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.

翻译：1998年,布拉戈和克莱纳以及(独立地)麦克穆伦列举了在欧克利底空间分离的网的例子,这些网在非比利普施奇茨空间中与整块板相当。我们研究了分离网等值的较弱概念,并证明这种概念也产生了不同的等值类别。换句话说,我们发现分离网与整块板差别特别大的情况。我们的方法概括了主要在连续环境中发生的布拉戈和克莱因勒以及麦克穆伦的做法。非正常分离网的存在是通过存在不可实现的密度功能来核实的。在目前的工作中,我们获得了更强的不真实密度。