# Lower-bounding the density of 3A in terms of that of 2A

Let $$A\subset\mathbb{N}$$ and $$2A=A+A=\{a+b \lvert a,b\in A\}$$ and $$3A=2A+A$$. I wonder how small the density of $$3A$$ can be, knowing that the density of $$2A$$ is, say, $$\beta >0$$, but not knowing anything about $$A$$ (assuming all three have densities).

I would be equally interested in the similar question for measurable sets in the circle $$\mathbb{R}/\mathbb{Z}$$, or finite sets of $$\mathbb{Z}/p\mathbb{Z}$$.

Thanks !