On the density of sumsets, II

TL;DR

This paper demonstrates that for any value between 0 and 1, there exists a set A such that its sumset with a specific set B has a prescribed density under various arithmetic quasi-densities, extending understanding of sumset densities.

Contribution

It constructs sets with sumsets having any desired density within [0,1] under a broad class of densities, using properties of a density introduced by Buck in 1946.

Findings

Existence of sets with sumset densities equal to any alpha in [0,1]

Sumset densities are flexible under arithmetic quasi-densities

Application of Buck's density properties to sumset analysis

Abstract

Arithmetic quasi-densities are a large family of real-valued set functions partially defined on the power set of $\mathbb{N}$, including the asymptotic density, the Banach density, the analytic density, etc. Let $B \subseteq \mathbb{N}$ be a non-empty set covering $o(n!)$ residue classes modulo $n!$ as $n\to \infty$ (e.g., the primes or the perfect powers). We show that, for each $\alpha \in [0,1]$, there is a set $A\subseteq \mathbb{N}$ such that, for every arithmetic quasi-density $\mu$, both $A$ and the sumset $A+B$ are in the domain of $\mu$ and, in addition, $\mu(A + B) = \alpha$. The proof relies on the properties of a little known density first considered by Buck in 1946.