What's known about $X$ when $|X(n) + X(n)| < kn$, $n \in \mathbb{N}$, absolute constant $k$?

Question

Let $X$ be an infinite sequence of integers$$x_1 < x_2 < x_3 < \ldots,$$and let $X(n)$ be the set$$\{x_1, x_2, \ldots, x_n\}.$$

Question. What is known about $X$ when we have$$|X(n) + X(n)| < kn$$for all $n \in \mathbb{N}$ and an absolute constant $k$?