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I came up with this question on my own while I was musing around reviewing notes. After unsuccessful Google search (thwarted by a deluge amount of webpages on basic calculus), I decided to ask here.

We know that radius of convergence characterize the behaviour of a power series at point inside and outside the radius, but not on it. Calculus book never take much of a careful look at that details. Thus the question is thus:

Does there exist a power series $s(z)=\sum\limits_{k=0}^{\infty}a_{k}z^{k}$ such that the set $$D=\{z\in\mathbb{C}:|z|=1, \;s(z) \text{ diverges } \}$$ is $\underline{not}$ a countable union of arcs (close, open or otherwise).

Hope someone can solve it. Thank you.

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Here is a variant of this question asked on MathOverflow: https://mathoverflow.net/q/49395/. In particular, your question is answered by the cited result that $D$ can be any $G_\delta$ subset of the circle (such a set can be uncountable with dense complement).

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  • $\begingroup$ Do you know any copies of those papers on the Internet that isn't too blurry? The paper seems readable enough - certainly within an undergraduate level, but I have some difficulty reading it due to the blur...Thanks for telling me about them though, so it's still an open problem... $\endgroup$ – Gina Jul 27 '14 at 20:49
  • $\begingroup$ @Gina: No, I don't know of any. $\endgroup$ – Jonas Meyer Jul 28 '14 at 1:43

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