I know of a remarkable result from a paper of
Matthew Kahle (PDF download), that there are arbitrarily low-density
*jammed packings* of congruent disks in $\mathbb{R}^2$:

In 1964 Böröczky used a subtle disk packing construction to disprove a conjecture of Fejes Tóth, that a locally jammed arrangement of disks in the plane, one where each disk is held in place by its neighbors, must have positive density.

Böröczky, K. "Über stabile Kreis- und Kugelsysteme."

Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 7, 79-82, 1964.

Surprisingly (to me), this natural candidate is not the lowest density:

My question is whether or not Böröczky's construction extends to $\mathbb{R}^3$
and higher dimensions. (I do not yet have access to Böröczky's 1964 paper.)

**Addendum**. (I add this prompted by user *j.c.*)
My question is answered: ** Yes**. First, through the help of
Wlodek Kuperberg and other commenters, we learned what exactly

*is*Böröczky's 1964 zero-density packing. Indeed it is "subtle" as Matthew Kahle said. Second, simply layering identical copies of that packing of spheres in $\mathbb{R}^3$ results in a jammed packing of zero density, as confirmed by Benoît Kloeckner. What remains is to correct Wikipedia's misleading entries on the topic, as detailed by Gerry Myerson.

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