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 $\xi$?