In a recent Math Stack Exchange [question](https://math.stackexchange.com/q/887671/137524) 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.



Distribution of zeroes of lacunary functions
In addition to the unique real zero at $z=-0.658626\ldots$, Mahler in a 1982 paper [*On the zeros of a special sequence of polynomials*, Math. Comp.] determined, to within eight decimal places, eight complex conjugate pairs of zeros of $f(z)$. This gives a total of $17$ fairly precisely located zeros. At the end of that paper he conjectures that there are infinitely many; in fact, in an earlier paper quoted [5] in *loc. cit.* (*On a special function*, J. Number Theory), he says that he expects every point on the unit circle to be a limit point of zeros of $f$. I do not know if his conjecture has since been proved or disproved. 


In a first approximation, one could look at the zeros of the polynomial truncations of $f$. Then it may help to know that those are equidistributed near the unit circle, a fact not noted by Mahler in his paper. (On page 211, he simply writes: "It seems that the arguments of the zeros are much more uniformly distributed over the values from $0$ to $360$ degrees." He notes instead the weaker statement that, by a general theorem of Jentzsch for power series having a natural boundary, every point of the boundary circle is a limit point of zeros of the truncations.) This follows by a theorem of Erdos and Turan; the same is true for any sequence of polynomials of degree $d \to \infty$ whose leading and free coefficients have non-zero absolute values at least $1$ and whose *lengths* (sums of the absolute values of all coefficients) are at most $e^{o(d)}$. For integer polynomials, this theorem has been refined by Yuri Bilu. For an exposition of those results and a sketch of their proofs I can refer you to Granville's article *The distribution of roots of a polynomial* in the volume *Equidistribution in Number Theory, An Introduction* (Edited by A. Granville and Z. Rudnick).


