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 $$. Two separated nets are called $$-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 $$. We show that the spectrum of $$-displacement equivalence spans from the established notion of bounded displacement equivalence, which corresponds to bounded $$, to the indiscrete equivalence relation, corresponding to $$, in which all separated nets are equivalent. In between the two ends of this spectrum, the notions of $$-displacement equivalence are shown to be pairwise distinct with respect to the asymptotic classes of $$ for $$. We further undertake a comparison of our notion of $$-displacement equivalence with previously studied relations on separated nets. Particular attention is given to the interaction of the notions of $$-displacement equivalence with that of bilipschitz equivalence.