A lower bound for additive complements to the primes

We study the size of sets B≥0 for which P+B contains every sufficiently large integer. We obtain B(X)log Xloglog X, where B(X)=#(B∩[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{o}s Problem 32. More generally, we construct a block of clog Xloglog X/(B(X)+1) consecutive integers outside P+B in $[X/2,X]$, whenever this number is positive.