Highly irregular separated nets

Michael Dymond, Vojtěch Kaluža

Research output: Working paper/PreprintPreprint

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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.
Original languageEnglish
Publication statusPublished - 14 Mar 2019

Bibliographical note

52 pages. A part of this work is an extensive refinement of a part of arXiv:1704.01940. v4: Changes according to referee's comments, correction of minor typos and errors, especially in the volume bound in Lemma 4.10. To appear in the Israel Journal of Mathematics


  • math.MG
  • cs.DM
  • math.FA
  • 51F99, 51M05, 52C99, 26B35, 26B10


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