# Divergence of separated nets with respect to displacement equivalence

Dymond M, Kaluza V. Divergence of separated nets with respect to displacement equivalence. arXiv, 2102.13046.

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Author

Dymond, Michael;
Kaluza, Vojtech

^{IST Austria}^{}Department

Abstract

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.

Publishing Year

Date Published

2021-02-25

Journal Title

arXiv

Article Number

2102.13046

IST-REx-ID

### Cite this

Dymond M, Kaluza V. Divergence of separated nets with respect to displacement equivalence.

*arXiv*.Dymond, M., & Kaluza, V. (n.d.). Divergence of separated nets with respect to displacement equivalence.

*arXiv*.Dymond, Michael, and Vojtech Kaluza. “Divergence of Separated Nets with Respect to Displacement Equivalence.”

*ArXiv*, n.d.M. Dymond and V. Kaluza, “Divergence of separated nets with respect to displacement equivalence,”

*arXiv*. .Dymond, Michael, and Vojtech Kaluza. “Divergence of Separated Nets with Respect to Displacement Equivalence.”

*ArXiv*, 2102.13046.**All files available under the following license(s):**

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arXiv 2102.13046