Let $A\subseteq\mathbb{N}$ be an infinite set such that $|A\cap \{1, ..., N\}| = o(N)$. Is it true that $$\limsup_{N\to\infty}\frac{|(A + A)\cap \{1, ..., N\}|}{|A \cap \{1, ..., N\}|} \geq 3?$$

Let $A\subseteq\mathbb{N}$ be an infinite set such that $|A\cap \{1, ..., N\}| = o(N)$. Is it true that $$\limsup_{N\to\infty}\frac{|(A + A)\cap \{1, ..., N\}|}{|A \cap \{1, ..., N\}|} \geq 3?$$

The answer is yes, proved by Freiman [Fr73].

[Fr73] Fre\u{\i}man, G. A., _Foundations of a structural theory of set addition_. (1973), vii+108.