On the density of sumsets

Abstract

Recently introduced by the authors in [Proc. Edinb. Math. Soc. 60 (2020), 139-167], quasi-densities form a large family of real-valued functions partially defined on the power set of the integers that serve as a unifying framework for the study of many known densities (including the asymptotic density, the Banach density, the logarithmic density, the analytic density, and the Pólya density). We further contribute to this line of research by proving that (i) for each $n \in \mathbf N^+$ and $α\in [0,1]$, there is $A \subseteq \mathbf{N}$ with $kA \in \text{dom}(μ)$ and $μ(kA) = αk/n$ for every quasi-density $μ$ and every $k=1,\ldots, n$, where $kA:=A+\cdots+A$ is the $k$-fold sumset of $A$ and $\text{dom}(μ)$ denotes the domain of definition of $μ$; (ii) for each $α\in [0,1]$ and every non-empty finite $B\subseteq \mathbf{N}$, there is $A \subseteq \mathbf{N}$ with $A+B \in \mathrm{dom}(μ)$ and $μ(A+B)=α$ for every quasi-density $μ$; (iii) for each $α\in [0,1]$, there exists $A\subseteq \mathbf{N}$ with $2A = \mathbf{N}$ such that $A \in \text{dom}(μ)$ and $μ(A) = α$ for every quasi-density $μ$. Proofs rely on the properties of a little known density first considered by R.C. Buck and the "structure" of the set of all quasi-densities; in particular, they are rather different than previously known proofs of special cases of the same results.