Divergence of separated nets with respect to displacement equivalence

Abstract

We introduce a hierachy 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 $\varphi (R)$. 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, coresponding to $\varphi (R)\in \Omega (R)$, 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 $\varphi (R)$ 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.

Final version, to appear in Geometriae Dedicata

References in corpus (3)

• Some examples of non-rectifiable, repetitive Delone sets
• A dichotomy for bounded displacement equivalence of Delone sets
• Mapping $n$ grid points onto a square forces an arbitrarily large Lipschitz constant

Cited by in corpus (1)

• A dichotomy for bounded displacement equivalence of Delone sets