Schnirelmann density and bases of finite order

Question

Let $\mathcal{A}$ be an additive set. We know that if the Schnirelmann density $\sigma_{\mathcal{A}}$ is positive then it is a basis of finite order. But it it not a necessary condition. My question is- how do we show that if $\sigma_{A+A}>0$ then $A$ is a basis of finite order? Is it true in general that if $kA$ has positive Schnirelmann density for some $k$ then $A$ is a basis of finite order?