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 $x\to \infty$?

1∀ (A B : Set ℕ),2  A.Infinite →3    B.Infinite →4      Erdos785.IsExactAdditiveComplement A B →5        ∀ (c : ℝ),6          0 < c →7            ¬(fun x => ↑(Erdos785.counting A x) * ↑(Erdos785.counting B x) - ↑x) =O[Filter.atTop] fun x =>8                ↑(Erdos785.counting A x) ^ c