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Consider a dense sphere packing in $\mathbb{R}^n$, i.e. an arrangement of mutually disjoint solid open spheres, all of the same radius.

In dimensions $2, 3, 8,$ and $24$, it is known that lattice packings (packings where center of the spheres form a discrete subgroup of $\mathbb{R}^n$) are optimal.

It is widely believed that, in high enough dimensions, the best packings will be non-lattice packings. Is anything known about the best non-lattice packings? Do the methods of Viazovska et al. and Hales give values for the density of the best non-lattice packing? Is anything known (or conjectured) about the the ratio of the density of the best non-lattice packing to the density of the best lattice packing?

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    $\begingroup$ Just a comment which you're probably aware of: in 3 dimensions, there are non-lattice packings with the same density as lattice ones. That's part of the reason that it seems unlikely that Viazovska's approach will work in 3D. $\endgroup$
    – Ian Agol
    Commented Sep 2, 2016 at 15:23
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    $\begingroup$ What qualifies for you as a non-lattice packing? If you take a lattice packing and remove one sphere, it has the same density and is not a lattice packing. Do you want it to be periodic? $\endgroup$
    – Yoav Kallus
    Commented Sep 2, 2016 at 17:47
  • $\begingroup$ I guess I mean not a trivial deformation of a lattice packing. I understand that this a little vague, but I think that the question "what is the most dense packing in $\mathbb{R}^{24}$ that is constructed by doing something to the leech lattice" make sense. $\endgroup$
    – Matt Larson
    Commented Sep 2, 2016 at 18:44
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    $\begingroup$ Do you perhaps mean "not constructed by doing something to the Leech lattice"? $\endgroup$
    – Noam D. Elkies
    Commented Sep 3, 2016 at 4:32
  • $\begingroup$ @NoamD.Elkies Right, I cannot edit the comment anymore. $\endgroup$
    – Matt Larson
    Commented Sep 3, 2016 at 15:08

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I recently posted a preprint together with Alexei Andreanov in which we enumerate all the locally optimal 2-periodic sphere packings (also known as double-lattice packings) in dimensions up to $d=5$. This enumeration includes lattices, since they can be represented as 2-periodic arrangements by using a sublattice of index 2 as the unit cell. For the optimal packing density (among 2-periodic packings), we found that in $d=3$, there is one non-lattice that achieves it (the hexagonal close packing), in $d=4$ there are no non-lattices that achieve it, and for $d=5$ there are two. The highest locally optimal non-lattice double-lattice density in $d=4$ is the intriguing $\delta = 1/(2 \sqrt{9+4\sqrt{5}}) \approx 0.118$ (compared to $\delta = 1/8 = 0.125$ for the $D_4$ lattice).

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