Abstract
AbstractWe introduce a hierarchy of equivalence relations on the set of separated nets of a given Euclidean space, indexed by concave increasing functions $$\phi :(0,\infty )\rightarrow (0,\infty )$$
ϕ
:
(
0
,
∞
)
→
(
0
,
∞
)
. Two separated nets are called $$\phi $$
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 $$\phi (R)$$
ϕ
(
R
)
. We show that the spectrum of $$\phi $$
ϕ
-displacement equivalence spans from the established notion of bounded displacement equivalence, which corresponds to bounded $$\phi $$
ϕ
, to the indiscrete equivalence relation, corresponding to $$\phi (R)\in \varOmega (R)$$
ϕ
(
R
)
∈
Ω
(
R
)
, in which all separated nets are equivalent. In between the two ends of this spectrum, the notions of $$\phi $$
ϕ
-displacement equivalence are shown to be pairwise distinct with respect to the asymptotic classes of $$\phi (R)$$
ϕ
(
R
)
for $$R\rightarrow \infty $$
R
→
∞
. We further undertake a comparison of our notion of $$\phi $$
ϕ
-displacement equivalence with previously studied relations on separated nets. Particular attention is given to the interaction of the notions of $$\phi $$
ϕ
-displacement equivalence with that of bilipschitz equivalence.
Funder
Austrian Science Fund
Institute of Science and Technology
Publisher
Springer Science and Business Media LLC