Growth rates of sequences governed by the squarefree properties of its translates

Abstract

We answer several questions of Erd\H{o}s regarding sequences of natural numbers $A$ whose translates $n+A$ intersect with the squarefree numbers in various specified ways. For instance, we show that if every translate only contains finitely many squarefree numbers, then $A$ has zero density, although the decay rate of this density can be arbitrarily slow. On the other hand, there exist sequences $A$ with optimal density $6/\pi^2$ for which infinitely many $n$ exist such that $n+a$ is squarefree for all $a \in A$ with $a < n$. In fact, infinitely many such $n$ exist for every exponentially increasing sequence, as long as the sequence avoids at least one residue class modulo $p^2$ for all primes $p$, a property we call admissible. If one instead requires infinitely many $n$ to exist such that $n+a$ is squarefree for all $a \in A$, then $A$ can have density arbitrarily close to, but not equal to, $6/\pi^2$. Finally, we prove bounds on the growth rate of sequences $A$ for which $a+a'$ is squarefree for all $a,a' \in A$, as well as bounds on the largest admissible subset of $\{1, 2, \ldots, N\}$.