[{"intvolume":"        51","keyword":["Lipschitz","bilipschitz","extension","separated net."],"author":[{"last_name":"Dymond","full_name":"Dymond, Michael","first_name":"Michael"},{"orcid":"0000-0002-2512-8698","first_name":"Vojtech","id":"21AE5134-9EAC-11EA-BEA2-D7BD3DDC885E","full_name":"Kaluza, Vojtech","last_name":"Kaluza"}],"file_date_updated":"2026-04-28T12:03:13Z","scopus_import":"1","publication":"Annales Fennici Mathematici","oa":1,"publication_identifier":{"eissn":["2737-114X"],"issn":["2737-0690"]},"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].","file":[{"content_type":"application/pdf","checksum":"442023926a3803d5d6ca8db8dbc4af1c","date_updated":"2026-04-28T12:03:13Z","file_id":"21772","date_created":"2026-04-28T12:03:13Z","creator":"dernst","file_name":"2026_AnnalesFenniciMath_Dymond.pdf","file_size":342082,"relation":"main_file","access_level":"open_access","success":1}],"corr_author":"1","language":[{"iso":"eng"}],"date_created":"2026-04-26T22:01:47Z","department":[{"_id":"UlWa"}],"external_id":{"arxiv":["2507.22007"]},"publication_status":"published","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."}],"license":"https://creativecommons.org/licenses/by-nc/4.0/","quality_controlled":"1","volume":51,"page":"237-260","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_updated":"2026-04-28T12:06:00Z","oa_version":"Published Version","year":"2026","has_accepted_license":"1","date_published":"2026-04-17T00:00:00Z","project":[{"_id":"fc35eaa2-9c52-11eb-aca3-88501ab155e9","grant_number":"M03100","name":"Spectra and topology of graphs and of simplicial complexes"}],"issue":"1","day":"17","publisher":"Finnish Mathematical Society","type":"journal_article","tmp":{"short":"CC BY-NC (4.0)","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)","image":"/images/cc_by_nc.png"},"status":"public","OA_type":"hybrid","article_type":"original","OA_place":"publisher","doi":"10.54330/afm.181562","ddc":["510"],"arxiv":1,"citation":{"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.","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.","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>","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>","ista":"Dymond M, Kaluza V. 2026. Extending bilipschitz mappings between separated nets. Annales Fennici Mathematici. 51(1), 237–260.","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>."},"article_processing_charge":"Yes (in subscription journal)","month":"04","title":"Extending bilipschitz mappings between separated nets","_id":"21766"},{"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."}],"volume":253,"quality_controlled":"1","date_created":"2021-07-14T07:01:28Z","external_id":{"arxiv":["1903.05923"],"isi":["000904950300003"]},"department":[{"_id":"UlWa"}],"language":[{"iso":"eng"}],"publication_status":"published","file":[{"access_level":"open_access","relation":"main_file","file_size":900422,"creator":"vkaluza","file_name":"separated_nets.pdf","date_created":"2021-07-14T07:41:50Z","file_id":"9653","date_updated":"2021-07-14T07:41:50Z","checksum":"6fa0a3207dd1d6467c309fd1bcc867d1","content_type":"application/pdf"}],"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.","oa":1,"publication_identifier":{"eissn":["1565-8511"]},"author":[{"first_name":"Michael","full_name":"Dymond, Michael","last_name":"Dymond"},{"orcid":"0000-0002-2512-8698","last_name":"Kaluza","full_name":"Kaluza, Vojtech","first_name":"Vojtech","id":"21AE5134-9EAC-11EA-BEA2-D7BD3DDC885E"}],"keyword":["Lipschitz","bilipschitz","bounded displacement","modulus of continuity","separated net","non-realisable density","Burago--Kleiner construction"],"scopus_import":"1","file_date_updated":"2021-07-14T07:41:50Z","publication":"Israel Journal of Mathematics","intvolume":"       253","isi":1,"article_processing_charge":"No","citation":{"ista":"Dymond M, Kaluza V. 2023. Highly irregular separated nets. Israel Journal of Mathematics. 253, 501–554.","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>.","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>","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>","ieee":"M. Dymond and V. Kaluza, “Highly irregular separated nets,” <i>Israel Journal of Mathematics</i>, vol. 253. Springer Nature, pp. 501–554, 2023.","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."},"_id":"9652","month":"03","title":"Highly irregular separated nets","article_type":"original","doi":"10.1007/s11856-022-2448-6","ddc":["515","516"],"arxiv":1,"publisher":"Springer Nature","type":"journal_article","day":"01","status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"501-554","has_accepted_license":"1","date_published":"2023-03-01T00:00:00Z","date_updated":"2023-08-14T11:26:34Z","year":"2023","oa_version":"Submitted Version"}]