Divergence of separated nets with respect to displacement equivalence

Abstract

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