Highly irregular separated nets

In 1998 Burago and Kleiner and (independently) McMullen gave examples of separated nets in Euclidean space which are bilipschitz non-equivalent to the integer lattice. We study weaker notions than bilipschitz equivalence and demonstrate that such notions also distinguish between separated nets. 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 ρ: [0,1]^d → (0, ∞). In the present work we obtain stronger types of non-realisable densities.