A lower bound for additive complements to the primes

Abstract

We prove that every set $\mathcal{B}\subseteq {\mathbb{Z}}_{\ge 0}$ for which $\mathbb{P}+\mathcal{B}$ contains all sufficiently large integers satisfies $B(X)\gg \log X\log \log X$, where $B(X)=\#(\mathcal{B}\cap [0,X])$ and the implied constant is absolute. This proves a quantitative form of Ruzsa's Conjecture 2 and gives a negative answer to the $O(\log X)$ question in Erdős Problem 32. The bound follows from a more general result: for an arbitrary set $\mathcal{B}\subseteq {\mathbb{Z}}_{\ge 0}$ and sufficiently large $X$, we construct a block of $\lfloor c\log X\log \log X/(B(X)+1)\rfloor$ consecutive integers in $[X/2,X]$ outside $\mathbb{P}+\mathcal{B}$, provided this length is positive.