---
_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'
...
---
_id: '10588'
abstract:
- lang: eng
  text: We prove the Sobolev-to-Lipschitz property for metric measure spaces satisfying
    the quasi curvature-dimension condition recently introduced in Milman (Commun
    Pure Appl Math, to appear). We provide several applications to properties of the
    corresponding heat semigroup. In particular, under the additional assumption of
    infinitesimal Hilbertianity, we show the Varadhan short-time asymptotics for the
    heat semigroup with respect to the distance, and prove the irreducibility of the
    heat semigroup. These results apply in particular to large classes of (ideal)
    sub-Riemannian manifolds.
acknowledgement: "The authors are grateful to Dr. Bang-Xian Han for helpful discussions
  on the Sobolev-to-Lipschitz property on metric measure spaces, and to Professor
  Kazuhiro Kuwae, Professor Emanuel Milman, Dr. Giorgio Stefani, and Dr. Gioacchino
  Antonelli for reading a preliminary version of this work and for their valuable
  comments and suggestions. Finally, they wish to express their gratitude to two anonymous
  Reviewers whose suggestions improved the presentation of this work.\r\n\r\nL.D.S.
  gratefully acknowledges funding of his position by the Austrian Science Fund (FWF)
  grant F65, and by the European Research Council (ERC, grant No. 716117, awarded
  to Prof. Dr. Jan Maas).\r\n\r\nK.S. gratefully acknowledges funding by: the JSPS
  Overseas Research Fellowships, Grant Nr. 290142; World Premier International Research
  Center Initiative (WPI), MEXT, Japan; JSPS Grant-in-Aid for Scientific Research
  on Innovative Areas “Discrete Geometric Analysis for Materials Design”, Grant Number
  17H06465; and the Alexander von Humboldt Stiftung, Humboldt-Forschungsstipendium."
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Lorenzo
  full_name: Dello Schiavo, Lorenzo
  id: ECEBF480-9E4F-11EA-B557-B0823DDC885E
  last_name: Dello Schiavo
  orcid: 0000-0002-9881-6870
- first_name: Kohei
  full_name: Suzuki, Kohei
  last_name: Suzuki
citation:
  ama: Dello Schiavo L, Suzuki K. Sobolev-to-Lipschitz property on QCD- spaces and
    applications. <i>Mathematische Annalen</i>. 2022;384:1815-1832. doi:<a href="https://doi.org/10.1007/s00208-021-02331-2">10.1007/s00208-021-02331-2</a>
  apa: Dello Schiavo, L., &#38; Suzuki, K. (2022). Sobolev-to-Lipschitz property on
    QCD- spaces and applications. <i>Mathematische Annalen</i>. Springer Nature. <a
    href="https://doi.org/10.1007/s00208-021-02331-2">https://doi.org/10.1007/s00208-021-02331-2</a>
  chicago: Dello Schiavo, Lorenzo, and Kohei Suzuki. “Sobolev-to-Lipschitz Property
    on QCD- Spaces and Applications.” <i>Mathematische Annalen</i>. Springer Nature,
    2022. <a href="https://doi.org/10.1007/s00208-021-02331-2">https://doi.org/10.1007/s00208-021-02331-2</a>.
  ieee: L. Dello Schiavo and K. Suzuki, “Sobolev-to-Lipschitz property on QCD- spaces
    and applications,” <i>Mathematische Annalen</i>, vol. 384. Springer Nature, pp.
    1815–1832, 2022.
  ista: Dello Schiavo L, Suzuki K. 2022. Sobolev-to-Lipschitz property on QCD- spaces
    and applications. Mathematische Annalen. 384, 1815–1832.
  mla: Dello Schiavo, Lorenzo, and Kohei Suzuki. “Sobolev-to-Lipschitz Property on
    QCD- Spaces and Applications.” <i>Mathematische Annalen</i>, vol. 384, Springer
    Nature, 2022, pp. 1815–32, doi:<a href="https://doi.org/10.1007/s00208-021-02331-2">10.1007/s00208-021-02331-2</a>.
  short: L. Dello Schiavo, K. Suzuki, Mathematische Annalen 384 (2022) 1815–1832.
date_created: 2022-01-02T23:01:35Z
date_published: 2022-12-01T00:00:00Z
date_updated: 2023-08-02T13:39:05Z
day: '01'
ddc:
- '510'
department:
- _id: JaMa
doi: 10.1007/s00208-021-02331-2
ec_funded: 1
external_id:
  arxiv:
  - '2110.05137'
  isi:
  - '000734150200001'
file:
- access_level: open_access
  checksum: 2593abbf195e38efa93b6006b1e90eb1
  content_type: application/pdf
  creator: alisjak
  date_created: 2022-01-03T11:08:31Z
  date_updated: 2022-01-03T11:08:31Z
  file_id: '10596'
  file_name: 2021_MathAnn_DelloSchiavo.pdf
  file_size: 410090
  relation: main_file
  success: 1
file_date_updated: 2022-01-03T11:08:31Z
has_accepted_license: '1'
intvolume: '       384'
isi: 1
keyword:
- quasi curvature-dimension condition
- sub-riemannian geometry
- Sobolev-to-Lipschitz property
- Varadhan short-time asymptotics
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: 1815-1832
project:
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '716117'
  name: Optimal Transport and Stochastic Dynamics
- _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2
  grant_number: F6504
  name: Taming Complexity in Partial Differential Systems
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
  name: IST Austria Open Access Fund
publication: Mathematische Annalen
publication_identifier:
  eissn:
  - 1432-1807
  issn:
  - 0025-5831
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Sobolev-to-Lipschitz property on QCD- spaces and applications
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: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 384
year: '2022'
...