On Jul 27, 2017, at 6:03 PM, James Propp <jamespropp@gmail.com> wrote:
How does one show that for a given packing body B (a finite set of integers) there is a periodic packing of the integers by disjoint translates of B that achieves as its density the supremum of the set of densities achieved by all periodic packings?
I know (via a compactness argument) that the supremum is achieved by some packing, but my argument does not give a packing that is periodic.
Surely this result is in the literature? Maybe it's easy but I'm just not seeing it. (It can't be completely trivial since corresponding assertions in higher dimensions are false, at least if we allow ourselves several packing bodies: consider aperiodic tilings, conceived of as packings of full density.) There's an "Einstein" in 2D, not a connected set, but you didn't care about that. In other words, there's even a single packing body counter example in 2D.
-Veit
Jim Propp
PS: I just posted this on MathOverflow ( https://mathoverflow.net/questions/277426/maximal-packings-of-the-integers ), so if you "go with the 'flow", consider posting there. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun