Sumsets avoiding squarefree integers

TL;DR

This paper investigates the structure of integer sets with positive density whose sumsets exclude squarefree integers, revealing a phase transition at specific density thresholds.

Contribution

It characterizes the structure of such sets and identifies critical density points where their behavior changes abruptly.

Findings

Behavior changes at density 0.0473 and 1/18.

Provides structural description of sets avoiding squarefree sums.

Identifies phase transition points in set behavior.

Abstract

We describe the structure of a set of integers $A$ of positive density $\delta$, such that $A+A$ contains no squarefree integer. It turns out that the behaviour changes abruptly at the values $\delta_0=1/4-\frac{2}{\pi^2}=0.0473...$ and $\delta_1=1/18$.