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A non-lattice periodic point set beating the optimal lattice packing-covering constant in dimension five

  • Sven Ahrend
  • Mathieu Dutour Sikirić
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Abstract

The packing-covering constant of a point set X⊆ℝd is γ(X)=μ(X)/ρ(X), the covering radius divided by the packing radius. Among lattices, its minimum γd is known for d≤ 5, attained by A2*, A3*, and Horváth's lattices Ho4, Ho5; Böröczky proved that γ3 is optimal without the lattice restriction, but for d=4,5 the non-lattice problem was open. We exhibit a $2$-periodic non-lattice point set of ℝ5 with γ= 9/√(40) = 1.423024… < γ5 = √(3/2+√(13)/6) = 1.4494568…, so that in dimension five the packing-covering problem is not solved by lattices.

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Cite

BibTeXRIS
BibTeX
@misc{ahrend2026nonlattice,
  title = {{A non-lattice periodic point set beating the optimal lattice packing-covering constant in dimension five}},
  author = {Sven Ahrend and Mathieu Dutour Sikirić},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30513},
  eprint = {2609.30513},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30513},
}
RIS
TY  - UNPB
TI  - A non-lattice periodic point set beating the optimal lattice packing-covering constant in dimension five
AU  - Sven Ahrend
AU  - Mathieu Dutour Sikirić
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.30513
UR  - https://arxiv.org/abs/2609.30513
AB  - The packing-covering constant of a point set X⊆ℝ^d is γ(X)=μ(X)/ρ(X), the covering radius divided by the packing radius. Among lattices, its minimum γ_d is known for d≤ 5, attained by A_2^*, A_3^*, and Horváth's lattices Ho_4, Ho_5; Böröczky proved that γ_3 is optimal without the lattice restriction, but for d=4,5 the non-lattice problem was open. We exhibit a $2$-periodic non-lattice point set of ℝ^5 with γ= 9/√(40) = 1.423024… < γ_5 = √(3/2+√(13)/6) = 1.4494568…, so that in dimension five the packing-covering problem is not solved by lattices.
ER  -