Timeline for Densest sphere packing for a given lattice

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Aug 2, 2017 at 10:12	comment	added	Campello		Right, so if $p$ is sufficiently large (say $\sqrt{n}$) then we can approximate the density of good ''random'' lattices. What happens in between might depend on $L$. My guess is that the question asked usually arises in ''algebraic'' contexts where $p$ is fairly small, say $2$ or $3$ for the Leech lattice. This is just a guess - it would be nice to have the motivation from the OP.
Aug 2, 2017 at 0:59	comment	added	Noah Stephens-Davidowitz		I'm not sure. One way to think about this is to consider a random sublattice of a given index. (It's mildly convenient to take the index to be a prime $p$, in which case these are sublattices are given by lattice vectors whose coordinates in some lattice basis satisfy a random linear equation mod $p$.) As the index gets large, this distribution looks more and more similar to the Haar measure over all lattices of the appropriate determinant. Morally, as this distribution gets closer to the Haar measure, the question becomes less interesting.
Aug 1, 2017 at 21:37	comment	added	Campello		The problem might became more interesting if we fix the index of the sub-lattice though.
Jun 13, 2017 at 5:58	history	answered	Noah Stephens-Davidowitz	CC BY-SA 3.0