On the density of sumsets

29 Oct 2020  ·  Leonetti Paolo, Tringali Salvatore ·

Quasi-densities are a large family of real-valued functions partially defined on the power set of the integers that were recently introduced by the authors in [Proc.~Edinb.~Math.~Soc.~\textbf{60} (2020), 139--167] to serve as a unifying framework for the study of many known densities, including the asymptotic, Banach, logarithmic, analytic, and P\'{o}lya densities. In the present paper, we further contribute to this line of research by proving that (i) for each $n \in \mathbf N^+$ and $\alpha \in [0,1]$, there exists $A \subseteq \mathbf{N}$ with $kA \in \text{dom}(\mu)$ and $\mu(kA) = \alpha k/n$ for every quasi-density $\mu$ and every $k=1,\ldots, n$, where $kA$ is the $k$-fold sumset of $A$ and $\text{dom}(\mu)$ denotes the domain of definition of $\mu$; (ii) for each $\alpha \in [0,1]$ and every non-empty finite $B\subseteq \mathbf{N}$, there exists $A\subseteq \mathbf{N}$ with $A+B \in \mathrm{dom}(\mu)$ and $\mu(A+B)=\alpha$ for every quasi-density $\mu$; (iii) for each $\alpha \in [0,1]$, there exists $A\subseteq \mathbf{N}$ with $2A = \mathbf{N}$ such that $A \in \text{dom}(\mu)$ and $\mu(A) = \alpha$ for every quasi-density $\mu$... Our approach relies on the properties of a little known density first considered by R.\,C.~Buck and the "structure" of the set of all quasi-densities. read more

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Number Theory Classical Analysis and ODEs Combinatorics