@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},
}

@article{10588,
  abstract     = {We prove the Sobolev-to-Lipschitz property for metric measure spaces satisfying the quasi curvature-dimension condition recently introduced in Milman (Commun Pure Appl Math, to appear). We provide several applications to properties of the corresponding heat semigroup. In particular, under the additional assumption of infinitesimal Hilbertianity, we show the Varadhan short-time asymptotics for the heat semigroup with respect to the distance, and prove the irreducibility of the heat semigroup. These results apply in particular to large classes of (ideal) sub-Riemannian manifolds.},
  author       = {Dello Schiavo, Lorenzo and Suzuki, Kohei},
  issn         = {1432-1807},
  journal      = {Mathematische Annalen},
  keywords     = {quasi curvature-dimension condition, sub-riemannian geometry, Sobolev-to-Lipschitz property, Varadhan short-time asymptotics},
  pages        = {1815--1832},
  publisher    = {Springer Nature},
  title        = {{Sobolev-to-Lipschitz property on QCD- spaces and applications}},
  doi          = {10.1007/s00208-021-02331-2},
  volume       = {384},
  year         = {2022},
}