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