A lower bound for additive complements to the primes

We study the size of sets $\mathcal{B}\subseteq\mathbb{Z}_{\ge0}$ for which $\mathbb{P}+\mathcal{B}$ contains every sufficiently large integer. We obtain $B(X)\gg\log X\log\log X$, where $B(X)=\#(\mathcal{B}\cap[0,X])$ 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 Erd\H{o}s Problem 32. More generally, we construct a block of $\lfloor c\log X\log\log X/(B(X)+1)\rfloor$ consecutive integers outside $\mathbb{P}+\mathcal{B}$ in $[X/2,X]$, whenever this number is positive.