article
Divergence of separated nets with respect to displacement equivalence
epub_ahead
yes
Michael
Dymond
author
Vojtech
Kaluza
author 21AE5134-9EAC-11EA-BEA2-D7BD3DDC885E0000-0002-2512-8698
UlWa
department
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 Nature2023
eng
Geometriae Dedicata
0046-5755
1572-9168
2102.13046
00110568150000110.1007/s10711-023-00862-3
M. Dymond, V. Kaluza, Geometriae Dedicata (2023).
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>.
Dymond M, Kaluza V. 2023. Divergence of separated nets with respect to displacement equivalence. Geometriae Dedicata., 15.
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>.
Dymond, M., & 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>
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>
M. Dymond and V. Kaluza, “Divergence of separated nets with respect to displacement equivalence,” <i>Geometriae Dedicata</i>. Springer Nature, 2023.
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