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Simultaneous visibility in cut-and-project sets

  • Rishi Kumar
  • Carlos Ospina
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Abstract

Visibility from the origin in integer lattices is a well-studied problem. More recently, the corresponding question for cut-and-project sets has attracted attention. We consider simultaneous visibility, which asks what points in a set are visible from every point of another set. This question was studied for the integer lattice by Rearick and Rumsey, among others. In this work, we investigate simultaneous visibility in cut-and-project sets associated with real quadratic fields satisfying the Hammarhjelm condition. Under suitable geometric conditions on the finite set, we prove that the density of the simultaneously visible points exists and derive an explicit formula for it. We also examine simultaneous visibility in more general cut-and-project sets under a condition on the associated lattice.

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Cite

BibTeXRIS
BibTeX
@misc{kumar2026simultaneous,
  title = {{Simultaneous visibility in cut-and-project sets}},
  author = {Rishi Kumar and Carlos Ospina},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.28119},
  eprint = {2610.28119},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.28119},
}
RIS
TY  - UNPB
TI  - Simultaneous visibility in cut-and-project sets
AU  - Rishi Kumar
AU  - Carlos Ospina
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.28119
UR  - https://arxiv.org/abs/2610.28119
AB  - Visibility from the origin in integer lattices is a well-studied problem. More recently, the corresponding question for cut-and-project sets has attracted attention. We consider simultaneous visibility, which asks what points in a set are visible from every point of another set. This question was studied for the integer lattice by Rearick and Rumsey, among others. In this work, we investigate simultaneous visibility in cut-and-project sets associated with real quadratic fields satisfying the Hammarhjelm condition. Under suitable geometric conditions on the finite set, we prove that the density of the simultaneously visible points exists and derive an explicit formula for it. We also examine simultaneous visibility in more general cut-and-project sets under a condition on the associated lattice.
ER  -