OPEN
We say that $A\subset \mathbb{N}$ has the translation property if, for every $n$, there exists some integer $t_n\geq 1$ such that, for all $1\leq a\leq n$,
\[a\in A\quad\textrm{ if and only if }\quad a+t_n\in A.\]
- Does the set of the sums of two squares have the translation property?
- If we partition all primes into $P\sqcup Q$, such that each set contains $\gg x/\log x$ many primes $\leq x$ 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 $c>0$?
Elementary sieve theory implies that the set of squarefree numbers has the translation property.
More generally, Brun's sieve can be used to prove that if $B\subseteq \mathbb{N}$ is a set of pairwise coprime integers with $\sum_{b<x}\frac{1}{b}=o(\log\log x)$ then $A=\{ n: b\nmid n\textrm{ for all }b\in A\}$ has the translation property. Erdős did not know what happens if the condition on $\sum_{b<x}\frac{1}{b}$ is weakened or dropped altogether.
