Let $A\subseteq \mathbb{N}$ be an additive basis (of any finite order) such that $\lvert A\cap \{1,\ldots,N\}\rvert=o(N)$. Is it true that $$\lim_{N\to \infty}\frac{\lvert (A+A)\cap \{1,\ldots,N\}\rvert} {\lvert A\cap \{1,\ldots,N\}\rvert}=\infty?$$

Let $A\subseteq \mathbb{N}$ be an additive basis (of any finite order) such that $\lvert A\cap \{1,\ldots,N\}\rvert=o(N)$. Is it true that $$\lim_{N\to \infty}\frac{\lvert (A+A)\cap \{1,\ldots,N\}\rvert} {\lvert A\cap \{1,\ldots,N\}\rvert}=\infty?$$

The answer is no, and a counterexample was provided by Turjányi [Tu84]. This was generalised (to the replacement of $A+A$ by the $h$-fold sumset $hA$ for any $h\geq 2$) by Ruzsa and Turjányi [RT85].

"Additive basis" is Set.IsAsymptoticAddBasis: some finite $h$ has $hA$ containing every sufficiently large integer. The exact notion Set.IsAddBasis, which asks that $hA$ be all of $\mathbb{N}$, would force $0, 1 \in A$ and is not the class these results are about.

The linked file states the basis hypothesis as ∃ N₀, Set.Ici N₀ ⊆ iterated_sumset A k and indexes both counting functions by a real $x$ through $\lfloor x\rfloor$, where the counting functions here are indexed by $N : \mathbb{N}$.