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

TL;DR

This paper investigates the growth rates and density properties of sequences related to squarefree numbers and their translates, answering longstanding questions posed by Erdős about the intersection patterns with squarefree integers.

Contribution

It provides new bounds and constructions for sequences with prescribed squarefree intersection properties, including sequences with optimal density and specific growth constraints.

Findings

Sequences with finitely many squarefree translates have zero density.

Existence of sequences with density close to 6/π^2 with infinitely many squarefree translates.

Bounds on growth rates of sequences where pairwise sums are squarefree.

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 $\{1, 2, \ldots, N\}$.