I don't think the initial set of "states" is finite: It is all periodic packings of the finite set B. I take this to mean all packings pack(B, n) of the form pack(B,n) = {B + K*n | K in Z} for some fixed n in Z-{0}, such that it is in fact a packing, i.e.: B + K*n is disjoint from B + L*n for all K <> L. However, it should be fairly easy to exclude all but a finite number of n as candidates for the *densest* packing. When there is a packing for a given n in Z-{0}, I guess the density is just dens(pack(B,n) = |B| / n . Let diam(B) = max{|i - j| | i,j in B} . Then it seems clear to me that if we let n_0 = diam(B) + 1 then any n > n_0 cannot possibly give rise to a denser packing than pack(B,n_0). (Which is a packing since the translates of B by multiples of n_0 must miss each other.) —Dan
From: Allan Wechsler <acwacw@gmail.com> Sent: Jul 27, 2017 3:28 PM
I think this would be approached via the same "finite state machine" technique that Rich sketched regarding Ed Pegg's "Mrs. Perkins's Quilt" phenomena. Because the set of states is finite, a search algorithm looking for an optimal packing will eventually enter a periodic regime. Obviously what I just said is not a proof, but rather a "reason to believe", with a possible pointer in the direction of a proof. I would be super-surprised if the conjecture is false.
On Thu, Jul 27, 2017 at 6:02 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.)
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.