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 $x\to \infty$?

1True ↔2  ∀ (A B : Set ℕ),3    A.Infinite →4      B.Infinite →5        IsAdditiveComplement A B →6          Filter.limsup (fun x => ↑(↑(Erdos785.counting A x) * ↑(Erdos785.counting B x)) / ↑↑x) Filter.atTop <7              ↑(3 / 2) →8            Filter.Tendsto (fun x => ↑(Erdos785.counting A x) * ↑(Erdos785.counting B x) - ↑x) Filter.atTop Filter.atTop