Are there locally jammed arrangements of spheres of zero density? 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.
 A: Wikipedia says, "The locally jammed sphere packing with the lowest density has a density of only 0.55536," and refers the reader to Martin Gardner's New Mathematical Diversions from Scientific American, Chicago: University of Chicago Press, 1983, pp. 82–90, ISBN 0-226-28247-3.
Aleksandar Donev has a presentation, Jamming in Hard-Sphere
Packings; on slide 49, it says, "There are locally jammed packings of vanishing density (covering fraction) (K. Baroczky)." It may be that this is intended to apply in two dimensions only, but this is not evident from the context. 
I am aware that the two paragraphs above appear to contradict each other. 
Another reference: Torquato and Stillinger, Jammed hard-particle packings: from Kepler to Bernal and beyond, write on page 38, "Another important
distinction is that it is possible to pack spheres subject only to the weak locally-jammed criterion, so that the resulting packing fraction is arbitrarily close to zero." They cite Boroczky, and also Stillinger, F. H., S. Torquato, and H. Sakai, 2003, Lattice-based random jammed configurations
for hard particles,” Phys. Rev. E 67, 031107.
EDIT: I'm looking at the Martin Gardner essay. On page 88, he wrote, 
"In his Geometry and the Imagination, first published in
Germany in 1932, David Hilbert describes what was then believed
to be the loosest [locally-jammed] packing [of spheres]: a structure with a density of
.123. In the following year, however, two Dutch mathematicians,
Heinrich Heesch and Fritz Laves, published the details of a much
looser packing with a density of only .0555 [see Fig. 34]. Whether
there are still looser packings is another intriguing question that,
like the question of the closest packing, remains undecided." 
This suggests that the number given in Wikipedia is a typo. 
Gardner doesn't give a reference for the Heesch-Laves result, but 
I think it's Z. Kristallogr., Mineral. Petrogr. 85, (1933) 443–453 Abt. A.
Probably worth a look is  O'Keeffe, M.,
Dense and rare four-connected nets, 
Z. Krist. 196 (1991), no. 1-4, 21–37, MR1132437 (92h:82125). The summary says, 
"A rare stable sphere packing that is rarer than that of Heesch and Laves is described." 
MORE EDIT: Lots of pretty pictures related to Heesch-Laves at George Hart's webpage,
e.g.:
    
A: Wlodek Kuperberg sent me this
'diagram from the "Lagerungen"' 
of Böröczky's packing, over which I overlaid several congruent disks:
 
(Yes, small cracks or overlays are visible between disks that should be tangent.
The diagram is not 100% metrically accurate.)

Added: The vertical red lines suggest a slight concavity.
