Divergence of separated nets with respect to displacement equivalence.

We introduce a hierarchy 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, corresponding 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. [ABSTRACT FROM AUTHOR]