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

Abstract

We answer several questions of Erdős 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 $a+a^{\prime }$ is squarefree for all $a,a^{\prime }\in A$, as well as bounds on the largest admissible subset of $\{1,2,\dots ,N\}$.

19 pages, 3 figures. Material on sets with squarefree sums shortened due to overlap with existing literature