@article{21766,
  abstract     = {We provide a new characterisation of the decades old open problem of extending bilipschitz mappings given on a Euclidean separated net. In particular, this allows for the complete positive solution of the open problem in dimension two. Along the way, we develop a set of tools for bilipschitz extensions of mappings between subsets of Euclidean spaces.},
  author       = {Dymond, Michael and Kaluza, Vojtech},
  issn         = {2737-114X},
  journal      = {Annales Fennici Mathematici},
  keywords     = {Lipschitz, bilipschitz, extension, separated net.},
  number       = {1},
  pages        = {237--260},
  publisher    = {Finnish Mathematical Society},
  title        = {{Extending bilipschitz mappings between separated nets}},
  doi          = {10.54330/afm.181562},
  volume       = {51},
  year         = {2026},
}

@article{9652,
  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 ρ:[0,1]d→(0,∞). In the present work we obtain stronger types of non-realisable densities.},
  author       = {Dymond, Michael and Kaluza, Vojtech},
  issn         = {1565-8511},
  journal      = {Israel Journal of Mathematics},
  keywords     = {Lipschitz, bilipschitz, bounded displacement, modulus of continuity, separated net, non-realisable density, Burago--Kleiner construction},
  pages        = {501--554},
  publisher    = {Springer Nature},
  title        = {{Highly irregular separated nets}},
  doi          = {10.1007/s11856-022-2448-6},
  volume       = {253},
  year         = {2023},
}