DISPROVED (LEAN)
This has been solved in the negative and the proof verified in Lean.
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]. Ruzsa and Turjányi do prove (under the same hypotheses) that\[\lim_{N\to \infty}\frac{\lvert (A+A+A)\cap \{1,\ldots,3N\}\rvert}{\lvert A\cap \{1,\ldots,N\}\rvert}=\infty,\]and conjecture that the same should be true with $(A+A)\cap \{1,\ldots,2N\}$ in the numerator.
Additional thanks to: Zachary Chase, Zach Hunter
