Consider a finite set $S$ of nonnegative integers.


What is the maximum natural density of an infinite subset of $\mathbb{Z}$ which does not contain any translation of $S$?


Of course, this will depend on $S$, but maybe there is a simple algorithm or characterization.
I am also interested about the same question in $\mathbb{Z}^k$.


Have the above questions been researched in any form? I didn't come up with a search query which returns anything.



Maximum density of a set without a fixed pattern
The question is equivalent to finding the minimum density of a covering of $\mathbb{Z}$ by translations of $-S$. This problem has been studied for the integers and also for other groups; see for example


Wolfgang M. Schmidt and David M. Tuller, Covering and packing in $\mathbb{Z}^n$ and $\mathbb{R}^n$,
<http://dx.doi.org/10.1007%2Fs00605-009-0099-x>


Béla Bollobás, Svante Janson and Oliver Riordan, On covering by translates of a set,
<https://doi.org/10.1002/rsa.20346>


