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 .