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Peer-reviewedCC BY 4.0Journal of Applied and Computational Topology

On the metric spaces of lattices and periodic point sets

  • Alexey Garber1
  • iga Virk2
  • Nicolò Zava3
  1. 1University of Texas Rio Grande Valley, Brownsville, TX USA
  2. 2University of Ljubljana and institute IMFM, Ljubljana, Slovenia
  3. 3Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria
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Abstract

Lattices and periodic point sets are well-known objects from discrete geometry. They are also used in crystallography as one of the models of atomic structure of periodic crystals. In this paper we study the embedding properties of spaces of lattices and periodic point sets equipped with optimal bijection metrics (i.e., bottleneck and Euclidean bottleneck metrics). We focus our treatment on embeddings into Hilbert space. On one hand this is motivated by modern data analysis, which is mostly based on statistical approaches developed on Euclidean or Hilbert spaces, hence such embeddings play a major role in applied pipelines. On the other hand there is a well-established methodology related to such questions in coarse geometry, arising from the work on the Novikov conjecture. The main results of this paper provide different conditions, under which the spaces of lattices or periodic point sets are Lipschitz or coarsely (non)embeddable into Hilbert space. The various conditions are phrased in terms of density, packing radius, covering radius, the cardinality of the motif, and the diameter of the unit cell.

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Cite

BibTeXRIS
BibTeX
@article{garber2026metric,
  title = {{On the metric spaces of lattices and periodic point sets}},
  author = {Alexey Garber and iga Virk and Nicolò Zava},
  journal = {Journal of Applied and Computational Topology},
  year = {2026},
  month = oct,
  doi = {10.1007/s41468-026-00258-8},
  url = {https://doi.org/10.1007/s41468-026-00258-8},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - On the metric spaces of lattices and periodic point sets
AU  - Alexey Garber
AU  - iga Virk
AU  - Nicolò Zava
PY  - 2026
DA  - 2026/10/29
JO  - Journal of Applied and Computational Topology
DO  - 10.1007/s41468-026-00258-8
UR  - https://doi.org/10.1007/s41468-026-00258-8
AB  - Lattices and periodic point sets are well-known objects from discrete geometry. They are also used in crystallography as one of the models of atomic structure of periodic crystals. In this paper we study the embedding properties of spaces of lattices and periodic point sets equipped with optimal bijection metrics (i.e., bottleneck and Euclidean bottleneck metrics). We focus our treatment on embeddings into Hilbert space. On one hand this is motivated by modern data analysis, which is mostly based on statistical approaches developed on Euclidean or Hilbert spaces, hence such embeddings play a major role in applied pipelines. On the other hand there is a well-established methodology related to such questions in coarse geometry, arising from the work on the Novikov conjecture. The main results of this paper provide different conditions, under which the spaces of lattices or periodic point sets are Lipschitz or coarsely (non)embeddable into Hilbert space. The various conditions are phrased in terms of density, packing radius, covering radius, the cardinality of the motif, and the diameter of the unit cell.
SN  - 2367-1726
SN  - 2367-1734
ER  -