Erdős Problem #35

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](https://www.erdosproblems.com/38)). Ruzsa has observed that this follows immediately from the stronger fact proved by Plünnecke [Pl70] that (under the same assumptions, and for \(\alpha>0\)) \[d_S(A+B)\geq \alpha^{1-1/k}.\]