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

Source 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.