[{"date_published":"2023-03-01T00:00:00Z","author":[{"last_name":"Dymond","first_name":"Michael","full_name":"Dymond, Michael"},{"orcid":"0000-0002-2512-8698","last_name":"Kaluza","id":"21AE5134-9EAC-11EA-BEA2-D7BD3DDC885E","first_name":"Vojtech","full_name":"Kaluza, Vojtech"}],"oa":1,"_id":"9652","type":"journal_article","language":[{"iso":"eng"}],"doi":"10.1007/s11856-022-2448-6","publisher":"Springer Nature","year":"2023","isi":1,"volume":253,"article_type":"original","month":"03","quality_controlled":"1","publication_identifier":{"eissn":["1565-8511"]},"article_processing_charge":"No","ddc":["515","516"],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","file_date_updated":"2021-07-14T07:41:50Z","keyword":["Lipschitz","bilipschitz","bounded displacement","modulus of continuity","separated net","non-realisable density","Burago--Kleiner construction"],"intvolume":"       253","date_created":"2021-07-14T07:01:28Z","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.","department":[{"_id":"UlWa"}],"oa_version":"Submitted Version","page":"501-554","publication_status":"published","day":"01","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."}],"status":"public","date_updated":"2023-08-14T11:26:34Z","publication":"Israel Journal of Mathematics","external_id":{"arxiv":["1903.05923"],"isi":["000904950300003"]},"title":"Highly irregular separated nets","file":[{"date_created":"2021-07-14T07:41:50Z","file_id":"9653","checksum":"6fa0a3207dd1d6467c309fd1bcc867d1","date_updated":"2021-07-14T07:41:50Z","file_name":"separated_nets.pdf","file_size":900422,"access_level":"open_access","relation":"main_file","content_type":"application/pdf","creator":"vkaluza"}],"scopus_import":"1","has_accepted_license":"1","arxiv":1,"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>","ista":"Dymond M, Kaluza V. 2023. Highly irregular separated nets. Israel Journal of Mathematics. 253, 501–554.","short":"M. Dymond, V. Kaluza, Israel Journal of Mathematics 253 (2023) 501–554.","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>","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>.","ieee":"M. Dymond and V. Kaluza, “Highly irregular separated nets,” <i>Israel Journal of Mathematics</i>, vol. 253. Springer Nature, pp. 501–554, 2023.","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>."}},{"publication":"Mathematische Annalen","external_id":{"isi":["000734150200001"],"arxiv":["2110.05137"]},"file":[{"file_name":"2021_MathAnn_DelloSchiavo.pdf","file_size":410090,"file_id":"10596","checksum":"2593abbf195e38efa93b6006b1e90eb1","date_created":"2022-01-03T11:08:31Z","date_updated":"2022-01-03T11:08:31Z","success":1,"creator":"alisjak","access_level":"open_access","relation":"main_file","content_type":"application/pdf"}],"title":"Sobolev-to-Lipschitz property on QCD- spaces and applications","scopus_import":"1","has_accepted_license":"1","project":[{"grant_number":"716117","call_identifier":"H2020","_id":"256E75B8-B435-11E9-9278-68D0E5697425","name":"Optimal Transport and Stochastic Dynamics"},{"grant_number":"F6504","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","name":"Taming Complexity in Partial Differential Systems"},{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"}],"citation":{"ista":"Dello Schiavo L, Suzuki K. 2022. Sobolev-to-Lipschitz property on QCD- spaces and applications. Mathematische Annalen. 384, 1815–1832.","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>","short":"L. Dello Schiavo, K. Suzuki, Mathematische Annalen 384 (2022) 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>.","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>","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."},"arxiv":1,"page":"1815-1832","oa_version":"Published Version","publication_status":"published","day":"01","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"ec_funded":1,"abstract":[{"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.","lang":"eng"}],"status":"public","license":"https://creativecommons.org/licenses/by/4.0/","date_updated":"2023-08-02T13:39:05Z","isi":1,"article_type":"original","volume":384,"month":"12","quality_controlled":"1","publication_identifier":{"issn":["0025-5831"],"eissn":["1432-1807"]},"article_processing_charge":"Yes (via OA deal)","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","ddc":["510"],"file_date_updated":"2022-01-03T11:08:31Z","keyword":["quasi curvature-dimension condition","sub-riemannian geometry","Sobolev-to-Lipschitz property","Varadhan short-time asymptotics"],"intvolume":"       384","date_created":"2022-01-02T23:01:35Z","department":[{"_id":"JaMa"}],"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.","date_published":"2022-12-01T00:00:00Z","author":[{"orcid":"0000-0002-9881-6870","last_name":"Dello Schiavo","id":"ECEBF480-9E4F-11EA-B557-B0823DDC885E","first_name":"Lorenzo","full_name":"Dello Schiavo, Lorenzo"},{"first_name":"Kohei","full_name":"Suzuki, Kohei","last_name":"Suzuki"}],"oa":1,"_id":"10588","type":"journal_article","language":[{"iso":"eng"}],"doi":"10.1007/s00208-021-02331-2","publisher":"Springer Nature","year":"2022"}]