In a recent Math Stack Exchange question I asked about the function $$f(z)=\sum_{n=0}^\infty z^{2^n},$$ and was informed of its status is a canonical example of a lacunary series with natural boundary at $|z|=1$. A phenomenon observed by the accepted answer was that this function has a multitude of zeroes within the unit disk; it was speculated but not proven that that this set is in fact infinite.

That raises the following questions, for which I've not been able to find appropriate literature:

- Does $f(x)$ have an infinitude of zeros within the unit circle? How can this be proven?
- How are the zeros distributed? (e.g. how many zeros are found within an annulus $0<a\leq |z|\leq b <1$.)
- How does this generalize to other lacunary functions? I am particularly interested in the case where the base in $f(x)$ is a different positive integer.

The main literature I could find online was a Costin and Huang paper from 2009 entitled " Behavior of Lacunary Series at the Natural Boundary". Unfortunately, I found this paper to be too beyond my level to get much out of it; if the paper is relevant, some exposition on it would be appreciated.