Boundary behavior of power series vs. boundedness of partial sums Let $f\left(z\right)=\sum_{n=0}^{\infty}a_{n}z^{n}$ be a power series with $0$s and $1$s as its coefficients ($a_{n}\in\left\{0,1\right\}$ for all $n$) with a radius of convergence of $1$. I call such holomorphic functions “digital" functions. I've been investigating the boundary values and limiting behaviors of digital functions, among other things, and have found myself confronted with the following question.
Suppose the unit circle is a natural boundary of $f\left(z\right)$ (meaning that there exists no analytic continuation of $f$ to a domain larger than $\mathbb{D}$). Letting $\xi$ denote any root of unity ($\xi=e^{2\pi i\frac{a}{b}}$ for some co-prime integers $a,b$), does it then follow that the sequence of partial sums: $$S_{N}\left(\xi\right)=\sum_{n=0}^{N}a_{n}\xi^{n}$$ is necessarily unbounded in magnitude (i.e., $\lim_{N\rightarrow\infty}\left|S_{N}\left(\xi\right)\right|=\infty$)? More generally, what, if anything, can be said about the behavior of a digital function $f\left(z\right)$ at a root of unity $\xi$ when it is known that the partial sums of $f$'s power series at $\xi$ are bounded in magnitude (i.e., $\limsup_{N\rightarrow\infty}\left|S_{N}\left(\xi\right)\right|<\infty$). Ex: is this a sufficient condition to imply the existence of an analytic continuation of $f$ to some open neighborhood of $\xi$?
 A: I think this question is odd because boundedness of $S_N(\xi)$  is a very weak condition whereas existence of an analytic continuation to some open neighborhood of $\xi$ (and even much less nice behavior) is a very strong condition. 
A "random" Taylor series has no analytic continuation beyond its circle of convergence. (See Why are lacunary series so badly behaved?, and especially the survey of J.-P. Kahane recommended in the accepted answer, Sec. 7.)  Imposing some  additional conditions, such that $a_n\in\{0,1\}$ and  $|S_N(\xi)|<C$ would not change very much.
Finding explicit (not random) examples might take some ingenuity.  Robert Israel's series is not bad. A still more simple one is the theta series $\sum z^{n^2}$ (which is known to be a lacunary series), again at $z=-1$.
A: Consider the case $a_n = 1$ if $n$ or $n-1$ is a power of $3$, $0$ otherwise, and $\xi = -1$.  Since  $f(z) = (1+z) \sum_{k=0}^\infty z^{3^k}$ and $\sum_{k=0}^\infty z^{3^k}$ is a lacunary series, the unit circle is a natural boundary.  But for $\xi = -1$ all partial sums are $-1$ or $0$.
