# Highly irregular separated nets

## TL;DR

This paper explores various weaker notions of equivalence among separated nets in Euclidean space, demonstrating the existence of strongly irregular nets that diverge significantly from the integer lattice, using advanced density function techniques.

## Contribution

It introduces stronger non-realisable density functions to establish the existence of irregular separated nets beyond previous examples.

## Key findings

- Existence of separated nets not equivalent to the integer lattice under weaker notions.
- Development of stronger non-realisable density functions.
- Extension of previous continuous setting approaches.

## Abstract

In 1998 Burago and Kleiner and (independently) McMullen gave examples of separated nets in Euclidean space which are non-bilipschitz equivalent to the integer lattice. We study weaker notions of equivalence of separated nets and demonstrate that such notions also give rise to distinct equivalence classes. Put differently, we find occurrences of particularly strong divergence of separated nets from the integer lattice. Our approach generalises that of Burago and Kleiner and McMullen which takes place largely in a continuous setting. Existence of irregular separated nets is verified via the existence of non-realisable density functions $\rho\colon [0,1]^{d}\to(0,\infty)$. In the present work we obtain stronger types of non-realisable densities.