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.

Preprint | Submitted | English
Author
Dymond, Michael; Kaluza, VojtechIST 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 M, Kaluza V. Divergence of separated nets with respect to displacement equivalence. arXiv, 2102.13046.
Dymond, Michael, and Vojtech Kaluza. “Divergence of Separated Nets with Respect to Displacement Equivalence.” ArXiv, 2102.13046.
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