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 Otto 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.