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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 !

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