On a question of Ruzsa about densities of sumsets

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 $00$ such that $$ \mathsf{d}_{\ast}(A+A)\ge c\bigl(\mathsf{d}^{