Let $A,B\subseteq \mathbb{N}$ be infinite sets such that $A+B$ contains all large integers. Let $A(x)=\lvert A\cap [1,x]\rvert$ and similarly for $B(x)$. Is it true that if $A(x)B(x)\sim x$ then $$A(x)B(x)-x\to \infty$$ as $x\to \infty$?

Let $A,B\subseteq \mathbb{N}$ be infinite sets such that $A+B$ contains all large integers. Let $A(x)=\lvert A\cap [1,x]\rvert$ and similarly for $B(x)$. Is it true that if $A(x)B(x)\sim x$ then $$A(x)B(x)-x\to \infty$$ as $x\to \infty$?

A conjecture of Erdős and Danzer. The answer is yes, proved by Sárközy and Szemerédi [SaSz94], who actually proved that it is impossible for $$A(x)B(x)-x=o(A(x)).$$