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.)



Maximal packings of the integers
Once you have it for some packing (and the standard mumbo-jumbo about achieving (as true density) the supremum of upper densities over all packings, periodic or not), the (upper) density of certain not too long intervals $A---A$ ($A$ here is some not too short packing pattern) is positive (because every sufficiently long interval contains an interval of that sort with some $A$ and there are only finitely many possible $A$ and finitely many possible pattern lengths). If $---A$ has strictly lower density than the supremum, you can improve the upper density of your packing by replacing each occurrence of $A---A$ by a single $A$, which we assumed impossible. Otherwise $---A---A---A---$ is the packing you want. 


