---
OA_place: publisher
OA_type: hybrid
_id: '21778'
abstract:
- lang: eng
  text: "We prove that every \U0001D43F-bilipschitz mapping ℤ 2 → ℝ2 canbe extended
    to a \U0001D436(\U0001D43F)-bilipschitz mapping ℝ2 → ℝ2,and we provide a polynomial
    upper bound for \U0001D436(\U0001D43F).Moreover, we extend the result to every
    separated netin ℝ2 instead of ℤ 2, with the upper bound gaininga polynomial dependence
    on the separation and netconstants associated to the given separated net. Thisanswers
    an Oberwolfach question of Navas from 2015and is also a positive solution of the
    two-dimensionalform of a decades old open (in all dimensions at leasttwo) problem
    due to Alestalo Trotsenko and Väisälä."
acknowledgement: The authors wish to thank Professor Leonid Kovalev for a valuable
  observation on the first versionof this work, which led to improved estimates and
  cleaner proofs in Section 6. The present workdeveloped from a research visit of
  Michael Dymond to Vojtěch Kaluža at IST Austria, funded by aLondon Mathematical
  Society Research in Pairs grant. This work was done whilst Vojtěch Kalužawas fully
  funded by the Austria Science Fund (FWF) [M 3100-N].
article_number: e70540
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. Planar bilipschitz extension from separated nets. <i>Journal
    of the London Mathematical Society</i>. 2026;113(4). doi:<a href="https://doi.org/10.1112/jlms.70540">10.1112/jlms.70540</a>
  apa: Dymond, M., &#38; Kaluza, V. (2026). Planar bilipschitz extension from separated
    nets. <i>Journal of the London Mathematical Society</i>. Wiley. <a href="https://doi.org/10.1112/jlms.70540">https://doi.org/10.1112/jlms.70540</a>
  chicago: Dymond, Michael, and Vojtech Kaluza. “Planar Bilipschitz Extension from
    Separated Nets.” <i>Journal of the London Mathematical Society</i>. Wiley, 2026.
    <a href="https://doi.org/10.1112/jlms.70540">https://doi.org/10.1112/jlms.70540</a>.
  ieee: M. Dymond and V. Kaluza, “Planar bilipschitz extension from separated nets,”
    <i>Journal of the London Mathematical Society</i>, vol. 113, no. 4. Wiley, 2026.
  ista: Dymond M, Kaluza V. 2026. Planar bilipschitz extension from separated nets.
    Journal of the London Mathematical Society. 113(4), e70540.
  mla: Dymond, Michael, and Vojtech Kaluza. “Planar Bilipschitz Extension from Separated
    Nets.” <i>Journal of the London Mathematical Society</i>, vol. 113, no. 4, e70540,
    Wiley, 2026, doi:<a href="https://doi.org/10.1112/jlms.70540">10.1112/jlms.70540</a>.
  short: M. Dymond, V. Kaluza, Journal of the London Mathematical Society 113 (2026).
date_created: 2026-05-03T22:01:37Z
date_published: 2026-04-01T00:00:00Z
date_updated: 2026-05-07T08:29:18Z
day: '01'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.1112/jlms.70540
external_id:
  arxiv:
  - '2410.22294'
file:
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  creator: dernst
  date_created: 2026-05-07T08:27:43Z
  date_updated: 2026-05-07T08:27:43Z
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file_date_updated: 2026-05-07T08:27:43Z
has_accepted_license: '1'
intvolume: '       113'
issue: '4'
language:
- iso: eng
month: '04'
oa: 1
oa_version: Published Version
project:
- _id: fc35eaa2-9c52-11eb-aca3-88501ab155e9
  grant_number: M03100
  name: Spectra and topology of graphs and of simplicial complexes
publication: Journal of the London Mathematical Society
publication_identifier:
  eissn:
  - 1469-7750
  issn:
  - 0024-6107
publication_status: published
publisher: Wiley
quality_controlled: '1'
scopus_import: '1'
status: public
title: Planar bilipschitz extension from separated nets
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 113
year: '2026'
...