For other dimensions, this is an open question; it seems unlikely to be true in general. For some dimensions the densest known irregular packing is denser than the densest known regular packing.
> For some dimensions the densest known irregular packing is denser than the densest known regular packing.
I thought that was one of the important results from the paper, the most efficient packing for all dimensions is symmetrical again and this increase was significant enough it seems unlikely that existing non-symmetrical methods will be able to beat it.
Perhaps so. If you hunt you might be able to find a new summary table somewhere (I didn't find one in a very brief skim around). My impression was this new work was more about high-dimensional cases than necessary a dramatic improvement for every low-dimensional example.
> His result has also revived a debate in the field about the nature of the optimal packing in arbitrarily high dimensions. For a while, mathematicians considered highly symmetric, lattice-based packings to be the best way to arrange spheres as densely as possible. But in 2023, a team found a packing that didn’t rely neatly on a repeating lattice; before Klartag’s result, it was the record to beat. Some mathematicians saw it as evidence that more disorder was needed in the search for an optimal sphere packing.
Clearly, this improvement doesn't apply to the few dimensions for which we already proven optimal packing, but the proof was general.
If you click through to the link from your quoted sentence, the 2023 paper "provided a new recipe for how to densely pack spheres in all arbitrarily high dimensions", and from the description the new paper also seems to address the same question.
This is different from asking about the best-known specific packing in n dimensions when n = 7, 10, 19, or whatever.
In a few of the specific low dimensional cases, my impression is that the best currently known packing is not regular, and I don't see any evidence that this has changed. Perhaps this new method will give people a way to beat those records, or perhaps not.
Not necessarily—in 3d there are uncountably many non-lattice packings. They all have the same density as the FCC lattice though. To construct these packings, shift horizontal layers of FCC horizontally with respect to each other.
It is conjectured that in higher dimensions, the densest packing is always non-lattice. The rationale being that there is just not enough symmetry in such spaces.
Well these new results (denser packings than before) are regular lattices which might suggest that the optimal packing could be a lattice. (Until the record is broken again by a irregular packing ;-)
