{"date_published":"2023-11-17T00:00:00Z","publication_status":"epub_ahead","main_file_link":[{"open_access":"1","url":"https://doi.org/10.1007/s10711-023-00862-3"}],"abstract":[{"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.","lang":"eng"}],"department":[{"_id":"UlWa"}],"day":"17","type":"journal_article","date_updated":"2024-01-11T13:06:32Z","article_processing_charge":"Yes (via OA deal)","month":"11","status":"public","quality_controlled":"1","title":"Divergence of separated nets with respect to displacement equivalence","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","date_created":"2021-07-14T07:01:27Z","oa":1,"oa_version":"Published Version","scopus_import":"1","publisher":"Springer Nature","citation":{"ieee":"M. Dymond and V. Kaluza, “Divergence of separated nets with respect to displacement equivalence,” <i>Geometriae Dedicata</i>. Springer Nature, 2023.","mla":"Dymond, Michael, and Vojtech Kaluza. “Divergence of Separated Nets with Respect to Displacement Equivalence.” <i>Geometriae Dedicata</i>, 15, Springer Nature, 2023, doi:<a href=\"https://doi.org/10.1007/s10711-023-00862-3\">10.1007/s10711-023-00862-3</a>.","short":"M. Dymond, V. Kaluza, Geometriae Dedicata (2023).","apa":"Dymond, M., &#38; Kaluza, V. (2023). Divergence of separated nets with respect to displacement equivalence. <i>Geometriae Dedicata</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s10711-023-00862-3\">https://doi.org/10.1007/s10711-023-00862-3</a>","ista":"Dymond M, Kaluza V. 2023. Divergence of separated nets with respect to displacement equivalence. Geometriae Dedicata., 15.","chicago":"Dymond, Michael, and Vojtech Kaluza. “Divergence of Separated Nets with Respect to Displacement Equivalence.” <i>Geometriae Dedicata</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s10711-023-00862-3\">https://doi.org/10.1007/s10711-023-00862-3</a>.","ama":"Dymond M, Kaluza V. Divergence of separated nets with respect to displacement equivalence. <i>Geometriae Dedicata</i>. 2023. doi:<a href=\"https://doi.org/10.1007/s10711-023-00862-3\">10.1007/s10711-023-00862-3</a>"},"article_type":"original","language":[{"iso":"eng"}],"article_number":"15","_id":"9651","doi":"10.1007/s10711-023-00862-3","publication":"Geometriae Dedicata","author":[{"last_name":"Dymond","first_name":"Michael","full_name":"Dymond, Michael"},{"full_name":"Kaluza, Vojtech","id":"21AE5134-9EAC-11EA-BEA2-D7BD3DDC885E","last_name":"Kaluza","first_name":"Vojtech","orcid":"0000-0002-2512-8698"}],"publication_identifier":{"eissn":["1572-9168"],"issn":["0046-5755"]},"external_id":{"isi":["001105681500001"],"arxiv":["2102.13046"]},"isi":1,"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.","year":"2023"}