Sphere Packing in High Dimensions

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

An equal-sphere packing in $\mathbb R^n$ is a union $P=\bigcup_{c\in C}B(c,1)$ of closed Euclidean unit balls whose interiors are pairwise disjoint; equivalently $\lVert c-c'\rVert\geq2$ for distinct centers. Let $B(x,R)$ be the radius-$R$ ball centered at $x$ and define \[ \overline\delta(P)=\limsup_{R\to\infty}\sup_{x\in\mathbb R^n} \frac{\operatorname{vol}_n(P\cap B(x,R))}{\operatorname{vol}_n(B(x,R))},\qquad \Delta_n=\sup_{P}\overline\delta(P). \] The supremum is over all packings, including nonlattice and nonperiodic ones.

Statement recorded in the supplied source.

For each integer $n\geq1$ with $n\notin\{1,2,3,8,24\}$, determine $\Delta_n$ and an equal-sphere packing attaining it. Determine also the asymptotic behavior of $\Delta_n$ as $n\to\infty$. [S2]

Short English statement

Find the densest sphere packings in the remaining dimensions and determine how their optimal densities behave as the dimension grows.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1227 (GEO-032). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Henry Cohn and Noam Elkies, New upper bounds on sphere packings I, Annals of Mathematics 157 (2003), 689--714; arXiv:math/0110009v3, Section 1 and Appendix A. https://arxiv.org/pdf/math/0110009.