On the density of sumsets, II

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 $n\to \mathrm{\infty }$ (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.