Let $H$ be a multiplicatively written monoid with identity $1_H$. An *atom* of $H$ is an element $x \in H \setminus  H^\times$ such that $a \ne xy$ for all $x, y \in H \setminus H^\times$, where $H^\times$ is the group of units of $H$. Accordingly, we define $\mathsf L_H(1_H) := \{0\} \subseteq \mathbf N$ and, for $x \in H \setminus \{1_H\}$, we take ${\sf L}_H(x)$ to be the set of all $k \in \mathbf N^+$ such that $x = a_1 \cdots a_k$ for some atoms $a_1, \ldots, a_k \in H$.

It follows by Proposition 1 in [Question #269229][1] and [Fekete's lemma][2] that the function
$$
\ell_H: H \to [0,\infty]: x \mapsto \lim_{n \to \infty} \frac{|\mathsf L_H(x^n)|}{n}
$$
is well defined, insomuch as the limit in the above exists, and is either a non-negative real number or $\infty$. My question is as follows:

> **Q.** Let $H$ be the multiplicative monoid of the ring of integers, $\mathbf Z_K$, of a number field $K$. Is it true that $\ell_H(H)$ is a compact subset of $\mathbf R$ (with the usual topology)?

Of course, $0 \in \ell_H(H)$. Moreover, it can be proved (and, as far as I can say, it is not so obvious) that, under the assumptions of this question, $\ell_H(H)$ is a bounded subset of $\mathbf R$, so the real point is whether or not $\ell_H(H)$ is closed. Lastly, let me observe that the answer is yes, if $\mathbf Z_K$ is half-factorial (in particular, a UFD), i.e., $|\mathsf L_H(x)| = 1$ for all $x \in H \setminus H^\times$: This is trivial, though it covers many non-trivial cases; in particular, it's known that $H$ is half-factorial iff the class number of $K$ is $\le 2$, see Theorem 1.7.3.5 in A. Geroldinger and F. Halter-Koch's, *Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory*, Pure Appl. Math. **278**, Chapman & Hall/CRC, 2006.


  [1]: https://mathoverflow.net/questions/269229/
  [2]: https://mathoverflow.net/questions/99122