Is anything known about functions of the type $f(z) = \sum_{n=0}^\infty z^{a_n}$, where $a_n$ is an integer sequence?

Question

Let's assume $a_n$ is strictly increasing. I know if $a_n$ grows too fast, the function has a natural boundary at $|z| = 1$ but I'm wondering if there are any other theorems or sources to look at for functions of this type, especially when $a_n$ grows more slowly?