pdf
  file_size: 900422
  relation: main_file
file_date_updated: 2021-07-14T07:41:50Z
has_accepted_license: '1'
intvolume: '       253'
isi: 1
keyword:
- Lipschitz
- bilipschitz
- bounded displacement
- modulus of continuity
- separated net
- non-realisable density
- Burago--Kleiner construction
language:
- iso: eng
month: '03'
oa: 1
oa_version: Submitted Version
page: 501-554
publication: Israel Journal of Mathematics
publication_identifier:
  eissn:
  - 1565-8511
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Highly irregular separated nets
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 253
year: '2023'
...