TITLE: Recursively enumerable sets as range sets of functions in Grzegorczyk-hierarchy
QUESTION [7 upvotes]: It is well known that recursively enumerable sets can be defined (among many other equivalent alternatives) as the range sets of primitive recusive functions (except for the trivial case of the empty set).
On the other hand, primitive recursive functions can be presented in an stratified way using Grzegorczyk-hierarchy.
I am wondering what it is known about whether recursively enumerable sets can be presented as the range sets of functions in some particular level of Grzegorczyk-hierarchy.

REPLY [7 votes]: Gregorczyk hierarchy is not suitable to classify functions of very low complexity. The answer here is that for any r.e. set, you can take the enumerating function from pretty much as low a complexity class as you can imagine, as long as the class can compute anything nontrivial.
For example, the function can be made uniform $\mathrm{AC}^0$, in other words, computable by a logarithmic-time alternating Turing machine with constantly many alternations (actually, 1 or 2 alternations should be enough, but I would have to check).
For another notion of simplicity, if $A$ is an r.e. set and $a_0$ is its least element, then $A$ is the range of a function $f\colon\mathbb N^k\to\mathbb N$ of the form
$$f(\vec x)=\max\{p(\vec x),a_0\}$$
where $p$ is a polynomial with integer coefficients. (This follows from the MRDP theorem.)