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 - if, roughly speaking, there is a bijection between them which, for large radii , displaces points of norm at most by something of order at most . We show that the spectrum of -displacement equivalence spans from the established notion of , which corresponds to bounded , to the indiscrete equivalence relation, corresponding to , in which all separated nets are equivalent.