---
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.
Annales Fennici Mathematici. 2026;51(1):237-260. doi:10.54330/afm.181562
apa: Dymond, M., & Kaluza, V. (2026). Extending bilipschitz mappings between
separated nets. Annales Fennici Mathematici. Finnish Mathematical Society.
https://doi.org/10.54330/afm.181562
chicago: Dymond, Michael, and Vojtech Kaluza. “Extending Bilipschitz Mappings between
Separated Nets.” Annales Fennici Mathematici. Finnish Mathematical Society,
2026. https://doi.org/10.54330/afm.181562.
ieee: M. Dymond and V. Kaluza, “Extending bilipschitz mappings between separated
nets,” Annales Fennici Mathematici, 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.” Annales Fennici Mathematici, vol. 51, no. 1, Finnish Mathematical
Society, 2026, pp. 237–60, doi:10.54330/afm.181562.
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
license: https://creativecommons.org/licenses/by-nc/4.0/
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'
...
---
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. Journal
of the London Mathematical Society. 2026;113(4). doi:10.1112/jlms.70540
apa: Dymond, M., & Kaluza, V. (2026). Planar bilipschitz extension from separated
nets. Journal of the London Mathematical Society. Wiley. https://doi.org/10.1112/jlms.70540
chicago: Dymond, Michael, and Vojtech Kaluza. “Planar Bilipschitz Extension from
Separated Nets.” Journal of the London Mathematical Society. Wiley, 2026.
https://doi.org/10.1112/jlms.70540.
ieee: M. Dymond and V. Kaluza, “Planar bilipschitz extension from separated nets,”
Journal of the London Mathematical Society, 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.” Journal of the London Mathematical Society, vol. 113, no. 4, e70540,
Wiley, 2026, doi:10.1112/jlms.70540.
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:
- access_level: open_access
checksum: 6dbfc7134f732d17c5c8467843a73e90
content_type: application/pdf
creator: dernst
date_created: 2026-05-07T08:27:43Z
date_updated: 2026-05-07T08:27:43Z
file_id: '21836'
file_name: 2026_JourLondonMathSoc_Dymond.pdf
file_size: 617569
relation: main_file
success: 1
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'
...
---
OA_place: publisher
OA_type: hybrid
_id: '9651'
abstract:
- lang: eng
text: We introduce a hierachy of equivalence relations on the set of separated nets
of a given Euclidean space, indexed by concave increasing functions ϕ:(0,∞)→(0,∞).
Two separated nets are called ϕ-displacement equivalent if, roughly speaking,
there is a bijection between them which, for large radii R, displaces points of
norm at most R by something of order at most ϕ(R). We show that the spectrum of
ϕ-displacement equivalence spans from the established notion of bounded displacement
equivalence, which corresponds to bounded ϕ, to the indiscrete equivalence relation,
coresponding to ϕ(R)∈Ω(R), in which all separated nets are equivalent. In between
the two ends of this spectrum, the notions of ϕ-displacement equivalence are shown
to be pairwise distinct with respect to the asymptotic classes of ϕ(R) for R→∞.
We further undertake a comparison of our notion of ϕ-displacement equivalence
with previously studied relations on separated nets. Particular attention is given
to the interaction of the notions of ϕ-displacement equivalence with that of bilipschitz
equivalence.
acknowledgement: 'Open access funding provided by Institute of Science and Technology
(IST Austria). This work was started while both authors were employed at the University
of Innsbruck and enjoyed the full support of Austrian Science Fund (FWF): P 30902-N35.
It was continued when the first named author was employed at University of Leipzig
and the second named author was employed at Institute of Science and Technology
of Austria, where he was supported by an IST Fellowship.'
article_number: '15'
article_processing_charge: Yes (via OA deal)
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. Divergence of separated nets with respect to displacement
equivalence. Geometriae Dedicata. 2024;218. doi:10.1007/s10711-023-00862-3
apa: Dymond, M., & Kaluza, V. (2024). Divergence of separated nets with respect
to displacement equivalence. Geometriae Dedicata. Springer Nature. https://doi.org/10.1007/s10711-023-00862-3
chicago: Dymond, Michael, and Vojtech Kaluza. “Divergence of Separated Nets with
Respect to Displacement Equivalence.” Geometriae Dedicata. Springer Nature,
2024. https://doi.org/10.1007/s10711-023-00862-3.
ieee: M. Dymond and V. Kaluza, “Divergence of separated nets with respect to displacement
equivalence,” Geometriae Dedicata, vol. 218. Springer Nature, 2024.
ista: Dymond M, Kaluza V. 2024. Divergence of separated nets with respect to displacement
equivalence. Geometriae Dedicata. 218, 15.
mla: Dymond, Michael, and Vojtech Kaluza. “Divergence of Separated Nets with Respect
to Displacement Equivalence.” Geometriae Dedicata, vol. 218, 15, Springer
Nature, 2024, doi:10.1007/s10711-023-00862-3.
short: M. Dymond, V. Kaluza, Geometriae Dedicata 218 (2024).
corr_author: '1'
date_created: 2021-07-14T07:01:27Z
date_published: 2024-02-01T00:00:00Z
date_updated: 2025-04-23T07:37:26Z
day: '01'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.1007/s10711-023-00862-3
external_id:
arxiv:
- '2102.13046'
isi:
- '001105681500001'
pmid:
- '38021107'
file:
- access_level: open_access
checksum: 9418534ac2f3d6f1f091a8b8ccaed01e
content_type: application/pdf
creator: dernst
date_created: 2024-07-16T10:14:13Z
date_updated: 2024-07-16T10:14:13Z
file_id: '17257'
file_name: 2024_GeometriaeDedicata_Dymond.pdf
file_size: 540981
relation: main_file
success: 1
file_date_updated: 2024-07-16T10:14:13Z
has_accepted_license: '1'
intvolume: ' 218'
isi: 1
language:
- iso: eng
month: '02'
oa: 1
oa_version: Published Version
pmid: 1
publication: Geometriae Dedicata
publication_identifier:
eissn:
- 1572-9168
issn:
- 0046-5755
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Divergence of separated nets with respect to displacement equivalence
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: 218
year: '2024'
...
---
_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. Israel Journal of Mathematics.
2023;253:501-554. doi:10.1007/s11856-022-2448-6
apa: Dymond, M., & Kaluza, V. (2023). Highly irregular separated nets. Israel
Journal of Mathematics. Springer Nature. https://doi.org/10.1007/s11856-022-2448-6
chicago: Dymond, Michael, and Vojtech Kaluza. “Highly Irregular Separated Nets.”
Israel Journal of Mathematics. Springer Nature, 2023. https://doi.org/10.1007/s11856-022-2448-6.
ieee: M. Dymond and V. Kaluza, “Highly irregular separated nets,” Israel Journal
of Mathematics, 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.” Israel
Journal of Mathematics, vol. 253, Springer Nature, 2023, pp. 501–54, doi:10.1007/s11856-022-2448-6.
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'
...
---
_id: '10335'
abstract:
- lang: eng
text: "Van der Holst and Pendavingh introduced a graph parameter σ, which coincides
with the more famous Colin de Verdière graph parameter μ for small values. However,
the definition of a is much more geometric/topological directly reflecting embeddability
properties of the graph. They proved μ(G) ≤ σ(G) + 2 and conjectured σ(G) ≤ σ(G)
for any graph G. We confirm this conjecture. As far as we know, this is the first
topological upper bound on σ(G) which is, in general, tight.\r\nEquality between
μ and σ does not hold in general as van der Holst and Pendavingh showed that there
is a graph G with μ(G) ≤ 18 and σ(G) ≥ 20. We show that the gap appears at much
smaller values, namely, we exhibit a graph H for which μ(H) ≥ 7 and σ(H) ≥ 8.
We also prove that, in general, the gap can be large: The incidence graphs Hq
of finite projective planes of order q satisfy μ(Hq) ∈ O(q3/2) and σ(Hq) ≥ q2."
acknowledgement: 'V. K. gratefully acknowledges the support of Austrian Science Fund
(FWF): P 30902-N35. This work was done mostly while he was employed at the University
of Innsbruck. During the early stage of this research, V. K. was partially supported
by Charles University project GAUK 926416. M. T. is supported by the grant no. 19-04113Y
of the Czech Science Foundation(GA ˇCR) and partially supported by Charles University
project UNCE/SCI/004.'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Vojtech
full_name: Kaluza, Vojtech
id: 21AE5134-9EAC-11EA-BEA2-D7BD3DDC885E
last_name: Kaluza
orcid: 0000-0002-2512-8698
- first_name: Martin
full_name: Tancer, Martin
id: 38AC689C-F248-11E8-B48F-1D18A9856A87
last_name: Tancer
orcid: 0000-0002-1191-6714
citation:
ama: Kaluza V, Tancer M. Even maps, the Colin de Verdière number and representations
of graphs. Combinatorica. 2022;42:1317-1345. doi:10.1007/s00493-021-4443-7
apa: Kaluza, V., & Tancer, M. (2022). Even maps, the Colin de Verdière number
and representations of graphs. Combinatorica. Springer Nature. https://doi.org/10.1007/s00493-021-4443-7
chicago: Kaluza, Vojtech, and Martin Tancer. “Even Maps, the Colin de Verdière Number
and Representations of Graphs.” Combinatorica. Springer Nature, 2022. https://doi.org/10.1007/s00493-021-4443-7.
ieee: V. Kaluza and M. Tancer, “Even maps, the Colin de Verdière number and representations
of graphs,” Combinatorica, vol. 42. Springer Nature, pp. 1317–1345, 2022.
ista: Kaluza V, Tancer M. 2022. Even maps, the Colin de Verdière number and representations
of graphs. Combinatorica. 42, 1317–1345.
mla: Kaluza, Vojtech, and Martin Tancer. “Even Maps, the Colin de Verdière Number
and Representations of Graphs.” Combinatorica, vol. 42, Springer Nature,
2022, pp. 1317–45, doi:10.1007/s00493-021-4443-7.
short: V. Kaluza, M. Tancer, Combinatorica 42 (2022) 1317–1345.
corr_author: '1'
date_created: 2021-11-25T13:49:16Z
date_published: 2022-12-01T00:00:00Z
date_updated: 2024-10-09T20:53:51Z
day: '01'
ddc:
- '514'
- '516'
department:
- _id: UlWa
doi: 10.1007/s00493-021-4443-7
external_id:
arxiv:
- '1907.05055'
isi:
- '000798210100003'
intvolume: ' 42'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: ' https://doi.org/10.48550/arXiv.1907.05055'
month: '12'
oa: 1
oa_version: Preprint
page: 1317-1345
publication: Combinatorica
publication_identifier:
issn:
- 0209-9683
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Even maps, the Colin de Verdière number and representations of graphs
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 42
year: '2022'
...