Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.30513

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

Sven Ahrend·Mathieu Dutour Sikirić
View original source ↗Read source document ↗Export metadata ↓

Abstract

The packing-covering constant of a point set $X\subseteq\mathbb{R}^d$ is $γ(X)=μ(X)/ρ(X)$, the covering radius divided by the packing radius. Among lattices, its minimum $γ_d$ is known for $d\leq 5$, attained by $\mathsf{A}_2^*$, $\mathsf{A}_3^*$, and Horváth's lattices $\mathsf{Ho}_4$, $\mathsf{Ho}_5$; 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 $\mathbb{R}^5$ with $γ= 9/\sqrt{40} = 1.423024\ldots < γ_5 = \sqrt{3/2+\sqrt{13}/6} = 1.4494568\ldots$, so that in dimension five the packing-covering problem is not solved by lattices.

Explore related subjects

math.MGmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sven Ahrend, Mathieu Dutour Sikirić. 2026-09-24. A non-lattice periodic point set beating the optimal lattice packing-covering constant in dimension five. https://arxiv.org/abs/2609.30513

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Braid spaces have no geodesic loops

The braid spaces are defined as universal metric coverings of the standard complex spaces, branched over certain hyperplane arrangements. We describe a natural family of convex functions on braid spaces and use them to prove that braid spaces contain no geodesic loops. This statement is a consequence of the well-known conjecture that every braid space is CAT(0).

math.MG↗
arXiv2026

An orthogonality inequality in $\mathbb{RP}^{n-1}$ and the double-cap conjecture

The orthogonal neighborhood of a set in real projective space is the union of the hyperplanes perpendicular to its points. We prove that caps minimize neighborhood measure among measurable sets of fixed measure in $\mathbb{RP}^{n-1}$, for $n \geq 3$, with rigidity up to null sets. This yields the double-cap conjecture: a measurable subset of $S^{n-1}$ containing no orthogonal pair has measure at most that of two opposite open caps of radius $π/4$. The argument combines compactness of the Funk transform, rotation comparisons, and projective variations. We also obtain qualitative stability and determine the measurable fractional chromatic number of the spherical orthogonality graph. AI played an assistive role in preparing this paper. The formalization and verification workflow for a separate Lean 4 proof of the main inequality and double-cap result, based on a different argument, were carried out almost entirely using AI-assisted tools.

math.MG↗
arXiv2026

A cut-and-project set which is not bi-Lipschitz equivalent to a lattice

We construct a cut-and-project set in a totally irrational plane in $\mathbb{R}^3$, with a compact window which is the closure of its interior and has countable boundary, which is not bi-Lipschitz equivalent to a lattice. This answers a question of Burago and Kleiner about cut-and-project sets, and shows in particular that the hypothesis of boundary measure zero is not enough to guarantee bi-Lipschitz equivalence to a lattice. Since our set has uniform density, it also shows that a non-rectifiable Delone set need not encode a non-realisable density, answering a question of Bhat and Dymond.

math.MG↗