Divergence of separated nets with respect to displacement equivalence
Dymond, Michael
Kaluza, Vojtech
ddc:510
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.
Springer Nature
2024
info:eu-repo/semantics/article
doc-type:article
text
http://purl.org/coar/resource_type/c_6501
https://research-explorer.ista.ac.at/record/9651
https://research-explorer.ista.ac.at/download/9651/17257
Dymond M, Kaluza V. Divergence of separated nets with respect to displacement equivalence. <i>Geometriae Dedicata</i>. 2024;218. doi:<a href="https://doi.org/10.1007/s10711-023-00862-3">10.1007/s10711-023-00862-3</a>
eng
info:eu-repo/semantics/altIdentifier/doi/10.1007/s10711-023-00862-3
info:eu-repo/semantics/altIdentifier/issn/0046-5755
info:eu-repo/semantics/altIdentifier/issn/1572-9168
info:eu-repo/semantics/altIdentifier/wos/001105681500001
info:eu-repo/semantics/altIdentifier/arxiv/2102.13046
info:eu-repo/semantics/openAccess