The question that I shall ask here has arisen in the context of Diophantine approximation. I find it rather interesting, and I have no idea how to answer it. Any help, advice, or suggestions for references would be much appreciated.

Let $S$ be a countably infinite subset of $\mathbb{R}_{>0}$ that satisfies the following conditions. Let $n$ be an arbitrary element of $\mathbb{Z}_{\geq 2}.$

1) For any integer $k \geq 1,$ we have $k \cdot S \subseteq S.$ (In particular, $S$ is unbounded.)

2) There exists a (necessarily unique) strictly increasing sequence $t_\bullet = (t_k)_{k \geq 1}$ such that $\{t_k : k \geq 1\} = S.$ Moreover, there exists a constant $C > 0$ such that for any $k \geq 1,$ we have $0 < t_{k+1} - t_k \leq C.$ (Notice that we clearly have $\lim_{k \to +\infty} t_k = +\infty$, in light of 1) above.)

3) There exists a constant $D > 0$ such that $$\limsup_{T \to +\infty} \frac{\mathrm{card}\{ k \in \mathbb{Z}_{\geq 1} : t_k \leq T\}}{T^n} \leq D.$$

Is it necessarily true that there exists a norm $\eta$ on $\mathbb{R}^n$ such that $\eta(\mathbb{Z}^n) - \{0\} = S$? (Clearly conditions 1), 2), and 3) are necessary in order for this to be true; I am wondering whether they are also, in fact, sufficient.)