# Divergence set at radius of convergence

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.

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).