{"date_updated":"2023-08-14T11:26:34Z","month":"03","article_processing_charge":"No","day":"01","file":[{"date_created":"2021-07-14T07:41:50Z","file_size":900422,"file_id":"9653","date_updated":"2021-07-14T07:41:50Z","file_name":"separated_nets.pdf","access_level":"open_access","creator":"vkaluza","content_type":"application/pdf","relation":"main_file","checksum":"6fa0a3207dd1d6467c309fd1bcc867d1"}],"type":"journal_article","abstract":[{"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.","lang":"eng"}],"department":[{"_id":"UlWa"}],"publication_status":"published","date_published":"2023-03-01T00:00:00Z","scopus_import":"1","publisher":"Springer Nature","date_created":"2021-07-14T07:01:28Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"oa_version":"Submitted Version","status":"public","title":"Highly irregular separated nets","quality_controlled":"1","publication_identifier":{"eissn":["1565-8511"]},"external_id":{"arxiv":["1903.05923"],"isi":["000904950300003"]},"page":"501-554","doi":"10.1007/s11856-022-2448-6","has_accepted_license":"1","publication":"Israel Journal of Mathematics","author":[{"full_name":"Dymond, Michael","last_name":"Dymond","first_name":"Michael"},{"orcid":"0000-0002-2512-8698","full_name":"Kaluza, Vojtech","id":"21AE5134-9EAC-11EA-BEA2-D7BD3DDC885E","last_name":"Kaluza","first_name":"Vojtech"}],"language":[{"iso":"eng"}],"_id":"9652","article_type":"original","citation":{"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.","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>","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>."},"year":"2023","volume":253,"intvolume":"       253","ddc":["515","516"],"file_date_updated":"2021-07-14T07:41:50Z","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.","isi":1,"keyword":["Lipschitz","bilipschitz","bounded displacement","modulus of continuity","separated net","non-realisable density","Burago--Kleiner construction"]}