We say that $A\subset \mathbb{N}$ has the translation property if, for every $n$, there exists some integer $t_n\ge 1$ such that, for all $1\le a\le n$,$$a\in A\text{ if and only if }a+t_n\in A.$$ Does the set of the sums of two squares have the translation property? If we partition all primes into $P\bigsqcup Q$, such that each set contains $\gg x/\log x$ many primes $\le x$ for all large $x$, then can the set of integers only divisible by primes from $P$ have the translation property? If $A$ is the set of squarefree numbers then how fast does the minimal such $t_n$ grow? Is it true that $t_n>\exp (n^c)$ for some constant $c>0$?