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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:…) – A Blumenthal Feb 14 '13 at 2:30
.... good thread! thanks. – user59671 Feb 14 '13 at 2:35

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