Divergence of separated nets with respect to displacement equivalence.

Abstract

We introduce a hierarchy of equivalence relations on the set of separated nets of a given Euclidean space, indexed by concave increasing functions $\varphi :(0,\infty )\to (0,\infty )$. Two separated nets are called $\varphi$-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 $\varphi \left(R\right)$. We show that the spectrum of $\varphi$-displacement equivalence spans from the established notion of bounded displacement equivalence, which corresponds to bounded $\varphi$, to the indiscrete equivalence relation, corresponding to $\varphi \left(R\right)\in \Omega \left(R\right)$, in which all separated nets are equivalent. In between the two ends of this spectrum, the notions of $\varphi$-displacement equivalence are shown to be pairwise distinct with respect to the asymptotic classes of $\varphi \left(R\right)$ for $R\to \infty$. We further undertake a comparison of our notion of $\varphi$-displacement equivalence with previously studied relations on separated nets. Particular attention is given to the interaction of the notions of $\varphi$-displacement equivalence with that of bilipschitz equivalence.