The growth function of S-recognizable sets

A set X ⊆ N is S-recognizable for an abstract numeration system S if the set repS(X) 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))n ) 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(n), 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(n). 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 Θ((log(n))n).