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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 and Fekete's lemma 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.

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 and Fekete's lemma 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, $0 \in \ell_H(H)$, and 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.

Also, let me note that the answer is yes if $\mathbf Z_K$ is half-factorial (in particular, a UFD), namely, $|\mathsf L_H(x)| = 1$ for all $x \in H \setminus H^\times$: This is trivial, though it covers many non-trivial cases. In particular, it is well known (at least in some circles) that $H$ is half-factorial iff the class number of $K$ is $\le 2$, see [2, Theorem 1.7.3.5]: The result was first established in [1], though the term "half-factorial" was introduced only later in [3].

Bibliography

[1] L. Carlitz, A characterization of algebraic number fields with class number two, Proc. Amer. Math. Soc. 11 (1960), 391-392.

[2] A. Geroldinger and F. Halter-Koch's, Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Pure Appl. Math. 278, Chapman & Hall/CRC, 2006.

[3] A. Zaks, Half factorial domains, Bull. Amer. Math. Soc. 82 (1976), 721-723.

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 and Fekete's lemma 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 of a number field. Is it true that $\ell_H(H)$ is a compact subset of $\mathbf R$ (with the usual topology)?

It can proved (and it is not so obvious, as far as I can say) 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. Moreover, let me observe that the answer is yes, if $\mathsf Z_K$ 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).

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 and Fekete's lemma 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.

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 and Fekete's lemma 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 of a number field. Is it true that $\ell_H(H)$ is a compact subset of $\mathbf R$ (with the usual topology)?

It is perhaps worth noting that, under the assumptions of this question, $\ell_H(H)$ is a bounded subset of $\mathbf R$ (this is not for free), so the real point is whether or not $\ell_H(H)$ is closed.

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 and Fekete's lemma 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 of a number field. Is it true that $\ell_H(H)$ is a compact subset of $\mathbf R$ (with the usual topology)?

It can proved (and it is not so obvious, as far as I can say) 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. Moreover, let me observe that the answer is yes, if $\mathsf Z_K$ 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).