Highly irregular separated nets

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