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Let $\mathbb{T}=\{z\in \mathbb{C}\mid |z|=1\}$. For which $S\subseteq \mathbb{T}$, is there a sequence $(a_n)\subseteq \mathbb{C}$ such that the series: $$\sum_{k=1}^\infty{a_kz^k}$$ is convergent on $S$ and is not convergent on $\mathbb{T}\setminus S$?

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    This is a difficult problem, and according to math overflow the answer is not known (the answer here has a lot of information, though: http://mathoverflow.net/questions/49395/behaviour-of-power-series-on-their-circle-of-convergence) – A Blumenthal Feb 14 '13 at 02:30
  • .... good thread! thanks. –  Feb 14 '13 at 02:35
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    See also: http://math.stackexchange.com/questions/82871 and http://math.stackexchange.com/questions/288765 and – mrf Feb 14 '13 at 08:53

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