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Let $B\subseteq\mathbb{N}$ be an additive basis of order $k$ with $0\in B$. Is it true that for every $A\subseteq\mathbb{N}$ we have \[d_s(A+B)\geq \alpha+\frac{\alpha(1-\alpha)}{k},\] where $\alpha=d_s(A)$ and \[d_s(A) = \inf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N}\] is the Schnirelmann density?
Erdős [Er36c] proved this is true with $k$ replaced by $2k$ in the denominator (in a stronger form that only considers $A\cup (A+b)$ for some $b\in B$, see [38]).

Ruzsa has observed that this follows immediately from the stronger fact proved by Plünnecke [Pl70] that (under the same assumptions) \[d_S(A+B)\geq \alpha^{1-1/k}.\]

Additional thanks to: Imre Ruzsa