Sphere Packings in Euclidean Space with Forbidden Distances

TL;DR

The paper studies sphere packings in Euclidean spaces with constraints that forbid certain inter-center distances. By extending Viazovska’s modular-form methods and introducing a dimension-dependent linear programming framework, it derives tight density bounds for constrained packings in dimension $48$ (where extremal even unimodular lattices maximize density) and, more generally, for dimensions up to $1200$ under spectral constraints. A key contribution is the construction of magic functions via modular forms and a Féjer-kernel-based LP bound that yields both existence and uniqueness results for optimal constrained packings, with periodicity guaranteed in many one-dimensional cases through a domino-tiling formulation. The results recover the unconstrained optimality in dimensions $8$ and $24$ and provide a broad, computer-assisted toolkit for identifying extremal lattices as optimal packings under various distance-forbidden constraints. The work thus links constrained discrete geometry, modular forms, and linear-programming bounds to advance understanding of extremal lattices and their role in high-dimensional packing problems.

Abstract

We study the sphere packing problem in Euclidean space where we impose additional constraints on the separations of the center points. We prove that any sphere packing in dimension $48$, with spheres of radii $r$, such that no two centers $x_1$ and $x_2$ satisfy $\sqrt{\tfrac{4}{3}} < \frac{1}{2r}|x_1-x_2| <\sqrt{\tfrac{5}{3}}$, has center density less or equal than $(3/2)^{24}$. Equality occurs for periodic packings if and only if the packing is given by a $48$-dimensional even unimodular extremal lattice. This shows that any of the lattices $P_{48p},P_{48q},P_{48m}$ and $P_{48n}$ are optimal for this constrained packing problem, and gives evidence towards the conjecture that extremal lattices are optimal unconstrained sphere packings in $48$ dimensions. We also provide results for packings up to dimension $d\leq 1200$, where we impose constraints on the distance between centers and on the minimal norm of the spectrum, showing that even unimodular extremal lattices are again uniquely optimal. Moreover, in the one-dimensional case, where it is not at all clear that periodic packings are among those with largest density, we nevertheless give a condition on the set of constraints that allows this to happen, and we develop an algorithm to find these periodic configurations by relating the problem to a question about dominos.

Figures (4)

• Figure 1: This is a plot of the functions $s\mapsto \widehat{H}(\sqrt{s})e^{\pi s}$ for $d=8$ (black), $d=24$ (blue) and $d=48$ (red), normalized so $\widehat{H}(0)=1$.
• Figure 2: This is a plot of the functions $s\mapsto \widehat{H}(\sqrt{s})e^{\pi s}$ for $d=72$ (red) and $d=80$ (blue), normalized so $\widehat{H}(0)=1$. For $d=80$, we have multiplied the function by $(s+1)^2$ for aesthetic reasons.
• Figure 3: Best configuration for $d=2$ and $K=[1,2]\cup \{4\}$.
• Figure 4: A visualization of the graph $\mathcal{G}$ for $K=\{1,2,\beta\}$ and $3<\beta \leq 4$ via higher-dimensional embedding.

Theorems & Definitions (40)

• Theorem 1
• Conjecture 1
• Conjecture 2
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Conjecture 3
• Theorem 6
• Conjecture 4