  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.
