Consider a power series $\sum_{n=0}^{\infty}c_{n}z^{n}$ for which $c_{n}\in\left\{ 0,1\right\}$ for all $n$. One can write this as: $$\varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}z^{v}$$ for some set $V$ of positive integers. I call this the “set-series” of $V$. There is a beautiful theorem due to Gabor Szëgo which, for the case of set-series, shows that $\varsigma_{V}\left(z\right)$ is either a rational function whose poles are simple and located at roots of unity, or that $\varsigma_{V}\left(z\right)$ is a transcendental function with the unit circle ($\partial\mathbb{D}$) as a natural boundary.

Natural boundaries generally occur as the result of singularities clustering arbitrarily close to one another. My intuition tells me that in the case where $\varsigma_{V}\left(z\right)$ has a natural boundary (example: $V=\left\{ 2^{n}:n\geq0\right\}$, $V=\left\{ n^{2}:n\geq0\right\}$, etc), the clustering singularities in question are simple poles.

I figure a good way to try to see this would be via Padé approximants. The “rigorous” statement of my intution would then be something along the lines of: for an appropriately chosen sequence of Padé approximants $\left\{ P_{n}\left(z\right)\right\} _{n\geq1}$ of $\varsigma_{V}\left(z\right)$ (where $\varsigma_{V}\left(z\right)$ has a natural boundary on $\partial\mathbb{D})$, for every $\epsilon>0$ and every $\xi\in\partial\mathbb{D}$, there is an $N_{\epsilon,\xi}\geq1$ so that, for all $n\geq N_{\epsilon,\xi}$, any pole $s$ of $P_{n}\left(z\right)$ satisfying $\left|s-\xi\right|<\epsilon$ is necessarily simple.

With the literature on Padé Approximants appears to be quite extensive (while the literature on natural boundaries appears to be comparatively paltry), I was wondering if anyone knew of anything about this question, or something similar. Insight and/or references would be most appreciated.