On the density of $A+B$ when $B$ has few elements

Abstract

Let $\mathsf{d}$ be the asymptotic density on the positive integers ${\mathbb{N}}^{+}$. We provide sufficient conditions on an infinite set $B\subseteq {\mathbb{N}}^{+}$ so that for each $\alpha \in [0,1]$ there exists a set $A\subseteq {\mathbb{N}}^{+}$ such that $\mathsf{d}(A)=0$ and $\mathsf{d}(A+B)=\alpha$. For instance, if $(b_n:n\in {\mathbb{N}}^{+})$ is the increasing enumeration of $B$, it is sufficient that ${\mathrm{liminf}}_n{b}_{n+1}/b_n>1$. Moreover, we show that, if $|B\cap [1,n]|\ll \log (n{)}^{1-\epsilon }$ for some $\epsilon >0$, then for each $\alpha \in [0,1]$ there exists a set $A\subseteq {\mathbb{N}}^{+}$ such that $\mathsf{d}(A)=\mathsf{d}(A+B)=\alpha$. Conversely, we show that, for each $\epsilon >0$, there exists an infinite set $B\subseteq {\mathbb{N}}^{+}$ such that $|B\cap [1,n]|\ll n^{2/3+\epsilon }$ and $\mathsf{d}(A+B)\in \{0,1\}$ for every $A\subseteq {\mathbb{N}}^{+}$ such that $A+B$ admits asymptotic density.