The growth function of S-recognizable sets

A set $X\subseteq \mathbb{N}$ is S-recognizable for an abstract numeration system S if the set ${\text{rep}}_S(X)$ of its representations is accepted by a finite automaton. We show that the growth function of an S-recognizable set is always either $\mathrm{\Theta }((\log (n){)}^{c-df}{n}^f)$ where $c,d\in \mathbb{N}$ and $f\ge 1$, or $\mathrm{\Theta }(n^r{\theta }^{\mathrm{\Theta }(n^q)})$, where $r,q\in \mathbb{Q}$ with $q\le 1$. If the number of words of length n in the numeration language is bounded by a polynomial, then the growth function of an S-recognizable set is $\mathrm{\Theta }(n^r)$, where $r\in \mathbb{Q}$ with $r\ge 1$. Furthermore, for every $r\in \mathbb{Q}$ with $r\ge 1$, we can provide an abstract numeration system S built on a polynomial language and an S-recognizable set such that the growth function of X is $\mathrm{\Theta }(n^r)$. For all positive integers k and l, we can also provide an abstract numeration system S built on a exponential language and an S-recognizable set such that the growth function of X is $\Theta((\log(n))^k n^l)$.