On a question of Ruzsa about densities of sumsets

Abstract

Let $\mathsf{d}$, ${\mathsf{d}}^{\ast }$, and ${\mathsf{d}}_{\ast }$ denote, respectively, the asymptotic density, the upper asymptotic density, and the lower asymptotic density on $\mathbb{N}$. We show that there are no constants $0<\nu <1$ and $c>0$ such that $${\mathsf{d}}_{\ast }(A+A)\ge c({\mathsf{d}}^{\ast }(A+A){)}^{1-\nu }\mathsf{d}(A{)}^{\nu }$$ for every set $A\subseteq \mathbb{N}$ which admits asymptotic density. This answers in the negative a question of I.Z. Ruzsa.