A lower bound for additive complements to the primes

We study the size of sets $\mathcal{B}\subseteq {\mathbb{Z}}_{\ge 0}$ for which $\mathbb{P}+\mathcal{B}$ contains every sufficiently large integer. We obtain $B(X)\gg \log X\log \log X$, where $B(X)=\mathrm{\#}(\mathcal{B}\cap [0,X])$ and the implied constant is absolute. This gives a quantitative form of Conjecture 2 of Ruzsa and excludes additive complements of size $O(\log X)$, as asked in one part of Erdős Problem 32. More generally, we construct a block of $\lfloor c\log X\log \log X\mathrm{/}(B(X)+1)\rfloor$ consecutive integers outside $\mathbb{P}+\mathcal{B}$ in $[X\mathrm{/}2,X]$, whenever this number is positive.