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it is well known that every power series has a (maybe infinite) $R$ s.t for every $|z|>R$ the series diverges and for every $|z|<R$ it converges. assume $R$ is finite and denote by $C$ the subset of the circle of radius $R$ on which the series converges. can $C$ be any arbitrary finite or co-finite set? are there any interesting family of sets that can be generated by a power series that way?

We know that $C$ can't be any set as it has to be measurable(as the points of which a series of functions converges), but is every measurable subset of the circle can be $C$. If so, is there a way given $C$ to generate a power series which converges only on $C$(and on the inner ball)?

terran
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RT1
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    https://math.stackexchange.com/q/206299/42969, https://math.stackexchange.com/q/288765/42969 – Martin R Mar 20 '24 at 13:44
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    Hello. Have you seen this and this? I think those pretty much address the state of the art of the answer to this question. In particular we know there are loads of measurable sets which do not appear this way, as $C$ can be described with a small number of quantifiers, so it must be quite early on in the Borel hierarchy. – Izaak van Dongen Mar 20 '24 at 13:44
  • thanks, this answers our question! – RT1 Mar 20 '24 at 13:49

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