Accumulation points of packings of $\mathbb{Z}^3$

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.

A sphere packing of the nearest-neighbor lattice graph $\mathbb Z^3$ consists of balls $B(c_v,r_v)\subset\mathbb R^3$, $r_v>0$, with disjoint interiors, such that two balls are tangent exactly when their indices differ by a standard unit vector. A point of the one-point compactification $\mathbb R^3\cup\{\infty\}$ is an accumulation point of the packing if each neighborhood of it meets infinitely many distinct balls.

Statement recorded in the supplied source.

Prove that every such packing has at most one accumulation point in $\mathbb R^3\cup\{\infty\}$. [S2]

Short English statement

Can a sphere packing realizing the three-dimensional lattice accumulate anywhere other than a single point, including infinity?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10000064 (AMR-099-0064). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Itai Benjamini, Euclidean vs. Graph Metric, version dated 16 July 2012; original author manuscript preserved in Arquivo.pt capture of 31 December 2020, Section 6, Question 6.5, p. 8. https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf.