The growth function of S-recognizable sets

A set X⊆N is S-recognizable for an abstract numeration system S, if the set of its representations is accepted by a finite automaton. We show that the growth function of an S-recognizable set is always either Θ((log(n))c−dfnf) where c,d∈N and f≥1, or Θ(nrθΘ(nq)), where r,q∈Q with q≤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 Θ(nr), where r∈Q with r≥1. Furthermore, for every r∈Q with r≥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 Θ(nr). For all positive integers k and ℓ, we can also provide an abstract numeration system S built on an exponential language and an S-recognizable set such that the growth function of X is Θ((log(n))knℓ).